dataset_name
stringclasses 4
values | dataset_version
timestamp[s] | qid
stringlengths 1
5
| queId
stringlengths 32
32
| competition_source_list
sequence | difficulty
stringclasses 5
values | qtype
stringclasses 1
value | problem
stringlengths 6
1.51k
| answer_option_list
list | knowledge_point_routes
sequence | answer_analysis
sequence | answer_value
stringclasses 7
values |
|---|---|---|---|---|---|---|---|---|---|---|---|
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2299 | 9779dd7f66c34a08938ee67dd57af11f | [
"其它"
] | 1 | single_choice | Fill in the blank:~\uline{~~~~~~~~~~}~is 2 tens 8 ones less than 5 tens 5 ones. | [
[
{
"aoVal": "A",
"content": "$$27$$ "
}
],
[
{
"aoVal": "B",
"content": "$$37$$ "
}
],
[
{
"aoVal": "C",
"content": "$$73$$ "
}
],
[
{
"aoVal": "D",
"content": "$$20$$ "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication->Law of Addition"
] | [
"55 - 28 = 27 "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2300 | 9c05cda4ab74421595036a965f31bc9e | [] | 1 | single_choice | In Tim\textquotesingle s class, there are $$20$$ students who can swim, $$25$$ students who can play basketball, and $$10$$ students who can do both. If everyone in the class plays at least one sport, how many students are there in Tim\textquotesingle s class?~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$32$$ "
}
],
[
{
"aoVal": "B",
"content": "$$33$$ "
}
],
[
{
"aoVal": "C",
"content": "$$34$$ "
}
],
[
{
"aoVal": "D",
"content": "$$35$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"$$20+25-10=35$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2307 | 9796c245051d418f92d4eee9e998c33c | [] | 1 | single_choice | Alice\textquotesingle s average score on the first four assignments is $$89$$. She got $$94$$ on the fifth assignment. What is Alice\textquotesingle s average score on all five assignments? | [
[
{
"aoVal": "A",
"content": "$$90$$ "
}
],
[
{
"aoVal": "B",
"content": "$$91$$ "
}
],
[
{
"aoVal": "C",
"content": "$$92$$ "
}
],
[
{
"aoVal": "D",
"content": "$$93$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Word Problem Modules->Average Problems ->Questions Involving Average->Questions Involving Average (ordinary type)"
] | [
"$$89\\times4=356$$ $$356+94=450$$ $$450\\div5=90$$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2311 | a0a592ed2df44c0cbb10e99c74e982b2 | [] | 1 | single_choice | Jack, Sarah, and Jimmy participated in a Maths competition. \textbf{Jack says: "I won the competition."} \textbf{Sarah says: "I didn\textquotesingle t win the competition."} \textbf{Jimmy says: "Jack didn\textquotesingle t win the competition."} Only one of them told the truth. Who won the Maths competition? | [
[
{
"aoVal": "A",
"content": "$$$$Jack "
}
],
[
{
"aoVal": "B",
"content": "Sarah "
}
],
[
{
"aoVal": "C",
"content": "Jimmy "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"We can spot that Jack\\textquotesingle s statement and Jimmy\\textquotesingle s statement contradict each other, so one of them is telling the truth. Therefore, Sarah tells a lie. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2313 | c96cc632f57b4425b11279a8f2aa29e2 | [
"其它"
] | 1 | single_choice | Maria had $$28$$ dreams last month. If $$16$$ of them involved monkeys, $$15$$ involved squirrels, and $$4$$ involved no animals, then at least how many dreams involved both monkeys and squirrels? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$11$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Inclusion-Exclusion Principle->Inclusion-Exclusion Principle for Two Sets"
] | [
"Maria had $$28$$ dreams last month, $$24$$ of which involved animals. Since $$16+ 15 =31$$ involved moneys or squirrels, then at least $$31 - 24 = 7$$ dreams involved both monkeys and squirrels. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2318 | 8eca05934a3e497da86abe0383d3f6ec | [
"其它"
] | 1 | single_choice | Joann and Sana have $20$ dresses in total. Joann gives half of her dresses to Claire, and then she and Sana have $14$ dresses in total. How many dresses does Sana have? | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$8$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12$$ "
}
],
[
{
"aoVal": "E",
"content": "$$14$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules"
] | [
"Joann gives $20-14=6$ dresses to Claire. Joann has $6+6=12$ dresses originally. Sana has $20-12=8$ dresses. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2321 | 9c3917ccb5d6457bb6192fb0e86e4002 | [
"其它"
] | 1 | single_choice | Vansen left for school at $6:30$ am. He took half an hour to walk from his house to school. Ron reached school $1$ hour later than Vansen. At what time did Ron reach school? (adapted from 2009 Math Kangaroo Problem, Level 3-4, Question \#8) | [
[
{
"aoVal": "A",
"content": "$7:00$ am. "
}
],
[
{
"aoVal": "B",
"content": "$8:00$ am. "
}
],
[
{
"aoVal": "C",
"content": "$8:45$ am. "
}
],
[
{
"aoVal": "D",
"content": "$9:00$ am. "
}
],
[
{
"aoVal": "E",
"content": "$10:00$ am. "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Time Problem->Time Calculation"
] | [
"Half an hour after $6:30$ is $7:00$ $7+1=8$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2324 | bbd5c995868249a89d7d1c6ac3c392d4 | [] | 1 | single_choice | In a certain country, a part of the residents can speak English only, a part can speak French only and the rest can speak both languages. It is known that $$85\textbackslash\%$$ residents can speak English and $$75\textbackslash\%$$ can speak French. What percent of the residents of this country can speak both English and French? ($$2002$$ Math kangaroo Problem, Level $$7-8$$, Question \#$$17$$) | [
[
{
"aoVal": "A",
"content": "$$50\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "B",
"content": "$$57\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "C",
"content": "$$25\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "D",
"content": "$$60\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "E",
"content": "$$40\\textbackslash\\%$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Inclusion-Exclusion Principle->Inclusion-Exclusion Principle for Two Sets"
] | [
"$85\\textbackslash\\%+75\\textbackslash\\%-100\\textbackslash\\%=60\\textbackslash\\%$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2326 | fbe3b83d7a0c42bb9511c5c95517d771 | [] | 1 | single_choice | Tim, an adventurer, found three treasure chests in the cave. Only one of the chests contains the treasure. The following clues are written on the three treasure chests respectively: \textbf{Treasure chest A: "The treasure is not in Treasure chest C!"} \textbf{Treasure chest B: "The treasure is not here!"} \textbf{Treasure chest C: "The treasure is here!"} Given that only one sentence of the above three is true, which chest should Tim open to get the treasure? | [
[
{
"aoVal": "A",
"content": "A "
}
],
[
{
"aoVal": "B",
"content": "B "
}
],
[
{
"aoVal": "C",
"content": "C "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning using Hypothesis"
] | [
"The clues on treasure chest A and C are conflicting, so either A or C is telling the truth. Thus, the clue on treasure chest B is a lie. Therefore, the treasure is in chest B. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2328 | b2d83d45a2ce435ea7e529e294e89be0 | [
"其它"
] | 1 | single_choice | Eddie, Avril and Pip want to share 10 cakes. Eddie only wants to get 3 cakes. Avril and Pip want to get at least 1 cake. How many ways are there to share the cakes? | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Questions Involving Enumeration->Enumeration"
] | [
"$$Omitted.$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2332 | a55b24a987114aa5a2ee64e4f5a0418e | [
"其它"
] | 0 | single_choice | The correct information James can get from the statement "there is a $11\textbackslash\%$ chance that it will rain tomorrow in Los Angeles"~is~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "It will rain $11\\textbackslash\\%$ of the time tomorrow in Los Angeles. "
}
],
[
{
"aoVal": "B",
"content": "It will rain in $11\\textbackslash\\%$ of the regions in Los Angeles tomorrow. "
}
],
[
{
"aoVal": "C",
"content": "It will definitely rain tomorrow in Los Angeles. "
}
],
[
{
"aoVal": "D",
"content": "The probability of raining in Los Angeles is high. "
}
],
[
{
"aoVal": "E",
"content": "The probability of raining in Los Angeles is low. "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"It is less likely to rain tomorrow in Los Angeles. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2334 | dbc93a3b6a1e435596c5021d882a4db5 | [
"其它"
] | 1 | single_choice | Mia has a date tomorrow. She has $2$ hats, $3$ dresses, and $3$ pairs of shoes. Now she wants to choose one dress and one pair of shoes. How many options does she have? | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$8$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12$$ "
}
],
[
{
"aoVal": "E",
"content": "$$18$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules"
] | [
"She does not need to choose a hat. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2335 | ce16f4810a404d9092b07c6bba10d330 | [
"其它"
] | 0 | single_choice | Amy picks some numbers as shown below: $13, 20, 14, 15, 19, 20, 20, 19, 19, 15, 19, 19, 20, 13, 15$. What is the difference between their mode and median? | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2$$ "
}
],
[
{
"aoVal": "D",
"content": "$$19$$ "
}
],
[
{
"aoVal": "E",
"content": "$$20$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"She picks two $13$s, one $14$, three $15$s, five $19$s, and four $20$s. The mode is $19.$ The median is $19.$ Thus, their difference should be $0.$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2337 | a0e9b31ab9bf41d081984672445be84e | [
"其它"
] | 1 | single_choice | How many multiples of 6 are there between 14 and 100? | [
[
{
"aoVal": "A",
"content": "$$16$$ "
}
],
[
{
"aoVal": "B",
"content": "$$15$$ "
}
],
[
{
"aoVal": "C",
"content": "$$14$$ "
}
],
[
{
"aoVal": "D",
"content": "$$13$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication"
] | [
"18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96 "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2340 | b776a11b77aa435c9f6cbdd3c7746604 | [] | 1 | single_choice | There are $$30$$ pupils in my class. $$20$$ pupils like Maths and $$18$$ pupils like English. Twice as many pupils like both subjects as those that like neither of them. How many pupils like only Maths? | [
[
{
"aoVal": "A",
"content": "$$20$$ "
}
],
[
{
"aoVal": "B",
"content": "$$16$$ "
}
],
[
{
"aoVal": "C",
"content": "$$12$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
],
[
{
"aoVal": "E",
"content": "$$4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"Let the number of pupils who like neither subject be $$x$$. Hence the number who like both subjects is $$2x$$. Therefore the number of pupils who like only Maths is $$20−2x$$ and the number who like only English is $$18−2x$$. Since there are $$30$$ pupils in my class, we have $$\\left( 20-2x \\right)+2x+\\left( 18-2x \\right)+x=30$$ and hence $$38−x = 30$$. This has solution $$x = 8$$ and hence the number of pupils who like only Maths is $$20-2\\times 8=4$$. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2342 | ae779cff3fa44d2fb7cccfd6f1c3b067 | [] | 2 | single_choice | Betty and Abby are playing a game. They take turns writing numbers from $$1$$ to $$52 $$ on a blackboard. Each person can only write $$1$$, $$2$$, $$3$$ or $$4$$ numbers at a time, and each number can only be written once. The person who has no more numbers to write loses. Should Betty go first or second in order to win? | [
[
{
"aoVal": "A",
"content": "First "
}
],
[
{
"aoVal": "B",
"content": "Second "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"$$52$$ is not a multiple of $$4+1$$, so the first player will win the game. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2345 | fbfa878ceb284cbbae63e7cf308c78d5 | [] | 2 | single_choice | Ashley and Elvis are playing a game that requires them to drink a total of $$12$$ cups of coffee. They take turns drinking and each can drink either $$1$$ or $$2$$ cups at a time. The person who finishes the last cup of coffee wins this game. Should Elvis go first or second to ensure victory? | [
[
{
"aoVal": "A",
"content": "Go first "
}
],
[
{
"aoVal": "B",
"content": "Go second "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Strategies and Operations->Game Strategy"
] | [
"$$12$$ is a multiple of $$2+1$$, so, the second player should make the total number for each round to be $$3$$ to ensure victory. Therefore, Elvis should go second to ensure his vitory. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2346 | aa058978e30e457390bb16522b4f758a | [] | 1 | single_choice | What is the smallest possible sum of two positive integers whose product is $$240$$? | [
[
{
"aoVal": "A",
"content": "$$30$$ "
}
],
[
{
"aoVal": "B",
"content": "$$31$$ "
}
],
[
{
"aoVal": "C",
"content": "$$32$$ "
}
],
[
{
"aoVal": "D",
"content": "$$34$$ "
}
],
[
{
"aoVal": "E",
"content": "$$38$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"Since the product of the two positive integers is $$240$$, the possible pairs of integers are $$\\left( 1,240 \\right)$$, $$\\left( 2,120 \\right)$$, $$\\left( 3,80 \\right)$$, $$\\left( 4,60 \\right)$$, $$\\left( 5,48 \\right)$$, $$\\left( 6,40 \\right)$$, $$\\left( 8,30 \\right)$$, $$\\left( 10,24 \\right)$$, $$\\left( 12,20 \\right)$$ and $$\\left( 15,16 \\right)$$. The respective sums of these pairs are $$241$$, $$122$$, $$83$$, $$64$$, $$53$$, $$46$$, $$38$$, $$34$$, $$32$$ and $$31$$. Of these, the smallest value is $$31$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2348 | a5820d815e3542f39c45ae3f7cd2512b | [
"其它"
] | 1 | single_choice | SASMO 2014 P2 Q9 Two $1 coins and ten 50c coins are randomly distributed among 4 children such that each child receives the same number of coins. What is the difference between the biggest amount and the smallest amount a child can receive? | [
[
{
"aoVal": "A",
"content": "$$50c$$ "
}
],
[
{
"aoVal": "B",
"content": "$1 "
}
],
[
{
"aoVal": "C",
"content": "$1.50 "
}
],
[
{
"aoVal": "D",
"content": "$2 "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication"
] | [
"There are a total of 2+10 = 12 coins. so each child receives 12/4 = 3 coins. Largest possible is $1 + $1 +50c= $2.50. smallest possible is 50c + 50c + 50c= $1.50 $2.50 - $1.50 = $1 "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2354 | c9b1f3a248b44867a71ce86ded57cf1a | [] | 1 | single_choice | Maria had $$28$$ dreams last month. If $$16$$ of them involved monkeys, $$15$$ involved squirrels, and $$4$$ involved no animals, then at least how many dreams involved both monkeys and squirrels? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$7$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
],
[
{
"aoVal": "E",
"content": "$$11$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Inclusion-Exclusion Principle->Inclusion-Exclusion Principle for Two Sets"
] | [
"Maria had $$28$$ dreams last month, $$24$$ of which involved animals. Since $$16+ 15 =31$$ involved moneys or squirrels, then at least $$31 - 24 = 7$$ dreams involved both monkeys and squirrels. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2357 | f76b6dbfbbfd4d34af39b0933c9af5fb | [
"其它"
] | 1 | single_choice | $6\times6$ =~\uline{~~~~~~~~~~}~groups of $4$. | [
[
{
"aoVal": "A",
"content": "$$8$$ "
}
],
[
{
"aoVal": "B",
"content": "$$9$$ "
}
],
[
{
"aoVal": "C",
"content": "$$24$$ "
}
],
[
{
"aoVal": "D",
"content": "$$36$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication"
] | [
"6x6=36; $$36\\div4=9$$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2359 | a12259d74b2940ecbb1e536d4f96c877 | [] | 1 | single_choice | Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true. $$\rm I$$. Bill is the oldest. $$\rm II$$. Amy is not the oldest. $$\rm III$$. Celine is not the youngest. Rank the friends from oldest to youngest. | [
[
{
"aoVal": "A",
"content": "Bill, Amy, Celine "
}
],
[
{
"aoVal": "B",
"content": "Amy, Bill, Celine "
}
],
[
{
"aoVal": "C",
"content": "Celine, Amy, Bill "
}
],
[
{
"aoVal": "D",
"content": "Celine, Bill, Amy "
}
],
[
{
"aoVal": "E",
"content": "Amy, Celine, Bill "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Conditions"
] | [
"If Bill is the oldest, then Amy is not the oldest, and both statements $$\\rm I$$ and $$\\rm II$$ are true, so statement $$\\rm I$$ is not the true one. If Amy is not the oldest, and we know Bill cannot be the oldest, then Celine is the oldest. This would mean she is not the youngest, and both statements $$\\rm II$$ and $$\\rm III$$ are true, so statement $$\\rm II$$ is not the true one. Therefore, statement $$\\rm III$$ is the true statement, and both $$\\rm I$$ and $$\\rm II$$ are false. From this, Amy is the oldest, Celine is in the middle, and lastly Bill is the youngest. This order is $$\\rm E$$. Amy, Celine, Bill. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2361 | a124f8e223ad47308b56e9dfe697cbfa | [] | 1 | single_choice | There are $$17$$ balls in a bag. Each ball has a number from $$1$$ to $$17$$ on it. We randomly pick a ball from the bag. What is the smallest number of balls we have to pick in order to be sure that we have at least one pair of balls with a sum equal to $$18$$? | [
[
{
"aoVal": "A",
"content": "$$7$$ "
}
],
[
{
"aoVal": "B",
"content": "$$8$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10$$ "
}
],
[
{
"aoVal": "D",
"content": "$$11$$ "
}
],
[
{
"aoVal": "E",
"content": "$$17$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Pigeonhole Principle->Worst Case in Pigeonhole Principle Problems"
] | [
"Among these numbers, there are $$8$$ pairs of numbers can get the sum of $$18$$($$1+17=2+16=3+15=4+14=5+13=6+12=7+11=$$$$8+10$$), and $$9$$ is useless. So in the worst case, after we choose $$9$$, we need $$8+1=9$$ more numbers to make sure a pair appears. Thus, the answer is $$1+8+1=10$$. Copyrighted material used with permission from Math Kangaroo in USA, NFP Inc. ($$2005$$ Math Kangaroo Problem, Level $$9-10$$, Question $$ \\textbackslash\\# $$$$15$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2363 | aeaa5ac24ae54bb698855a055f3d265a | [] | 1 | single_choice | Pip multiplies all the digits in the number $$145$$ to get $$20$$. What's the biggest three digit number whose digits multiply together to give $$30$$?~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$999$$ "
}
],
[
{
"aoVal": "B",
"content": "$$922$$ "
}
],
[
{
"aoVal": "C",
"content": "$$651$$ "
}
],
[
{
"aoVal": "D",
"content": "$$532$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations->Basic Operations of Combinations"
] | [
"$$651$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2365 | a5b3e8f27d1d4344917251f9120b41f1 | [] | 1 | single_choice | Vicky is ordering lunch at a fast food restaurant that has sandwiches and burgers on the lunch menu, along with coffee, milk, and tea as drink options. If Vicky chooses one food item and one drink item from the lunch menu, she has~\uline{~~~~~~~~~~}~different ways to order lunch. | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"$$2\\times 3=6$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2366 | 93d81ea6b4dc41f1855426d2ed7c8350 | [] | 1 | single_choice | In the following events, the certain event is~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "Ame will get $$100$$ points in the final math exam. "
}
],
[
{
"aoVal": "B",
"content": "Claire will eat breakfast tomorrow. "
}
],
[
{
"aoVal": "C",
"content": "The news is on when you open the TV. "
}
],
[
{
"aoVal": "D",
"content": "There are two red balls and one white ball in a bag. If two balls are taken out, one must be red. "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"$$\\text{A}$$. \"Ame will get $100$ points in the final math exam\" is a random event. $$\\text{B}$$. \"Claire will eat breakfast tomorrow.\" is a random event. $$\\text{C}$$. \"The news is on when you open the TV\" is a random event. $$\\text{D}$$. There are two red balls and one white ball in a bag. If two balls are taken out, one must be red. It is a certain event. So $$\\text{D}$$ is the answer. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2374 | c5501eef7cec4125ac04849014b374e3 | [
"其它"
] | 1 | single_choice | In a two-digit number, the digit in the ones place is 3. The digit in the tens place is 3 more than the digit in the ones place. What is the number? | [
[
{
"aoVal": "A",
"content": "$$63$$ "
}
],
[
{
"aoVal": "B",
"content": "$$36$$ "
}
],
[
{
"aoVal": "C",
"content": "$$33$$ "
}
],
[
{
"aoVal": "D",
"content": "$$66$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Questions Involving Enumeration->Enumeration"
] | [
"NA "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2375 | bc4b07a56d4242418a4ac08dd38d6d05 | [
"其它"
] | 1 | single_choice | Two tiles numbered $1$ and $2$ are turned face down, respectively. One tile is turned up at random, and throw a die to get a number from $1$ to $6$. What is the probability that the product of the numbers on the tile and the die is greater than or equal to $10$? | [
[
{
"aoVal": "A",
"content": "$\\frac {1}{2}$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac {1}{6}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac {1}{3}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\frac {1}{4}$ "
}
],
[
{
"aoVal": "E",
"content": "$\\frac {1}{12}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"There are $12$ different combinations. The product of two numbers is greater than or equal to $10$ will be $2\\times5$ and $2\\times6$. Thus, the probability is $\\frac 2{12}$ = $\\frac 16$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2381 | a5eb55733af14fc5be1a65ed43c0ac59 | [] | 1 | single_choice | The Dragonvale Middle School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible? | [
[
{
"aoVal": "A",
"content": "$$12$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"Boy, Girl, Girl, Girl, Boy.There are $$2\\times 1=2$$ ways of arranging two boys; there are $$3\\times 2\\times 1=6\\textasciitilde$$ways of arranging $3$ girls. Therefore, there are $$2\\times 6=12$$ ways of arranging all students. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2382 | e9cd7929a30c4bb497735c00e81a5047 | [] | 1 | single_choice | Martina and Linda had $$68$$ seashells. After Martina gave $$8$$ seashells to Linda, they had an equal amount of seashells. How many seashells did Martina have at first? ． | [
[
{
"aoVal": "A",
"content": "$$30$$ "
}
],
[
{
"aoVal": "B",
"content": "$$32$$ "
}
],
[
{
"aoVal": "C",
"content": "$$40$$ "
}
],
[
{
"aoVal": "D",
"content": "$$42$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"$$ 68\\div 2=34$$, $$34+8=42$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2383 | c569d2401d0f48f6858865b7262402a0 | [] | 1 | single_choice | There are $$60$$ pupils in Arnold\textquotesingle s class. $$35$$ pupils like Maths and $$38$$ pupils like English. Twice as many pupils like both subjects as those that like neither of them. How many pupils like only Maths? | [
[
{
"aoVal": "A",
"content": "$$36$$ "
}
],
[
{
"aoVal": "B",
"content": "$$27$$ "
}
],
[
{
"aoVal": "C",
"content": "$$18$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Word Problem Modules->Equation Word Problems->Word Problems for Linear Equations with One Variable"
] | [
"Let the number of pupils who like neither subject be $$x$$. Hence the number who like both subjects is $$2x$$. Therefore the number of pupils who like only Maths is $$35−2x$$ and the number who like only English is $$38−2x$$. Since there are $$60$$ pupils in Arnold\\textquotesingle s class, we have: $$\\left( 35-2x \\right)+2x+\\left( 38-2x \\right)+x=60$$ and hence $$x = 13$$ and hence the number of pupils who like only Maths is $$35-(2\\times 13)=9$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2386 | a5f647926dbc420399f3d5a9c31e5e66 | [] | 1 | single_choice | Julia, Kasia, Zuzanna, and Helena have their birthdays on March $$1^{\rm st}$$, May $$17^{\rm th}$$, July $$20^{\rm th}$$, and March $$20^{\rm th}$$. Kasia and Zuzanna were born in the same month. Julia and Zuzanna were born on the same day of a month. Which of the girls was born on May $$17^{\rm th}$$? | [
[
{
"aoVal": "A",
"content": "Julia "
}
],
[
{
"aoVal": "B",
"content": "Kasia "
}
],
[
{
"aoVal": "C",
"content": "Zuzanna "
}
],
[
{
"aoVal": "D",
"content": "Helena "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Conditions"
] | [
"Given that Kasia and Zuzanna were born in the same month, their birth month must be March. Given that Julia and Zuzanna were born on the same day of a month, they must be born on the $$20^{\\rm th}$$. Hence, Zuzanna was born on March $$20$$; Kasia was born on March $$1$$; and Julia was born on July $$20$$. Helena is therefore the one born on May $$17$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2387 | ce7df7e183e1440b858a6062bd5d3d6c | [] | 1 | single_choice | A box contains blue marbles. Another two boxes contain only white marbles. Label on Box $$\rm A$$: white marbles Label on Box $$\rm B$$: blue marbles Label on Box $$\rm C$$: Box $$\rm B$$ contains blue marbles Which box contains blue marbles if two of the above labels are wrong?． | [
[
{
"aoVal": "A",
"content": "Box A "
}
],
[
{
"aoVal": "B",
"content": "Box B "
}
],
[
{
"aoVal": "C",
"content": "Box C "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning using Hypothesis"
] | [
"If two of the labels are wrong, \\textbf{one label is correct.} If the label on Box A is correct, Box A contains white marbles. The label on Box B is wrong, so Box B contains white marbles. The label on Box C is wrong, so Box B contains white marbles. Therefore, Box A and Box B contain white marbles, while \\textbf{Box C} contains blue marbles. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2394 | e0bfced69d294a35bdb29ccdd8aa58b9 | [] | 1 | single_choice | Three kids line up to play games. How many different ways can they form the line? | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication->Queuing Problems"
] | [
"There are six different ways for three kids to line up. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2396 | bc8b0dd2a6fa4b168cd0772f89bc9a56 | [] | 1 | single_choice | An elementary school is arranging students\textquotesingle{} schedules. For the first class in the morning, it has to be Chinese or Maths or English. Students won\textquotesingle t start the day with the same class for two days in a row. If Wednesday starts with Chinese, then Thursday can start with~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "Chinese or Maths "
}
],
[
{
"aoVal": "B",
"content": "Maths or English "
}
],
[
{
"aoVal": "C",
"content": "English or Chinese "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Questions Involving Enumeration->Tree Diagrams"
] | [
"Students won\\textquotesingle t start the day with the same class for two days in a row. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2398 | a61d91c2f3554c36a848ff4b32e30423 | [] | 1 | single_choice | Granny has $$10$$ grandchildren. Alice is the oldest. One day, Granny notices that her grandchildren all have different ages. If the sum of her grandchildren\textquotesingle s ages is $$180$$, what is the youngest age that Alice can be． | [
[
{
"aoVal": "A",
"content": "$$19$$ "
}
],
[
{
"aoVal": "B",
"content": "$$20$$ "
}
],
[
{
"aoVal": "C",
"content": "$$21$$ "
}
],
[
{
"aoVal": "D",
"content": "$$22$$ "
}
],
[
{
"aoVal": "E",
"content": "$$23$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Combinatorics Involving Extreme Values->Extreme Value with Fixed Sums"
] | [
"Try to consider the problem of the youngest Alice in such way: make each grandchild has a similar age as possible, and the age of each grandchild should be different, that is, $$1 + 2 + 3 +\\cdots 9 + 10 = 55$$; then $$180 - 55 = 125$$, $$125 \\div 10 = 12\\cdots\\cdots5$$, and $$5$$ is left. If every child\\textquotesingle s age is added by $$12$$, then $$5$$ is left. Give the extra year to each of the $$5$$ children who are the oldest. In doing so, the minimum number of Alice\\textquotesingle s age is $$10+12+1=23$$. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2401 | ce9f35d71ed640a786178e2a7f43155e | [] | 1 | single_choice | $$\dfrac{1}{2}$$ of the pens in a box are red, $$\dfrac{1}{4}$$ of the remainder are blue and the rest are green. What percentage of the pens in the box are green? | [
[
{
"aoVal": "A",
"content": "$$12.5\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "B",
"content": "$$25\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "C",
"content": "$$37.5\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "D",
"content": "$$75\\textbackslash\\%$$ "
}
]
] | [
"Overseas In-curriculum->Knowledge Point->Operations of Numbers ->Word Problems Involving Fractions and Percentages->Finding the Percentage Given a Part and a Whole",
"Overseas Competition->Knowledge Point->Counting Modules"
] | [
"Remainder $=1-\\frac{1}{2}=\\frac{1}{2}$ Green $=1-\\frac{1}{4}=\\frac{3}{4}$ of remainder $=\\frac{3}{4}\\times\\frac{1}{2}$ of total $=\\frac{3}{8}$ of total $=\\frac{3}{8}\\times100\\textbackslash\\%=37.5\\textbackslash\\%$ of total "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2402 | cea2680dc2644c8ca9e12ed694d8df3b | [] | 1 | single_choice | Using the digits $$1$$, $$3$$, and $$9$$, we can form~\uline{~~~~~~~~~~}~different 3-digit numbers without repeating digits. | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"Three-digit:$$139$$, $$193$$, $$319$$, $$391$$ , $$913$$, $$931$$ for a total of $$6$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2403 | cea3c62e291d457eb01dae8ca219c1b9 | [] | 1 | single_choice | Four students asked their teacher, Mr Carter, to line up with them to take a picture. ① If Mr Carter does not want to stand on either end, how many different ways can they line up for the picture? ② If Mr Carter insists on standing on one of the $2$ ends, how many different ways can they line up for the picture? | [
[
{
"aoVal": "A",
"content": "$72$ , $24$ "
}
],
[
{
"aoVal": "B",
"content": "$96$ , $24$ "
}
],
[
{
"aoVal": "C",
"content": "$72$ , $48$ "
}
],
[
{
"aoVal": "D",
"content": "$96$ , $48$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication->Queuing Problems"
] | [
"①$$3\\times 4\\times 3\\times 2\\times 1=72$$ , ②$$2\\times 4\\times 3\\times 2\\times 1=48$$ . "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2406 | c5a63070999b4b91a54935ddf8bbcc64 | [
"其它"
] | 1 | single_choice | When Pinocchio lies, his nose gets 6 cm longer. When he tells the truth, his nose gets 2 cm shorter. When his nose was 9cm long, he told three lies and made twotrue statments. How long was Pinnochio\textquotesingle s nose afterwards? | [
[
{
"aoVal": "A",
"content": "$$14$$cm "
}
],
[
{
"aoVal": "B",
"content": "$$15$$cm "
}
],
[
{
"aoVal": "C",
"content": "$$19$$cm "
}
],
[
{
"aoVal": "D",
"content": "$$23$$cm "
}
],
[
{
"aoVal": "E",
"content": "$$31$$cm "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication"
] | [
"9 + (3 x 6) - (2 x 2) = 23cm. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2407 | dc54ca8438214cde85383b78815ff4e7 | [
"其它"
] | 1 | single_choice | Four boys each prepared $1$ gift for a party. How many ways can these $4$ gifts be \textsubscript{} distributed among the $4$ kids so that no one receives his own gift? | [
[
{
"aoVal": "A",
"content": "$$12$$ "
}
],
[
{
"aoVal": "B",
"content": "$$9$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$1$$ "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Questions Involving Enumeration"
] | [
"Let the boys be A, B,~C and D and their gifts be 1, 2, 3 and 4. List down all possible arrangements: (A2, B1, C4, D3), (A2, B3, C4, D1), (A2, B4, C1, D3), (A3, B1, C4, D2), (A3, B4, C2, D1), (A3, B4, C1, D2), (A4, B1, C2, D3), (A4, B3, C2, D1), (A4, B3, C1, D2) There are \\textbf{9} possible arrangements. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2409 | a63e9ccc9ebe41a8a22286a7f6b279ca | [] | 1 | single_choice | How many three-digit whole numbers have a ones digit equal to the sum of the hundreds digit and the tens digit? | [
[
{
"aoVal": "A",
"content": "$$10$$ "
}
],
[
{
"aoVal": "B",
"content": "$$25$$ "
}
],
[
{
"aoVal": "C",
"content": "$$30$$ "
}
],
[
{
"aoVal": "D",
"content": "$$45$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"If the ones digit is $$2$$, the $$2$$ numbers are $$202$$ and $$112$$. For each ones digit, the number of possible numbers is the same as the ones digit. In all, there are $$1+2+3+\\cdots+8+9 =45$$ numbers. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2412 | bcb32ca7ba83490fa0745c83abd5f8c7 | [
"其它"
] | 1 | single_choice | What is the sum of all the digits in the number $2023$? | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
],
[
{
"aoVal": "E",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules"
] | [
"$2+0+2+3=7$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2414 | af43aa09c7134446b78f5960fc214021 | [
"其它"
] | 0 | single_choice | The correct information James can get from the statement ``there is a $11\textbackslash\%$ chance that it will rain tomorrow in Los Angeles'' is~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "It will rain $11\\textbackslash\\%$ of the time tomorrow in Los Angeles. "
}
],
[
{
"aoVal": "B",
"content": "It will rain in $11\\textbackslash\\%$ of the regions in Los Angeles tomorrow. "
}
],
[
{
"aoVal": "C",
"content": "It will definitely rain tomorrow in Los Angeles. "
}
],
[
{
"aoVal": "D",
"content": "The probability of raining in Los Angeles is high. "
}
],
[
{
"aoVal": "E",
"content": "The probability of raining in Los Angeles is low. "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"It is less likely to rain tomorrow in Los Angeles. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2415 | d7d60869cbeb41ebab2d03b0d102f2e2 | [] | 1 | single_choice | Ron\textquotesingle s work day is half over at $$1:15$$ P.M. if he starts work at $$9:30$$ A.M., his work day ends at． | [
[
{
"aoVal": "A",
"content": "$$4:30$$ P.M. "
}
],
[
{
"aoVal": "B",
"content": "$$5:00$$ P.M. "
}
],
[
{
"aoVal": "C",
"content": "$$5:30$$ P.M. "
}
],
[
{
"aoVal": "D",
"content": "$$9:30$$ P.M. "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Time Problem->Time Calculation"
] | [
"From $$9:30$$ A.M. until $$1:15$$ P.M. is $$3$$ hours and $$45$$ minutes. Add $$3$$ hours and $$45$$ minutes to $$1:15$$ P.M. to get $$5:00$$ P.M.. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2416 | e580d20ee3e447eeb3fea060defb06b3 | [] | 0 | single_choice | What time is $$11999$$ hours after $$3$$ P.M.? | [
[
{
"aoVal": "A",
"content": "$$2$$ A.M. "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ P.M. "
}
],
[
{
"aoVal": "C",
"content": "$$4$$ A.M. "
}
],
[
{
"aoVal": "D",
"content": "$$4$$ P.M. "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"$$12000$$ hours after $$3$$ P.M. is $$3$$ P.M. $$11999$$ hours after $$3$$ P.M is $$2$$ P.M. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2417 | a65eafe014e64617a3346747ac7d190b | [] | 3 | single_choice | Moon and Archie played chess competitively. Both of them are of the same level in terms of skill. They agreed to play in a best-of-seven games, where the one who wins four games first would be the ultimate winner. They have already played three games, with Moon winning two games and Archie winning just one game. What is the probability that Moon will be the ultimate winner? | [
[
{
"aoVal": "A",
"content": "$$\\frac{3}{8}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{11}{16}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{3}{16}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{7}{16}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"$$\\rm B$$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2418 | eea371b244b34ad7842f480d2f34b255 | [] | 1 | single_choice | Ms. Osborne asks each student in her class to draw a rectangle with integral side lengths and a perimeter of $$50$$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles? | [
[
{
"aoVal": "A",
"content": "$$76$$ "
}
],
[
{
"aoVal": "B",
"content": "$$120$$ "
}
],
[
{
"aoVal": "C",
"content": "$$128$$ "
}
],
[
{
"aoVal": "D",
"content": "$$132$$ "
}
],
[
{
"aoVal": "E",
"content": "$$136$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Combinatorics Involving Extreme Values->Extreme Value with Fixed Sums"
] | [
"As we know, the sum of the length and width is $$25$$. The largest area is $$13\\times12=156$$ and the smallest area is $$24\\times1=24$$, so the difference is $$156-24=132$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2419 | dc6dcb8d42a748ae8d20fbb83b097956 | [] | 0 | single_choice | Sandy, Sam, and Steve are observing a cat in the distance. The cat is eating something but it\textquotesingle s hard to tell what it is. Sandy says:"The cat is eating fish." Sam says:"I agree with Sandy!" Steve says: "The cat is eating meat." Then they come closer and realize only one of them guessed right. The cat is eating . | [
[
{
"aoVal": "A",
"content": " meat "
}
],
[
{
"aoVal": "B",
"content": " fish "
}
],
[
{
"aoVal": "C",
"content": " Not certain "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"We can directly find that Sandy\\textquotesingle s point and Sam\\textquotesingle s point are identical,so both of them tell lies.Therefore, Steve tells the truth. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2422 | b84b8086106340249e6b7cd7a9ba4b42 | [
"其它"
] | 1 | single_choice | Today is Tuesday, and Cindy eats cakes today. She will eat cakes again after $5$ days. What day will it be? | [
[
{
"aoVal": "A",
"content": "Sunday "
}
],
[
{
"aoVal": "B",
"content": "Saturday "
}
],
[
{
"aoVal": "C",
"content": "Friday "
}
],
[
{
"aoVal": "D",
"content": "Thursday "
}
],
[
{
"aoVal": "E",
"content": "Wednesday "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Counting the Number of Figures"
] | [
"Today is Tuesday, and $5$ days later will be Sunday. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2428 | f34ccda45217476381bb9ca89f85248f | [] | 1 | single_choice | One day, Pip asks his parents: "What day is it today?" His mother says: "Today is Monday." His father says: "Today is Tuesday." From the options below, which one do you agree with? | [
[
{
"aoVal": "A",
"content": "One of these two sentences is definitely wrong and the other one is correct. "
}
],
[
{
"aoVal": "B",
"content": "It is possible that both of Pip\\textquotesingle s parents are wrong. "
}
],
[
{
"aoVal": "C",
"content": "It is possible that both of Pip\\textquotesingle s parents are right. "
}
],
[
{
"aoVal": "D",
"content": "If Pip\\textquotesingle s mother is wrong, then his father must be right "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Comparing"
] | [
"\"Today is Monday\" is not the opposite of \"Today is Tuesday\".i.e. they can both be false. \"Today is Monday\" is the direct opposite of \"Today is not Monday\". One must be true and the other must be false. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2433 | d37cad20ddaa4cefb601bb3dbc0c4276 | [] | 1 | single_choice | How many different three-digit numbers can we make using the digits $$2$$, $$4$$, and $$6$$? (Each digit can be used only once.) | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations->Basic Operations of Combinations"
] | [
"$246, 264, 426, 462, 624, 642$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2435 | b879e45163ce4a22b695ad1d9c34da20 | [
"其它"
] | 1 | single_choice | What is $$542$$ hundredths? | [
[
{
"aoVal": "A",
"content": "$$0.0542$$ "
}
],
[
{
"aoVal": "B",
"content": "$$0.542$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5.42$$ "
}
],
[
{
"aoVal": "D",
"content": "$$54.02$$ "
}
],
[
{
"aoVal": "E",
"content": "$$54.2$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules"
] | [
"$$542$$ hundredths $=5$ ones, $4$ tenths and $2$ hundredths $=5.42$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2436 | fc9caa5fc3334cfa9b37b390a55ae5e0 | [
"其它"
] | 1 | single_choice | Two tiles numbered $1$ and $2$ are turned face down, respectively. One tile is turned up at random, and throw a die to get a number from $1$ to $6$. What is the probability that the product of the numbers on the tile and the die is greater than or equal to $12$? | [
[
{
"aoVal": "A",
"content": "$\\frac {1}{2}$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac {1}{6}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac {1}{3}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\frac {1}{4}$ "
}
],
[
{
"aoVal": "E",
"content": "$\\frac {1}{12}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"There are $12$ different combinations. The product of two numbers is greater than $12$ will be $2\\times6$. Thus, the probability is $\\frac 1{12}$ . "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2439 | ab21b4645a044e75aa66f47bd474ad08 | [] | 1 | single_choice | How many two-digit numbers are there where the ones digit is greater than the tens digit?． | [
[
{
"aoVal": "A",
"content": "$$26$$ "
}
],
[
{
"aoVal": "B",
"content": "$$18$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$30$$ "
}
],
[
{
"aoVal": "E",
"content": "$$36$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"$$8+7+6+5+4+3+2+1=36$$. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2442 | ab251de94f1a4ce0a1a7a702241294c8 | [
"其它"
] | 2 | single_choice | How many different four-digit numbers can be formed by rearranging the four digits in $2004$?~ | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$16$$ "
}
],
[
{
"aoVal": "D",
"content": "$$24$$ "
}
],
[
{
"aoVal": "E",
"content": "$$81$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations"
] | [
"0 shouldn\\textquotesingle t be put in the highest digit. There are two 0s The solutions are: 2400, 2040, 2004, 4200, 4020, 4002. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2444 | ab2c17c6c6494a07b392bc009fd7a8c4 | [
"其它"
] | 1 | single_choice | When Coco the Caterpillar is not sleeping, he eats $$5$$ grams of leaves per hour. Yesterday he slept $$20$$ hours. How many grams of leaves did he eat yesterday? | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10$$ "
}
],
[
{
"aoVal": "D",
"content": "$$20$$ "
}
],
[
{
"aoVal": "E",
"content": "$$40$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication"
] | [
"Yesterday he did not sleep for $$24 - 20 = 4$$ hours, so he ate $$4 \\times 5 = 20$$ grams of leaves. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2446 | dca8ee477c0c4f62be97c544e454b510 | [] | 1 | single_choice | Eight pupils from Victory Primary School take a Mathematics test, but none of the pupils wrote his/her name on the test. The tests are therefore handed back to the pupils at random. In how many ways can exactly $5$ of the $8$ pupils get the correct test back? | [
[
{
"aoVal": "A",
"content": "$$30$$ "
}
],
[
{
"aoVal": "B",
"content": "$$36$$ "
}
],
[
{
"aoVal": "C",
"content": "$$56$$ "
}
],
[
{
"aoVal": "D",
"content": "$$112$$ "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication->Principle of Multiplication"
] | [
"If exactly $5$ pupils get the correct test, then exactly $3$ pupils must get the wrong test. No. of ways to choose $5$ pupils to get the correct test is $$\\frac{8 \\times 7 \\times 6 \\times 5 \\times 4}{5 \\times 4 \\times 3 \\times 2 \\times 1}-56.$$ To make sure that the other $3$ pupils get the wrong tests, the correct number is $2$. Hence, the total no, of ways $=56 \\times2=112$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2448 | dcad7eb7c4a34404823f2c333ad628d4 | [
"其它"
] | 2 | single_choice | In how many ways can the letters in $BEEKBBPER$ be rearranged so that two or more $E$s do not appear together? | [
[
{
"aoVal": "A",
"content": "$$4200$$ "
}
],
[
{
"aoVal": "B",
"content": "$$900$$ "
}
],
[
{
"aoVal": "C",
"content": "$$800$$ "
}
],
[
{
"aoVal": "D",
"content": "$$720$$ "
}
],
[
{
"aoVal": "E",
"content": "$$700$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations"
] | [
"There are $3$ $E$s in total now with other $6$ letters remaining. But pay attention to $B$: there are $3$ $B$s here. There are $\\_6P\\_3$ ways for us to arrange the $6$ letters\\textquotesingle{} positions. Then, we can put the $3$ $E$s in the $7$ intervals. So the answer is $\\_6P\\_3 \\times \\_7C\\_3=4200$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2451 | bd20e4135a1843e2a7cc3bf68b67d32b | [] | 1 | single_choice | Pip multiplies all the digits in the number $$145$$ to get $$20$$. How many three digit numbers are there whose digits multiply to give $$20$$? | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations->Basic Operations of Combinations"
] | [
"$$145, 154, 415, 451, 514, 541, 225, 252, 522$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2452 | cf1e8441163d46f992fac984ca91d12f | [] | 1 | single_choice | Ranson holds $3$ white and $9$ purple jelly beans in his hand. Nancy holds $2$ white, $5$ yellow, and $2$ purple jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match? (adapted from $2013$ AMC $8$ Problem, Question \# $14$) | [
[
{
"aoVal": "A",
"content": "$\\dfrac{1}{6}$ "
}
],
[
{
"aoVal": "B",
"content": "$\\dfrac{1}{3}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\dfrac{2}{9}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\dfrac{1}{2}$ "
}
],
[
{
"aoVal": "E",
"content": "$\\dfrac{2}{3}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability->Questions Involving Probability->Basic Concepts of Probability"
] | [
"The probability that both show a white bean is $\\dfrac{3}{12}\\times \\dfrac{2}{9}=\\dfrac{1}{18}$. The probability that both show a purple bean is $\\dfrac{9}{12}\\times \\dfrac{2}{9}=\\dfrac{1}{6}$. Therefore, the probability is $\\dfrac{1}{18}+\\dfrac{1}{6}=\\frac29$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2454 | cf23660aa74b4175a1af0dd2f69671a4 | [] | 3 | single_choice | Linda picks $3$ different numbers from $$1-15$$. To make the sum of the three numbers divisible by $3$, how many different groups are there for Linda to pick? | [
[
{
"aoVal": "A",
"content": "$$100$$ "
}
],
[
{
"aoVal": "B",
"content": "$$910$$ "
}
],
[
{
"aoVal": "C",
"content": "$$91$$ "
}
],
[
{
"aoVal": "D",
"content": "$$155$$ "
}
],
[
{
"aoVal": "E",
"content": "$$30$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations"
] | [
"Group $A$: $$1$$, $$4$$, $$7$$, $$10$$, $13$; Group $B$: $$2$$, $$5$$, $$8$$, $$11$$, $$14$$; Group $C$: $$3$$, $$6$$, $$9$$, $$12$$, $$15$$. Linda can choose three numbers from the same group, or choose each number from a different group to get the sum she needs. There are $3\\times\\_5C\\_3+5\\times5\\times5=155$ groups. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2455 | eef21b75d2494366bb82308b1151deb8 | [] | 1 | single_choice | There are $$5$$ dancers. Every $$2$$ dancers will have a dance. How many dances would they have in total? | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$25$$ "
}
],
[
{
"aoVal": "E",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules"
] | [
"$4+3+2+1=10$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2461 | d3b94fd0e3644f609ae3840c205b2115 | [] | 1 | single_choice | If your average score on your first six mathematics tests was $84$ and your average score on your first seven mathematics tests was $85$, then your score on the seventh test was ． | [
[
{
"aoVal": "A",
"content": "$$86$$ "
}
],
[
{
"aoVal": "B",
"content": "$$88$$ "
}
],
[
{
"aoVal": "C",
"content": "$$90$$ "
}
],
[
{
"aoVal": "D",
"content": "$$91$$ "
}
],
[
{
"aoVal": "E",
"content": "$$92$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Word Problem Modules->Average Problems ->Questions Involving Average->Questions Involving Average (ordinary type)"
] | [
"The total score of the first six tests was $84\\times6=504$, and the total score of the first seven tests was $85\\times7=595$. Therefore, the score of the seventh test equals to the difference: $595-504=91$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2472 | c6527faea2834d1196949ba2dba9a447 | [] | 1 | single_choice | Take out $$4$$ digits from $101112131415$ to make the rest of the digits into a new $8-$digit number without changing the order of digits. The least possible value of the new number is~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$10111111$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10111110$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10111100$$ "
}
],
[
{
"aoVal": "D",
"content": "$$10111011$$ "
}
],
[
{
"aoVal": "E",
"content": "$$10111112$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"To make it the least, you should make the digit in the first place as small as possible. Thus the $$2$$, $$3$$, $4$ and $$5$$ should be taken out. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2473 | fcef3f4c348346119f05bbbe0cb3458f | [
"其它"
] | 1 | single_choice | How many two-digit numbers have digits whose sum is a perfect square? (2006 AMC 8 Problem, Question \#11) | [
[
{
"aoVal": "A",
"content": "$$13$$ "
}
],
[
{
"aoVal": "B",
"content": "$$16$$ "
}
],
[
{
"aoVal": "C",
"content": "$$17$$ "
}
],
[
{
"aoVal": "D",
"content": "$$18$$ "
}
],
[
{
"aoVal": "E",
"content": "$$19$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules"
] | [
"There is 1 integer whose digits sum to $1: 10$. There are 4 integers whose digits sum to $4: 13,22,31$, and 40 . There are 9 integers whose digits sum to $9: 18,27,36,45,54,63,72,81$, and 90 . There are 3 integers whose digits sum to $16: 79,88$, and 97 . Two digits cannot sum to 25 or any greater square since the greatest sum of digits of a twodigit number is $9+9=18$ Thus, the answer is $1+4+9+3=$ (C)17. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2474 | c6562aa5cfcd403faee3b5e3f12d0904 | [] | 1 | single_choice | Three students go apple-picking. Eddie picks $30$ apples, Avril picks $27$ apples, and Mike picks $33$ apples. How many apples does each of them pick on average? | [
[
{
"aoVal": "A",
"content": "$$90$$ "
}
],
[
{
"aoVal": "B",
"content": "$$45$$ "
}
],
[
{
"aoVal": "C",
"content": "$$31$$ "
}
],
[
{
"aoVal": "D",
"content": "$$30$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"They pick $30+27+33=90$ apples in total. Therefore, on average, each of them picks $90\\div3=30$ apples. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2476 | bd5e640710994bc08a5ab9b8c4b46b1f | [] | 1 | single_choice | How many two-digit numbers are there where the ones digit is greater than the tens~ digit?． | [
[
{
"aoVal": "A",
"content": "$$26$$ "
}
],
[
{
"aoVal": "B",
"content": "$$18$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$30$$ "
}
],
[
{
"aoVal": "E",
"content": "$$36$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication->Law of Addition"
] | [
"$$8+7+6+5+4+3+2+1=36$$. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2478 | fcfd4633aa54404eb36925fdd3fcaef5 | [
"其它"
] | 1 | single_choice | SASMO 2014 P2 Q1 What is 2014 + 2 x 0 x 1 x 4 equal to? | [
[
{
"aoVal": "A",
"content": "$$2014$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2016$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2021$$ "
}
],
[
{
"aoVal": "D",
"content": "$$2022$$ "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication"
] | [
"anything x 0 = 0 "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2479 | c1e5322370724acab3745a635ce09e1f | [] | 1 | single_choice | There is a ball in a box and three kids are guessing what colour it is. Val says: "The ball is~red." John says: "The ball is~green." Elvis says: "I agree with~John." They open the box and find only one of them guessed right. What colour is the ball? | [
[
{
"aoVal": "A",
"content": "red "
}
],
[
{
"aoVal": "B",
"content": "green "
}
],
[
{
"aoVal": "C",
"content": "Uncertain "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Comparing"
] | [
"We can spot that John\\textquotesingle s guess and Elvis\\textquotesingle{} guess are the same, so both of them must be wrong. Therefore, Val guessed it right. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2484 | b4763e9384324301b0d20a819a1de17a | [
"其它"
] | 2 | single_choice | Each of the 20 balls is tossed independently and at random into one of the 5 bins. Let $p$ be the probability that some bin ends up with 3 balls, another with 5 balls, and the other three with 4 balls each. Let $q$ be the probability that every bin ends up with 4 balls. What is $\frac{p}{q}$ ? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12$$ "
}
],
[
{
"aoVal": "E",
"content": "$$16$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"Solution 1: For simplicity purposes, we assume that the balls and the bins are both distinguishable. Recall that there are $5^{20}$ ways to distribute $20$ balls into $5$ bins. We have $$ p=\\frac{5 \\cdot 4 \\cdot\\left(\\begin{array}{c} 20 \\textbackslash\\textbackslash{} 3,5,4,4,4 \\end{array}\\right)}{5^{20}} \\text { and } q=\\frac{\\left(\\begin{array}{c} 20 \\textbackslash\\textbackslash{} 4,4,4,4,4 \\end{array}\\right)}{5^{20}} \\text {. } $$ Therefore, the answer is $$ \\frac{p}{q}=\\frac{5 \\cdot 4 \\cdot\\left(\\begin{array}{c} 20 \\textbackslash\\textbackslash{} 3,5,4,4,4 \\end{array}\\right)}{\\left(\\begin{array}{c} 20 \\textbackslash\\textbackslash{} 4,4,4,4,4 \\end{array}\\right)}=\\frac{5 \\cdot 4 \\cdot \\frac{20 !}{3 ! \\cdot 5 ! \\cdot 4 ! \\cdot 4 ! \\cdot 4 !}}{\\frac{20 !}{4 ! \\cdot 4 ! \\cdot 4 ! \\cdot 4 ! \\cdot 4 !}}=\\frac{5 \\cdot 4 \\cdot(4 ! \\cdot 4 ! \\cdot 4 ! \\cdot 4 ! \\cdot 4 !)}{3 ! \\cdot 5 ! \\cdot 4 ! \\cdot 4 ! \\cdot 4 !}=\\frac{5 \\cdot 4 \\cdot 4}{5}=(\\mathbf{E}) 16 . $$ Solution 2: For simplicity purposes, we assume that the balls and the bins are both distinguishable. Let $q=\\frac{x}{a}$, where $a$ is the total number of combinations and $x$ is the number of cases where every bin ends up with 4 balls. We can take 1 ball from one bin and place it in another bin so that some bin ends up with 3 balls, another with 5 balls, and the other three with 4 balls each. Note that one configuration of $4$-$4$-$4$-$4$-$4$ corresponds to $5 \\cdot 4 \\cdot 4=80$ configurations of $3$-$5$-$4$-$4$-$4$. On the other hand, one configuration of $3$-$5$-$4$-$4$-$4$ corresponds to 5 configurations of $4$-$4$-$4$-$4$-$4$. Therefore, we have $$ p=\\frac{80}{5} \\cdot \\frac{x}{a}=16 \\cdot \\frac{x}{a}, $$ from which $\\frac{p}{q}=$ (E) 16 . "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2485 | fd0d8d6173de4a41adf8252c46bde370 | [] | 1 | single_choice | Yuki and Claire are looking for a place to eat dinner. They know there are four Chinese restaurants, three French restaurants, and two Peruvian restaurants nearby. There are~\uline{~~~~~~~~~~}~different choices in total for them to eat one meal in one place. | [
[
{
"aoVal": "A",
"content": "$$9$$ "
}
],
[
{
"aoVal": "B",
"content": "$$24$$ "
}
],
[
{
"aoVal": "C",
"content": "$$27$$ "
}
],
[
{
"aoVal": "D",
"content": "$$18$$ "
}
],
[
{
"aoVal": "E",
"content": "$$14$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"They can only choose one place, so it can only be either Chinese, French, or Peruvian restaurants. Therefore, we can add each one up to get $$4+3+2 = 9$$.~ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2487 | eabeb8c5c5fe40779c651eb626f9426e | [
"其它"
] | 1 | single_choice | There are $12$ black balls, $27$ red balls, and $11$ blue balls in an opaque box. The balls are the same except for their colors. Bob adds several red balls in the box and mixes the balls. Now, if the probability of taking out a black ball is $\frac15$, how many red balls does Bob add? | [
[
{
"aoVal": "A",
"content": "$$5$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$15$$ "
}
],
[
{
"aoVal": "D",
"content": "$$20$$ "
}
],
[
{
"aoVal": "E",
"content": "$$25$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"$\\frac15=\\frac{12}{60}$, which means there are $60$ balls in total. Thus, Bob adds $60-12-27-11=10$ red balls. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2491 | c20cb81f61c64476aaee692c0191796b | [] | 1 | single_choice | There are $$10$$ players in a chess tournament. If each game is played by $$2$$ players, and each player plays every other player exactly once, what is the total number of games played in the tournament? | [
[
{
"aoVal": "A",
"content": "$$100$$ "
}
],
[
{
"aoVal": "B",
"content": "$$90$$ "
}
],
[
{
"aoVal": "C",
"content": "$$50$$ "
}
],
[
{
"aoVal": "D",
"content": "$$45$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Sports Competition"
] | [
"There are $10$ players and each player plays $9$ games, so there are $10 \\times 9 \\div 2 = 45$ games in total. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2492 | e63d6b2dd6754002a17adb214e6604b9 | [] | 1 | single_choice | Among $$30$$ children, there are at least~\uline{~~~~~~~~~~}~children born in the same month. | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$3$$ "
}
],
[
{
"aoVal": "D",
"content": "$$4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"There are $$12$$ months. Thus, $$30\\div 12=2R6$$ and hence $$2+1=3$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2493 | d417d5dca1e34d569125b63827cc8532 | [] | 1 | single_choice | If the sum of $$9$$ numbers is $$1998$$, then their average is ． | [
[
{
"aoVal": "A",
"content": "$$9+1998$$ "
}
],
[
{
"aoVal": "B",
"content": "$$9\\times 1998$$ "
}
],
[
{
"aoVal": "C",
"content": "$$1998\\div 9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9\\div 1998$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"The average of any $$9$$ numbers is their sum divided by $$9$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2496 | eadab1d59ac24d1e9ae00cde0a91c8d5 | [
"其它"
] | 2 | single_choice | In the Coin Game, you toss three coins at the same time. You win only if the 3 coins are all showing heads, or if the 3 coins are all showing tails. If you play the game once only, what is the probability of winning? | [
[
{
"aoVal": "A",
"content": "$\\frac{1}{6}$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac{1}{3}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac{2}{27}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\frac{2}{3}$ "
}
],
[
{
"aoVal": "E",
"content": "$\\frac{1}{4}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability->Questions Involving Probability->Typical Probability Problems->Tossing Coins"
] | [
"When tossing a single coin, there are two possible outcomes, a head (H) or a tail (T). When tossing 2 coins, there are $$2 \\times 2 = 4 $$possible outcomes. These are HH, HT, TH, and TT. When tossing 3 coins, there are $$2 \\times 2 \\times 2 = 8 $$possible outcomes. These are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. Of these 8 possible outcomes, there are 2 winning outcomes, HHH and TTT. Thus, the probability of winning the Coin Game is $\\frac{1}{4}$ . Answer: E "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2500 | ef70ffef212f4a8f81175aa644b19a19 | [
"其它"
] | 2 | single_choice | How many different four-digit numbers can be formed by rearranging the four digits in $2021$?~(Adapted from $2004$ AMC $8$ Problem, Question \#2) | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$18$$ "
}
],
[
{
"aoVal": "E",
"content": "$$24$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations"
] | [
"When the thousand digit is $2$, there are $\\_3P\\_3=6$ ways. When the thousand digit is $1$, there are $\\_3C\\_1=3$ ways. So the answer is $3+6=9$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2504 | d435b88f8f31478fa6111cff560271c7 | [
"其它"
] | 1 | single_choice | Paula shoots arrows at the following target. When she misses, she obtains zero points. Paula shoots two arrows and adds the number of points. Which of the following sums cannot be her score? | [
[
{
"aoVal": "A",
"content": "$$60$$ "
}
],
[
{
"aoVal": "B",
"content": "$$70$$ "
}
],
[
{
"aoVal": "C",
"content": "$$80$$ "
}
],
[
{
"aoVal": "D",
"content": "$$90$$ "
}
],
[
{
"aoVal": "E",
"content": "$$100$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication->Law of Addition"
] | [
"A is possible as 30+30 = 60 B is possible as 70+0+0 = 70 C is possible as 30 + 50 = 80 E is ossible as 30 + 70 = 100. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2508 | e1d7d09be42f49949fdfce15b31d1210 | [] | 1 | single_choice | How many different natural numbers can be formed with the numbers $$1$$, $$2$$, $$3$$?（without using the same number two times like $$33$$） | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$9$$ "
}
],
[
{
"aoVal": "C",
"content": "$$12$$ "
}
],
[
{
"aoVal": "D",
"content": "$$15$$ "
}
],
[
{
"aoVal": "E",
"content": "$$18$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Questions Involving Enumeration->Dictionary Ordering"
] | [
"Classify the number first by the digits and then enumerate． One-digit number：$$1$$、$$2$$、$$3$$， Two-digit number：$$12$$、$$13$$、$$21$$、$$23$$、$$31$$、$$32$$； Three-digit number：$$123$$、$$132$$、$$213$$、$$231$$、$$312$$、$$321$$； So totally $$3+6+6=15$$ different natural numbers can be formed． "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2509 | e6692e2ba24646fa8c2161e8572310c1 | [] | 2 | single_choice | Three kids $$A$$、$$B$$、$$C$$ are playing the game "pass the ball". If it starts with $$A$$, he can pass the ball to~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$A$$ or $$B$$ or $$C$$ "
}
],
[
{
"aoVal": "B",
"content": "$$B$$ or $$C$$ "
}
],
[
{
"aoVal": "C",
"content": "$$A$$ or $$C$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Questions Involving Enumeration->Tree Diagrams"
] | [
"A cannot pass the ball back to himself. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2511 | f8c6d87d8300475dafe4573626ecfd6f | [] | 1 | single_choice | An acronym is a word formed from the first one or more letters of each word in a group of words. If "UFO"is an acronym for "unidentified flying object," then for how many of the following word groups could MATH be an acronym? $$\text{I}$$. Multiply All Those Hundreds $$\text{II}$$. MArtians Take Hostages $$\text{III}$$. MATthew Hides $$\text{IV}$$. Minutes After The Hour | [
[
{
"aoVal": "A",
"content": "one "
}
],
[
{
"aoVal": "B",
"content": "two "
}
],
[
{
"aoVal": "C",
"content": "three "
}
],
[
{
"aoVal": "D",
"content": "four "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Conditions"
] | [
"For $$\\text{I}$$ and $$\\text{IV}$$, use first letter of each word; for $$\\text{II}$$, use first two letters of first word and first letter of other words; and for $$\\text{III}$$, use first three letters of first word and first letter of second word. MATH could be an acronym for all four word groups. $$\\text{I}$$. Multiply All Those Hundreds $$\\text{II}$$. MArtians Take Hostages $$\\text{III}$$. MATthew Hides $$\\text{IV}$$. Minutes After The Hour "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2517 | e68631c9420a403eb5bed8bc4ce61599 | [] | 1 | single_choice | There are $$10$$ players in a chess tournament. If each game is played by $$2$$ players, and each player plays every other player exactly once, what is the total number of games played in the tournament? | [
[
{
"aoVal": "A",
"content": "$$100$$ "
}
],
[
{
"aoVal": "B",
"content": "$$90$$ "
}
],
[
{
"aoVal": "C",
"content": "$$50$$ "
}
],
[
{
"aoVal": "D",
"content": "$$45$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"There are $10$ players and each player plays $9$ games, so there are $10 \\times 9 \\div 2 = 45$ games in total. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2519 | efa5efe78a404b5e81d42cef66fe920b | [] | 2 | single_choice | Gregor forms two numbers with digits $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, and $$6$$. Both numbers have three digits, and each digit is used only once. He adds these two numbers. What is the greatest sum Gregor can get? | [
[
{
"aoVal": "A",
"content": "$$975$$ "
}
],
[
{
"aoVal": "B",
"content": "$$999$$ "
}
],
[
{
"aoVal": "C",
"content": "$$1083$$ "
}
],
[
{
"aoVal": "D",
"content": "$$1173$$ "
}
],
[
{
"aoVal": "E",
"content": "$$1221$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication->Complex Forming Numbers"
] | [
"If we want to get the sum as large as possible, the hundred digits for both numbers should be as large as possible. Therefore, they should be $$6$$ and $$5$$. The sum of the hundreds digits is $$11$$. For the same reason, the sum of the tens digits should be $$3+4=7$$ and the sum of the ones digits should be $$2 + 1 = 3$$. Therefore, $$\\rm D$$ is the correct answer. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2524 | d47a71fa532d410fb495431263d2fe77 | [
"其它"
] | 1 | single_choice | A fair $6$-sided die is rolled once. What is the probability that the number on the top is an odd number? (adapted from 2011 AMC 8 Problem, Question \#18) | [
[
{
"aoVal": "A",
"content": "$\\frac16$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac13$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac12$ "
}
],
[
{
"aoVal": "D",
"content": "$\\frac23$ "
}
],
[
{
"aoVal": "E",
"content": "$\\frac56$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"There are $3$ odd numbers out of $6$, so the probability is $\\frac36=\\frac12$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2526 | d90a8963a65b4431a2b9f5d3afbb089b | [] | 1 | single_choice | How many two-digit numbers are there where the ones digit is greater than the tens digit? (2008 Math Kangaroo Problem, Level 3-4, Question \#21) | [
[
{
"aoVal": "A",
"content": "$$26$$ "
}
],
[
{
"aoVal": "B",
"content": "$$18$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$30$$ "
}
],
[
{
"aoVal": "E",
"content": "$$36$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations"
] | [
"$$8+7+6+5+4+3+2+1=36$$. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2530 | d92127155ba2428bac61aae5756fa92a | [] | 1 | single_choice | Amy, Bill and Celine are friends with different ages. $$\rm I$$. Bill is the oldest. $$\rm II$$. Amy is not the oldest. $$\rm III$$. Celine is not the youngest. Rank the friends from youngest to oldest. | [
[
{
"aoVal": "A",
"content": "Bill, Amy, Celine "
}
],
[
{
"aoVal": "B",
"content": "Amy, Bill, Celine "
}
],
[
{
"aoVal": "C",
"content": "Celine, Amy, Bill "
}
],
[
{
"aoVal": "D",
"content": "Celine, Bill, Amy "
}
],
[
{
"aoVal": "E",
"content": "Amy, Celine, Bill "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning by Conditions"
] | [
"If Bill is the oldest, then Amy is not the oldest, and both statements $$\\rm I$$ and $$\\rm II$$ are true, so statement $$\\rm I$$ is not the true one. If Amy is not the oldest, and we know Bill cannot be the oldest, then Celine is the oldest. This would mean she is not the youngest, and both statements $$\\rm II$$ and $$\\rm III$$ are true, so statement $$\\rm II$$ is not the true one. Therefore, statement $$\\rm III$$ is the true statement, and both $$\\rm I$$ and $$\\rm II$$ are false. From this, Amy is the oldest, Celine is in the middle, and lastly Bill is the youngest. This order is $$\\rm E$$. Amy, Celine, Bill. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2534 | d025e59893d246d6b42f72bb2b139be0 | [] | 1 | single_choice | $$30$$ children going to Adventure Park took part in at least one of two events. $$15$$ of them took part in the "moving bridge" contest, and $$20$$ of them went down the zip-wire. How many children from Adventure Park took part in both events? ($$2013$$ Math Kangaroo Problems, Level $$3-4$$, Question \#$$14$$) | [
[
{
"aoVal": "A",
"content": "$$25$$ "
}
],
[
{
"aoVal": "B",
"content": "$$15$$ "
}
],
[
{
"aoVal": "C",
"content": "$$30$$ "
}
],
[
{
"aoVal": "D",
"content": "$$10$$ "
}
],
[
{
"aoVal": "E",
"content": "$$5$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Inclusion-Exclusion Principle->Inclusion-Exclusion Principle for Two Sets"
] | [
"$15+20-30=5$ "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2537 | f92c2e0062c94f90b5de6ce08f77136e | [
"其它"
] | 2 | single_choice | The faces of each of two fair dice are numbered $1$, $2$, $3$, $5$, $7$, and $8$. When the two dice are tossed, what is the probability that their sum will be an even number? (2019 AMC 8 Problems, Question \#18) | [
[
{
"aoVal": "A",
"content": "$\\dfrac{4}{9}$ "
}
],
[
{
"aoVal": "B",
"content": "$\\dfrac{1}{2}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\dfrac{5}{9}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\dfrac{3}{5}$ "
}
],
[
{
"aoVal": "E",
"content": "$\\dfrac{2}{3}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability->Questions Involving Probability->Basic Concepts of Probability"
] | [
"We have a 2 die with 2 evens and 4 odds on both dies. For the sum to be even, the 2 rolls be 2 odds or 2 evens. Ways to roll 2 odds (Case 1 ): The total number of ways to obtain 2 odds on 2 rolls is $4 * 4=16$, as there are 4 possible odds on the first roll and 4 possible odds on the second roll. Ways to roll 2 evens (Case 2 ): Similarly, we have $2 * 2=4$ ways to obtain 2 evens. Probability is $\\frac{20}{36}=\\frac{5}{9}$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2538 | f48d47fdc56c4bf696a9f1930c0cc651 | [] | 1 | single_choice | How many different four-digit odd numbers can be made by using digits $$1$$, $$2$$, $$3$$, $$4$$, and $$5$$ (without digits be repeated)? | [
[
{
"aoVal": "A",
"content": "$$24$$ "
}
],
[
{
"aoVal": "B",
"content": "$$36$$ "
}
],
[
{
"aoVal": "C",
"content": "$$48$$ "
}
],
[
{
"aoVal": "D",
"content": "$$72$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication->Complex Forming Numbers"
] | [
"Rule of product: $$3\\times 4\\times 3\\times 2=72$$; therefore we can choose D. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2539 | f930f04b14ea48798ce974aee4142547 | [
"其它"
] | 2 | single_choice | How many ways are there of making a total of 10 using three different positive numbers? | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2$$ "
}
],
[
{
"aoVal": "D",
"content": "$$1$$ "
}
],
[
{
"aoVal": "E",
"content": "$$0$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Questions Involving Enumeration->Splitting Whole Numbers"
] | [
"$$1+2+7=10$$， $$1+3+6=10$$， $$1+4+5=10$$， $$2+3+5=10$$． So the answer is $$\\text{A}$$． "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2540 | e6efaac23bf74e2c9aebef6ef6381f64 | [] | 1 | single_choice | A conductor wanted to make a trio consisting of a fiddler, a pianist, and a drummer. He had to choose one of two fiddlers, one of two pianists, and one of two drummers. He decided to try each of the possible trios. How many attempts did he have to make? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$24$$ "
}
],
[
{
"aoVal": "E",
"content": "$$25$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Law of Addition and Multiplication->Principle of Multiplication"
] | [
"$$2\\times2\\times2=8$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2541 | d9572184f9db488c9fd3385df1a4f731 | [
"其它"
] | 1 | single_choice | Eve brings $$12$$ pieces of candy, Alice brings $$9$$ pieces of candy and Irene doesn\textquotesingle t bring any candy. They put all the pieces of candy together on a table and divide them equally among themselves. How many pieces of candy does each of the girls get? (2012 Math Kangaroo Problem, Levels 1-2, Question \#15） | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
],
[
{
"aoVal": "E",
"content": "$$12$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"They have $12+9=21$ pieces of candy in total. $21\\div3=7$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2543 | d95d1f305f804593b5f612f66cf11261 | [] | 2 | single_choice | Given that only one of the following statement is corect, which one is correct? ($$1$$) All of the statements below are correct. ($$2$$) None of the statement below is corect. ($$3$$) One of the statements above is corect. ($$4$$) All the statements above are correct. ($$5$$) None of the statement above is corect. | [
[
{
"aoVal": "A",
"content": "(1) "
}
],
[
{
"aoVal": "B",
"content": "(2) "
}
],
[
{
"aoVal": "C",
"content": "(3) "
}
],
[
{
"aoVal": "D",
"content": "(4) "
}
],
[
{
"aoVal": "E",
"content": "(5) "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Logical Reasoning->Reasoning using Hypothesis"
] | [
"Suppose ($$1$$) is correct, then ($$2$$) must be wrong which contradicts that only one statement is correct. Suppose ($$2$$) is correct, then ($$5$$) is correct which contradicts that only one statement is correct. Suppose ($$3$$) is correct, it also contradicts that only one statement is correct. Suppose ($$4$$) is correct, it also contradicts that only one statement is correct. Hence ($$5$$) is the correct statement. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2544 | e271e8538b984252b78fc49194c8bf83 | [] | 1 | single_choice | The farmer has a fish, a cat, and a dog. He wants to take them cross the river by boat, and he can only take one animal each time. When the farmer is away, the cat cannot be put with the fish or the dog, or they will fight. The boat makes one trip from one side of the river to the other side, so it takes~\uline{~~~~~~~~~~}~trips in total to carry all the animals and farmers cross the river. | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
],
[
{
"aoVal": "E",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules"
] | [
"We can use $A$ to represent the side of the river where the farmer is at the beginning, and use $B$ to represent the other side. The first time, take the cat from $A$ to $B$. The second time, the farmer go back to $A$. The third time, take the fish from $A$ to $B$. The fourth time, take the cat from $B$ to $A$. The fifth time, take the dog from $A$ to $B$. The sixth time, the farmer go back to $A$. The seventh time, take the cat from $A$ to $B$. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2548 | d4ec3b8cda7148c0a16cc37e42785c43 | [] | 1 | single_choice | Two dice are thrown. What is the probability that the product of the two numbers is a multiple of $$5$$? ($$2001$$ AMC $$8$$ Problem, Question \#$$ 18$$) | [
[
{
"aoVal": "A",
"content": "$$\\frac{1}{36}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{1}{18}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{1}{6}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{11}{36}$$ "
}
],
[
{
"aoVal": "E",
"content": "$$\\frac{1}{3}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability->Questions Involving Probability->Basic Concepts of Probability"
] | [
"$$5$$ is the only multiple of $$5$$ on a die, so one of the numbers rolled must be a $$5$$. To find the probability of rolling at least one $$5$$, we can find the probability of not rolling a $$5$$ and subtract that from $$1$$, since you either roll a $$5$$ or not roll a $$5$$. The probability of not rolling a $$5$$ on either dice is $$\\left( \\frac{5}{6} \\right)\\times\\left( \\frac{5}{6} \\right)=\\frac{25}{36}$$. Therefore, the probability of rolling at least one five, and thus rolling two numbers whose product is a multiple of $$5$$, is $$1-\\frac{25}{36}=\\frac{11}{36}$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2551 | fe14ff795f714e9795b2ec756be29b35 | [
"其它"
] | 1 | single_choice | Mom has $190$ coins in total and wants to give them to Sana as the pocket money. She gives Sana $30$ coins for the first month, and $40$ coins for each of the following months. How many months in total can Sana get the pocket money? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
],
[
{
"aoVal": "E",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules"
] | [
"$190-30=160$ $40+40+40+40=160$ $4+1=5$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2554 | f98009c9471f4e868f9580e0dd2a76e3 | [
"其它"
] | 1 | single_choice | A top hat contains $$3$$ red chips and $$2$$ green chips. Chips are drawn randomly, one at a time without replacement, until all $$3$$ of the reds are drawn or until both green chips are drawn. What is the probability that the $$3$$ reds are drawn? (2016 AMC 8 Problems, Question \#21) | [
[
{
"aoVal": "A",
"content": "$$\\frac{3}{10}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{2}{5}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{1}{2}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{3}{5}$$ "
}
],
[
{
"aoVal": "E",
"content": "$$\\frac{7}{10}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability->Questions Involving Probability"
] | [
"There are two ways of ending the game, either you picked out all the red chips or you picked out all the green chips. We can pick out 3 red chips, 3 red chips and 1 green chip, 2 green chips, 2 green chips and 1 red chip, and 2 green chips and 2 red chips. Because order is important in this problem, there are $1+4+1+3+6=15$ ways to pick out the chip. But we noticed that if you pick out the three red chips before you pick out the green chip, the game ends. So we need to subtract cases like that to get the total number of ways a game could end, which $15-5=10$. Out of the 10 ways to end the game, 4 of them ends with a green chip. The answer is $\\frac{4}{10}=\\frac{2}{5}$, or $(\\mathbf{B}) \\frac{2}{5}$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2555 | fe1b1d8608f44dee933e5311baf20726 | [] | 1 | single_choice | My train leaves Southampton and arrives in Birmingham at $$08:48$$ that morning, the duration of the whole journey is $$52$$ minutes, what time does the train leave Southampton? | [
[
{
"aoVal": "A",
"content": "$$09:40$$ "
}
],
[
{
"aoVal": "B",
"content": "$$09:30$$ "
}
],
[
{
"aoVal": "C",
"content": "$$07:56$$ "
}
],
[
{
"aoVal": "D",
"content": "$$07:04$$ "
}
],
[
{
"aoVal": "E",
"content": "$$07:40$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Combinatorics->Time Problem->Time Calculation"
] | [
"The departure time is $$52$$ minutes forward from $$8:48$$. You can push it forward $$48$$ minutes, which is $$8:00$$, and then push it forward $$4$$ minutes, which is $$7:56$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 2561 | de29deaa143a4ba6abcf1ac368fea1c3 | [] | 1 | single_choice | In the following events, the certain event is~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "Toss two identical coins and both land on heads. "
}
],
[
{
"aoVal": "B",
"content": "Throw a fair die and number of the dot shown is $$3$$. "
}
],
[
{
"aoVal": "C",
"content": "Sun sets in the west. "
}
],
[
{
"aoVal": "D",
"content": "It must rain in cloudy days. "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"$$\\text{A}$$. Throw a coin randomly. It has a $$50\\textbackslash\\%$$ chance of landing head up and a $$50\\textbackslash\\%$$ chance of landing tail up. $$\\text{B}$$. Throw a die, and number of the dot shown may be any number from $$1$$ to $$6$$. $$\\text{C}$$. Sun sets in the west. $$\\text{D}$$. It may not rain in cloudy days. So $$\\text{C}$$ is the answer. "
] | C |
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