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 | 5659 | a48cb9e695ce4433a7d565fbead933af | [
"其它"
] | 2 | single_choice | For how many integers $x$ is the number $x^{4}-51 x^{2}+50$ negative? (2014 AMC 10B Problems, Question \#20) | [
[
{
"aoVal": "A",
"content": "$$8$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$12$$ "
}
],
[
{
"aoVal": "D",
"content": "$$14$$ "
}
],
[
{
"aoVal": "E",
"content": "$$16$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"First, note that $50+1=51$, which motivates us to factor the polynomial as $\\left(x^{2}-50\\right)\\left(x^{2}-1\\right)$. Since this expression is negative, one term must be negative and the other positive. Also, the first term is obviously smaller than the second, so $x^{2}-50\\textless0\\textless x^{2}-1$. Solving this inequality, we find $1\\textless x^{2}\\textless50$. There are exactly $12$ integers $x$ that satisfy this inequality, $\\pm\\textbackslash{2,3,4,5,6,7\\textbackslash}$. Thus our answer is $(\\mathbf{C}) 12$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5660 | e476430d44274efa86feeccefc2acc40 | [] | 1 | single_choice | $$2015$$ Rosyth School, First Continual Assessment, Primary $$5$$, Question \#$$11$$ What is the value of $$\left( 84\div 7 \right)+7\times 6-3$$? | [
[
{
"aoVal": "A",
"content": "$$33$$ "
}
],
[
{
"aoVal": "B",
"content": "$$51$$ "
}
],
[
{
"aoVal": "C",
"content": "$$57$$ "
}
],
[
{
"aoVal": "D",
"content": "$$111$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations"
] | [
"$84\\div7=12$ $7\\times6=42$ $$12+42-3=51$$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5664 | bfdc46f4cdde4a8b9418162ba90debef | [
"其它"
] | 1 | single_choice | Which of the following fractions is equivalent to $$0.3\dot{8}$$？ | [
[
{
"aoVal": "A",
"content": "$$\\frac{7}{18}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{38}{990}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{35}{99}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{19}{45}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Decimals"
] | [
"$$0.3\\dot{8}=\\frac{38-3}{90}$$ $$\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde=\\frac{35}{90}$$ $$\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde=\\frac{7}{18}$$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5667 | 9b7d54f832ab44cbb4ce5c82f1e6c550 | [
"其它"
] | 1 | single_choice | $$100\div \dfrac{2}{3}=$$~\uline{~~~~~~~~~~}~，$$\dfrac{15}{16}\div 5=$$~\uline{~~~~~~~~~~}~． | [
[
{
"aoVal": "A",
"content": "$$\\dfrac{200}{3}$$，$$\\dfrac{75}{16}$$． "
}
],
[
{
"aoVal": "B",
"content": "$$150$$，$$\\dfrac{75}{16}$$． "
}
],
[
{
"aoVal": "C",
"content": "$$\\dfrac{200}{3}$$，$$\\dfrac{3}{16}$$． "
}
],
[
{
"aoVal": "D",
"content": "$$150$$，$$\\dfrac{3}{16}$$． "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"$$100\\div \\dfrac{2}{3}=\\dfrac{50\\times 2}{1}\\times \\dfrac{3}{2}=150$$． $$\\dfrac{15}{16}\\div 5=\\dfrac{3\\times 5}{16}\\times \\dfrac{1}{5}=\\dfrac{3}{16}$$． "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5669 | 815c6566fc81438cbf1242bbd47632a0 | [
"其它"
] | 0 | single_choice | Bruce is a talented writer and graphic artist who enjoys both types of work equally. Instead of earning 45,000 dollars as a writer, Bruce now earns 25,000 dollars in accounting profits as a graphic artist using the same computer equipment he would have used as a writer. What is Bruce\textquotesingle s economic profit from choosing to work as a graphic artist? | [
[
{
"aoVal": "A",
"content": "-45,000 dollars "
}
],
[
{
"aoVal": "B",
"content": "-20,000 dollars "
}
],
[
{
"aoVal": "C",
"content": "20,000 dollars "
}
],
[
{
"aoVal": "D",
"content": "45,000 dollars "
}
],
[
{
"aoVal": "E",
"content": "70,000 dollars "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"Economic profit = total revenue - economic costs = total revenue - (explicit costs + implicit costs) "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5682 | b6bd8e2984064826a58775982ac176e8 | [
"其它"
] | 1 | single_choice | If~$\dfrac{3}{5}=\dfrac{M}{45}=\dfrac{60}{N}$ , what is $M+N$? （2008 AMC 8,7） | [
[
{
"aoVal": "A",
"content": "$$27$$ "
}
],
[
{
"aoVal": "B",
"content": "$$29$$ "
}
],
[
{
"aoVal": "C",
"content": "$$45$$ "
}
],
[
{
"aoVal": "D",
"content": "$$105$$ "
}
],
[
{
"aoVal": "E",
"content": "$$127$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions->Proportions->Simplifying Continued Ratios"
] | [
"Separate into two equations~$\\dfrac{3}{5}=\\dfrac{M}{45}$~and~$\\dfrac{3}{5}=\\dfrac{60}{N}$~and solve for the unknowns.~$M=27$~and $N=100$, therefore $M+N=\\boxed{\\left( E\\right)127}.$. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5693 | f25021cd25824f77809d10f215d48068 | [] | 1 | single_choice | How many digits are there in the correct answer to the calculation $$1234123412340\div 1234$$? | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$10$$ "
}
],
[
{
"aoVal": "E",
"content": "$$12$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Calculation of Multi-digit Numbers"
] | [
"The correct answer to the calculation $$1234123412340\\div1234=1000100010$$. This has $$10$$ digits. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5696 | b6c4ebdbfd6345e687805c5627b28131 | [
"其它"
] | 1 | single_choice | Four numbers are written in a row. The average of the first two is $21$, the average of the middle two is $26$, and the average of the last two is $30$. What is the average of the first and last of the numbers? | [
[
{
"aoVal": "A",
"content": "$$24$$ "
}
],
[
{
"aoVal": "B",
"content": "$$25$$ "
}
],
[
{
"aoVal": "C",
"content": "$$26$$ "
}
],
[
{
"aoVal": "D",
"content": "$$27$$ "
}
],
[
{
"aoVal": "E",
"content": "$$28$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"NA "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5697 | 8e03114b986b4f988a8e063f2e29f6a8 | [] | 1 | single_choice | Compare these fractions. $$\frac{3}{8}$$~\uline{~~~~~~~~~~}~$$\frac{1}{2}$$, $$\frac{5}{18}$$~\uline{~~~~~~~~~~}~$$\frac{1}{3}$$ | [
[
{
"aoVal": "A",
"content": "$$\\textgreater$$, $$\\textgreater$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\textgreater$$, $$\\textless$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\textless$$, $$\\textgreater$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\textless$$, $$\\textless$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating->Comparing and Ordering"
] | [
"$$\\frac{3}{8}\\textless\\frac{4}{8}$$;~$$\\frac{5}{18}\\textless\\frac{6}{18}$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5703 | 8e05a065704149cfadeb4aad0eda272a | [
"其它"
] | 1 | single_choice | Solve the equation: $30-9x=28-5x$. | [
[
{
"aoVal": "A",
"content": "$x=1$ "
}
],
[
{
"aoVal": "B",
"content": "$x=0.5$ "
}
],
[
{
"aoVal": "C",
"content": "$x=2$ "
}
],
[
{
"aoVal": "D",
"content": "$x=1.5$ "
}
],
[
{
"aoVal": "E",
"content": "$x=1.6$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation"
] | [
"$30-28=9x-5x$ $2=4x$ $x=0.5$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5707 | a01c42b5cb1a4882b086db898edcc585 | [
"其它"
] | 0 | single_choice | Find the next number in the following sequence $$21, 24, 19, 26, 17, \cdots $$ | [
[
{
"aoVal": "A",
"content": "$$18$$ "
}
],
[
{
"aoVal": "B",
"content": "$$19$$ "
}
],
[
{
"aoVal": "C",
"content": "$$27$$ "
}
],
[
{
"aoVal": "D",
"content": "$$28$$ "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns in Number Sequences"
] | [
"$$21, 19, 17, \\cdots $$ (pattern of $$-2$$) $$24, 26, ?, \\cdots $$ (pattern of $$+2$$) Thus, the \"?\" is equal to $$26+2=28$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5708 | a931ad6215944ebaa73de9f4e285bad1 | [
"其它"
] | 1 | single_choice | The ones digit of $$1!+2!+3!+···+100!$$ is~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
],
[
{
"aoVal": "E",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations through Formulas"
] | [
"$$\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde1+1\\times 2+1\\times 2\\times 3+1\\times 2\\times 3\\times 4$$ $$=1+2+6+24$$ $$=33$$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5711 | 85ee6f4a15d44c0ea64d66607d6c2bbb | [
"其它"
] | 1 | single_choice | There are two girls and two cats in the room. How many legs are there in this room? (Adapted from 2011 Math Kangaroo Problem, Level 1-2, Question \#4) | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12$$ "
}
],
[
{
"aoVal": "E",
"content": "$$16$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$$2+2+4+4=12$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5713 | 971071f93e9947c18f78e491eb8fb3ed | [
"其它"
] | 1 | single_choice | Peter Rabbit eats cabbages and carrots. Each day he eats either $$10$$ carrots or $$2$$ cabbages. Last week Peter ate $$6$$ cabbages. How many carrots did he eat last week? (2014 Math Kangaroo Problem, Level 1-2, Question \#18) | [
[
{
"aoVal": "A",
"content": "$$20$$ "
}
],
[
{
"aoVal": "B",
"content": "$$30$$ "
}
],
[
{
"aoVal": "C",
"content": "$$34$$ "
}
],
[
{
"aoVal": "D",
"content": "$$40$$ "
}
],
[
{
"aoVal": "E",
"content": "$$50$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations"
] | [
"Peter ate $6$ cabbages in $6\\div2=3$ days. In $7-3=4$ days, Peter ate $4\\times10=10+10+10+10=40$ carrots. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5715 | 74221ffccc5d44688752cadb12382f47 | [
"其它"
] | 1 | single_choice | An amusement park has a collection of scale models, with ratio $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number? (2018 AMC 8, Question 1) | [
[
{
"aoVal": "A",
"content": "$$14$$ "
}
],
[
{
"aoVal": "B",
"content": "$$15$$ "
}
],
[
{
"aoVal": "C",
"content": "$$16$$ "
}
],
[
{
"aoVal": "D",
"content": "$$18$$ "
}
],
[
{
"aoVal": "E",
"content": "$$20$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions"
] | [
"You can see that since the ratio of real building\\textquotesingle s heights to the model building\\textquotesingle s height is $1: 20$. We also know that the U.S Capitol is 289 feet in real life, so to find the height of the model, we divide by 20 . That gives us $14.45$ which rounds to 14 . Therefore, to the nearest whole number, the duplicate is (A) 14 feet. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5716 | c479a4443c134efa9478531cd1a98edd | [] | 1 | single_choice | Seventy $$-$$ seven hundred is equal to． | [
[
{
"aoVal": "A",
"content": "$$70\\times 70$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7\\times 7000$$ "
}
],
[
{
"aoVal": "C",
"content": "$$7\\times 10$$ "
}
],
[
{
"aoVal": "D",
"content": "$$11 \\times 700$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"Seventy $$-$$ seven hundred $$= 7700 = 11\\times700$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5726 | b2495473faed44c194cfb4bc28a8b727 | [] | 1 | single_choice | $$180\div 6=6\times $$． | [
[
{
"aoVal": "A",
"content": "$$180$$ "
}
],
[
{
"aoVal": "B",
"content": "$$36$$ "
}
],
[
{
"aoVal": "C",
"content": "$$30$$ "
}
],
[
{
"aoVal": "D",
"content": "$$5$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"$$180\\div 6=30=6\\times 5$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5736 | 860264f32c03437680a1edbb0cd8053d | [
"其它"
] | 1 | single_choice | Alen and Austin share $60$ apples between them. Alen has twice as much as Austin. How many apples does Alen have? | [
[
{
"aoVal": "A",
"content": "$$20$$ "
}
],
[
{
"aoVal": "B",
"content": "$$30$$ "
}
],
[
{
"aoVal": "C",
"content": "$$40$$ "
}
],
[
{
"aoVal": "D",
"content": "$$45$$ "
}
],
[
{
"aoVal": "E",
"content": "$$50$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$60 \\div (2 + 1) \\times 2 = 40$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5738 | 860465da932d483b9fd8d9c651ca2c12 | [
"其它"
] | 4 | single_choice | Fifteen integers $a\_1, a\_2, a\_3, \cdots , a\_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that $$ 1 \leq a\_1 \leq 10$$,~~$$13 \leq a\_2 \leq 20$$, and $$241 \leq a\_{15} \leq 250$$. What is the sum of the digits of $a\_{14}$? | [
[
{
"aoVal": "A",
"content": "$$8$$ "
}
],
[
{
"aoVal": "B",
"content": "$$9$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10$$ "
}
],
[
{
"aoVal": "D",
"content": "$$11$$ "
}
],
[
{
"aoVal": "E",
"content": "$$12$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"A "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5741 | a02f33af2807415586c2899695763e08 | [] | 2 | single_choice | Given a twelve-digit number $$370592318436$$. Brandon wants to delete $4$ odd digits from the number to create an eight-digit number. What is the smallest eight-digit number that Brandon can form? | [
[
{
"aoVal": "A",
"content": "$$30214836$$ "
}
],
[
{
"aoVal": "B",
"content": "$$30128436$$ "
}
],
[
{
"aoVal": "C",
"content": "$$30231436$$ "
}
],
[
{
"aoVal": "D",
"content": "$$30523136$$ "
}
],
[
{
"aoVal": "E",
"content": "$$30218436$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"NA "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5743 | c4846080de1e4721a24aed4e18b7ec27 | [
"其它"
] | 1 | single_choice | How many inequalities are there in the following options? 1. $x = 1$ 2. $y \textless{} 2$ 3. $1 + a \geq g + p$ 4. $1 \textgreater{} 2$ | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3$$ "
}
],
[
{
"aoVal": "E",
"content": "$$4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities"
] | [
"$23$ and $4$ are inequalities. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5750 | 819d016f869d4fa88daf92bab3ae7ece | [] | 1 | single_choice | Express $108:9$ in simplest form:． | [
[
{
"aoVal": "A",
"content": "$12:1$ "
}
],
[
{
"aoVal": "B",
"content": "$24:6$ "
}
],
[
{
"aoVal": "C",
"content": "$16:4$ "
}
],
[
{
"aoVal": "D",
"content": "$4:1$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions->Ratio"
] | [
"$(108\\div9):(9\\div9)=12:1$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5751 | b6dce72e23e5457d87358ee9cbc14037 | [] | 1 | single_choice | Calculate: $$5 -- 4 + 3 -- 2 + 1 =$$． | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$3$$ "
}
],
[
{
"aoVal": "D",
"content": "$$4$$ "
}
],
[
{
"aoVal": "E",
"content": "$$5$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction "
] | [
"$$= (5 -- 4) + (3 -- 2) + 1 = 1 + 1 + 1 = 3$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5755 | 8a9551f7735340ec94587b30c49fcafb | [] | 1 | single_choice | Devi is $$9$$ years old and Valli is $$13$$ years old now. What is the ratio of Devi\textquotesingle s age to Valli\textquotesingle s age in $$5$$ years\textquotesingle{} time? | [
[
{
"aoVal": "A",
"content": "$$3:7$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7:9$$ "
}
],
[
{
"aoVal": "C",
"content": "$$13:14$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9:18$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions->Ratio"
] | [
"$5$ years later, Devi is $9+5=14$ years old and Valli is $13+5=18$ years old. $D:V$ $=$ $14:18$ $\\to$ $7:9$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5762 | 7d3509ba702e41dea831472bbc90de6e | [
"其它"
] | 2 | single_choice | Four fifths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 10 cups. What percent of the total capacity of the pitcher did each cup receive?~ (adapted from 2020 AMC 8, Question \#5) | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12$$ "
}
],
[
{
"aoVal": "E",
"content": "$$20$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Percentage Calculation"
] | [
"The pitcher is $\\frac{4}{5}$ full, i.e. $80 \\textbackslash\\%$ full. Therefore each cup receives $\\frac{80}{10}=(\\mathbf{C}) 8$ percent of the total capacity. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5765 | 8e362b3f4149423194ff167e23aa68f3 | [
"其它"
] | 1 | single_choice | If Anna takes one step forward, it means she walks $"+1"$ step; If Anna takes one step backward, it means she walks $"-1"$ step. Now, Anna walks $"-5"$ steps, then she walks $"+4"$ steps, then she walks another $"-1"$ step. Anna\textquotesingle s position is from her original location. | [
[
{
"aoVal": "A",
"content": "$1$ step forward "
}
],
[
{
"aoVal": "B",
"content": "$1$ step backward "
}
],
[
{
"aoVal": "C",
"content": "$2$ step forward "
}
],
[
{
"aoVal": "D",
"content": "$2$ step backward "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Negative Numbers"
] | [
"Since walking one step forward and one step backward gets you back to your original location. Anna takes $4$ steps foward and $1+5=6$ steps backward. $6-4=2$(steps) Therefore, Anna is now $2$ steps backward from her original location. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5768 | 8a9ce4c8d1e54ab284ca9b36e2df85a7 | [] | 1 | single_choice | $$(12 \times7)-(12 + 12 + 12 + 12) =$$． | [
[
{
"aoVal": "A",
"content": "$$36$$ "
}
],
[
{
"aoVal": "B",
"content": "$$48$$ "
}
],
[
{
"aoVal": "C",
"content": "$$72$$ "
}
],
[
{
"aoVal": "D",
"content": "$$108$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations"
] | [
"$$(12 \\times7)-(12 + 12 + 12 + 12) =83-48=36$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5770 | d6d9639a0a2c4f9d8c6c354865bf3df0 | [
"其它"
] | 1 | single_choice | The last five digits of Zoe\textquotesingle s phone number are $3,4,8,5,7.$ If these $5$ digits are filled in the square~$$\huge\square+\square =\square +\square $$, which number is not used?~(adapted from $$2017$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$8$$) | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
],
[
{
"aoVal": "E",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Addition of Whole Numbers->Addition in Horizontal Form"
] | [
"$4+7=3+8$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5772 | 81b2e0d7f53d4b478268194aadb42708 | [] | 1 | single_choice | When $$115$$ is divided by~ \uline{?~} , the quotient is 4 and remainder is $$3$$. | [
[
{
"aoVal": "A",
"content": "$$27$$ "
}
],
[
{
"aoVal": "B",
"content": "$$28$$ "
}
],
[
{
"aoVal": "C",
"content": "$$460$$ "
}
],
[
{
"aoVal": "D",
"content": "$$463$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$115 - 3 = 112$ $? \\times 4 = 112$ $? = 112 \\div ~4 = 28$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5776 | dffe7934778e4dceb3eca43c1122b699 | [] | 1 | single_choice | $$\left( {} \right.$$The number of seconds in a week$$\left. {} \right)$$$$\div$$$$\left( {} \right.$$the number of minutes in a week$$\left. {} \right)=$$． | [
[
{
"aoVal": "A",
"content": "$$60$$ "
}
],
[
{
"aoVal": "B",
"content": "$$420$$ "
}
],
[
{
"aoVal": "C",
"content": "$$3600$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7200$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions"
] | [
"There are $$60$$ seconds in each minute, so the quotient of the two quantities is $$60$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5781 | 974209e1ef2e40c1bf8d9710f1c7c064 | [
"其它"
] | 1 | single_choice | There are $12$ boys in the class. The ratio of boys to girls is $1:2$. How many students are there in the class? | [
[
{
"aoVal": "A",
"content": "$$18$$ "
}
],
[
{
"aoVal": "B",
"content": "$$24$$ "
}
],
[
{
"aoVal": "C",
"content": "$$30$$ "
}
],
[
{
"aoVal": "D",
"content": "$$36$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions"
] | [
"$12-\\textgreater1$ $24-\\textgreater2$ $12+24=36$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5788 | 81c6279f5a76407c96e2ccc91019da24 | [
"其它"
] | 2 | single_choice | \textbf{A news organization conducted a survey about preferred methods for obtaining the news. A random sample of 1,605 adults living in a certain state was selected, and 16.2 percent of the adults in the sample reported that television was their preferred method. Which of the following is an appropriate margin of error for a 90 percent confidence interval to estimate the population proportion of all adults living in the state who would report that television is their preferred method for obtaining the news?} | [
[
{
"aoVal": "A",
"content": "\\textbf{$1.645 \\sqrt{\\frac{(0.162)(1-0.162)}{1605}}$} "
}
],
[
{
"aoVal": "B",
"content": "\\textbf{$1.645 \\sqrt{\\frac{(0.5)(1-0.5)}{1605}}$} "
}
],
[
{
"aoVal": "C",
"content": "\\textbf{$1.96 \\sqrt{\\frac{(0.162)(1-0.162)}{1605}}$} "
}
],
[
{
"aoVal": "D",
"content": "\\textbf{$1.96 \\sqrt{\\frac{(0.5)(1-0.5)}{1605}}$} "
}
],
[
{
"aoVal": "E",
"content": "\\textbf{$1.83 \\sqrt{\\frac{(0.162)(1-0.162)}{1605}}$} "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"\\textbf{$\\hat{p} = 0.162$} \\textbf{MOE = $Z\\_{0.05}\\sqrt{\\frac{0.162*(1-0.162)}{1605}}$ = 1.65*$\\sqrt{\\frac{0.162*(1-0.162)}{1605}}$} "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5789 | f70d9974f97a4cfcaecc4e3dcad37d3e | [
"其它"
] | 2 | single_choice | Cassandra sets her watch to the correct time at noon. At the actual time of $1: 00$ PM, she notices that her watch reads $12: 57$ and $36$ seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads $$10:00$$ PM? (2003 AMC 12B Problems, Question \#11) | [
[
{
"aoVal": "A",
"content": "$10:22$ PM and $24$ seconds "
}
],
[
{
"aoVal": "B",
"content": "$10:24$ PM "
}
],
[
{
"aoVal": "C",
"content": "$10:25$ PM "
}
],
[
{
"aoVal": "D",
"content": "$10:27$ PM "
}
],
[
{
"aoVal": "E",
"content": "$10:30$ PM "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"For every $60$ minutes that pass by in actual time, $57+\\frac{36}{60}=57.6$ minutes pass by on Cassandra\\textquotesingle s watch. When her watch first reads, $10: 00$ PM, $10(60)=600$ minutes have passed by on her watch. Setting up a proportion, $$ \\frac{57.6}{60}=\\frac{600}{x} $$ where $x$ is the number of minutes that have passed by in actual time. Solve for $x$ to get $625$ minutes, or $10$ hours and $25$ minutes $$ \\Rightarrow \\text { (C) 10:25} $$ PM. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5794 | 7d56933ba1534f95843e118e6ae3d3d4 | [
"其它"
] | 1 | single_choice | Yesterday is Saturday, what day is the day after tomorrow? (Adapted from 2019 Math Kangaroo Problem, Level 3-4, Question \#3) | [
[
{
"aoVal": "A",
"content": "Monday "
}
],
[
{
"aoVal": "B",
"content": "Tuesday "
}
],
[
{
"aoVal": "C",
"content": "Wednesday "
}
],
[
{
"aoVal": "D",
"content": "Thursday "
}
],
[
{
"aoVal": "E",
"content": "Sunday "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"Yesterday: Saturday Today: Sunday Tomorrow: Monday The day after tomorrow: Tuesday "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5795 | ade9a46d154e4f8795be7786a405f366 | [
"其它"
] | 2 | single_choice | \textbf{Free-response questions on the AP Statistics Exam are graded 4, 3, 2, 1, or~ 0. Question 2 on the exam was of moderate difficulty. The average score on question 2 was 2.05 with a standard deviation of 1. To the nearest tenth, what score was achieved by a student who was at the 90th percentile of all students on the test? You may assume that the scores on the question were approximately normally distributed.} | [
[
{
"aoVal": "A",
"content": "$$3.5$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3.3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2.9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3.7$$ "
}
],
[
{
"aoVal": "E",
"content": "$$3.1$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"\\textbf{Z = 1.28} \\textbf{$$\\frac{x-2.05}{1}$$ = 1.28} \\textbf{x= 3.33} "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5801 | 8e54d0f77a634a9d9c837cb96cf764da | [
"其它"
] | 0 | single_choice | What is the simplest form of $\frac{15}{20}$? | [
[
{
"aoVal": "A",
"content": "$\\frac{4}{5}$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac{3}{4}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac{2}{3}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\frac{15}{20}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Basic Understanding of Fractions->Using Common Factors to Simplify Fractions"
] | [
"Factors of numerator $15$ are $1, 3, 5, 15$ and factors of denominator $20$ are $1, 2, 4, 5, 10, 20$. The highest common factor(HCF) is $5$. Divide $15$ and $20$ by the HCF: $\\frac{15 \\div 5}{20 \\div 5}$=$\\frac{3}{4}$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5805 | 81d254e2509c48a0a681a9b5306604e7 | [] | 1 | single_choice | Which of the following sequences is different from the rest? | [
[
{
"aoVal": "A",
"content": "$$1$$，$$1$$，$$2$$，$$3$$，$$5$$，$$8$$，$$13$$，$$\\cdots $$ "
}
],
[
{
"aoVal": "B",
"content": "$$0$$，$$2$$，$$2$$，$$4$$，$$6$$，$$10$$，$$16$$，$$\\cdots $$ "
}
],
[
{
"aoVal": "C",
"content": "$$1$$，$$3$$，$$4$$，$$7$$，$$11$$，$$18$$，$$29$$，$$\\cdots $$ "
}
],
[
{
"aoVal": "D",
"content": "$$1$$，$$2$$，$$3$$，$$6$$，$$11$$，$$20$$，$$37$$，$$\\cdots $$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns in Number Sequences"
] | [
"All sequences are Fibonacci numbers except for ($$\\text{D}$$). （Maths Olympiad 《Looking for a Pattern》 Pr$$3$$\\&$$4$$ Question \\#$$18$$） "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5814 | c4a6559641bc486eb2b85af6b89b6639 | [] | 1 | single_choice | What is the product of $286$ and $4$? | [
[
{
"aoVal": "A",
"content": "$$1144$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1124$$ "
}
],
[
{
"aoVal": "C",
"content": "$$844$$ "
}
],
[
{
"aoVal": "D",
"content": "$$824$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"omitted "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5815 | c01f83bac32c42bd9f899292ec53b461 | [] | 1 | single_choice | Let $ a◆b=a+(2\times b)$, then $1◆(2◆3) =$~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$9$$ "
}
],
[
{
"aoVal": "B",
"content": "$$11$$ "
}
],
[
{
"aoVal": "C",
"content": "$$17$$ "
}
],
[
{
"aoVal": "D",
"content": "$$36$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Operating Directly"
] | [
"$1◆(2◆3)=1◆[2+(2\\times3)]=1◆8=1+16=17$. So the answer is $\\rm C$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5832 | db8658d1e3274cf5bc56fe4081b0ddc0 | [] | 1 | single_choice | There is a total of $3567$ dollars in Thompson\textquotesingle s wallet. What is the sum of the digits of his money? (adapted from $$2011$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$7$$) | [
[
{
"aoVal": "A",
"content": "$$356$$ "
}
],
[
{
"aoVal": "B",
"content": "$$57$$ "
}
],
[
{
"aoVal": "C",
"content": "$$22$$ "
}
],
[
{
"aoVal": "D",
"content": "$$21$$ "
}
],
[
{
"aoVal": "E",
"content": "$$24$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Addition of Whole Numbers->Addition in Horizontal Form"
] | [
"$3+5+6+7=21$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5833 | e94521167e2e443a9c0f2d95ea9b4542 | [
"其它"
] | 1 | single_choice | Sreshtha needs to estimate the quantity $\frac{a}{b}-c$, where $a, b$, and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\frac{a}{b}-c ?$ ．(2015 AMC 12A Problem, Question \#5) | [
[
{
"aoVal": "A",
"content": "She rounds all three numbers up. "
}
],
[
{
"aoVal": "B",
"content": "She rounds $a$ and $b$ up, and she rounds $c$ down. "
}
],
[
{
"aoVal": "C",
"content": "She rounds $a$ and $c$ up, and she rounds $b$ down. "
}
],
[
{
"aoVal": "D",
"content": "She rounds $a$ up, and she rounds $b$ and $c$ down. "
}
],
[
{
"aoVal": "E",
"content": "She rounds $c$ up, and she rounds $a$ and $b$ down. "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"To maximize our estimate, we want to maximize $\\frac{a}{b}$ and minimize $c$, because both terms are positive values. Therefore we round $c$ down. To maximize $\\frac{a}{b}$, round $a$ up and $b$ down. $\\Rightarrow(\\mathbf{D})$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5834 | bb9416c5d7984a8ea40c018554e3377c | [
"其它"
] | 1 | single_choice | Let $a, b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$? | [
[
{
"aoVal": "A",
"content": "$$15$$ "
}
],
[
{
"aoVal": "B",
"content": "$$15.5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$16$$ "
}
],
[
{
"aoVal": "D",
"content": "$$16.5$$ "
}
],
[
{
"aoVal": "E",
"content": "$$17$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Unary Quadratic Equations"
] | [
"Expanding the equation and combining like terms results in $2 x^{2}-(a+2 b+c) x+(a b+b c)=0$. By Vieta\\textquotesingle s formula the sum of the roots is $\\frac{-[-(a+2 b+c)]}{2}=\\frac{a+2 b+c}{2}$. To maximize this expression we want $b$ to be the largest, and from there we can assign the next highest values to $a$ and $c$. So let $b=9, a=8$, and $c=7$. Then the answer is $\\frac{8+18+7}{2}=16.5$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5835 | 7d7c5fb8d3024bbcb8a66316217fdce1 | [] | 1 | single_choice | $$10+11+12+13+14=\left(30+31+32+33+34\right)-$$． | [
[
{
"aoVal": "A",
"content": "$$5$$ "
}
],
[
{
"aoVal": "B",
"content": "$$20$$ "
}
],
[
{
"aoVal": "C",
"content": "$$3\\times 20$$ "
}
],
[
{
"aoVal": "D",
"content": "$$5\\times 20$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction "
] | [
"Add five $$20$$\\textquotesingle s to $$10+11+12+13+14$$ to get $$30+31+32+33+34$$. The answer is $$\\text{D}$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5841 | eddcf3ede7774204823315506c7811c4 | [] | 1 | single_choice | Which of the following is the correct order of the fractions $$\dfrac{15}{11}$$, $$\dfrac{19}{15}$$ and $$\dfrac{17}{13}$$, from least to greatest? ($$2019$$ AMC $$8$$ Problem, Question \#$$3$$) | [
[
{
"aoVal": "A",
"content": "$$\\dfrac{15}{11}\\textless{} \\dfrac{17}{13}\\textless{} \\dfrac{19}{15}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\dfrac{15}{11}\\textless{} \\dfrac{19}{15}\\textless{} \\dfrac{17}{13}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\dfrac{17}{13}\\textless{} \\dfrac{19}{15}\\textless{} \\dfrac{15}{11}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\dfrac{19}{15}\\textless{} \\dfrac{15}{11}\\textless{} \\dfrac{17}{13}$$ "
}
],
[
{
"aoVal": "E",
"content": "$$\\dfrac{19}{15}\\textless{} \\dfrac{17}{13}\\textless{} \\dfrac{15}{11}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating->Comparing and Ordering"
] | [
"Method $$1$$: Each one is $$x+\\dfrac{4}{x}$$ so we are really comparing $$\\dfrac{4}{11}$$, $$\\dfrac{4}{15}$$, and $$\\dfrac{4}{13}$$, where you can see $$\\dfrac{4}{11}$$ $$\\textgreater$$ $$\\dfrac{4}{13}$$ $$\\textgreater$$ $$\\dfrac{4}{15}$$. So, $$\\dfrac{19}{15}\\textless{} \\dfrac{17}{13}\\textless{} \\dfrac{15}{11}$$. Method $$2$$: We take a common denominator: $$\\dfrac{15}{11}, \\dfrac{19}{15}, \\dfrac{17}{13}= \\dfrac{15 \\cdot 15 \\cdot 13}{11 \\cdot 15 \\cdot 13}, \\dfrac{19 \\cdot 11 \\cdot 13}{15 \\cdot 11 \\cdot 13}, \\dfrac{17 \\cdot 11 \\cdot 15}{13 \\cdot 11 \\cdot 15}= \\dfrac{2925}{2145}, \\dfrac{2717}{2145}, \\dfrac{2805}{2145}$$. $$2717 \\textless{} 2805\\textless{} 2925$$. So, $$\\dfrac{19}{15}\\textless{} \\dfrac{17}{13}\\textless{} \\dfrac{15}{11}$$. Method $$3$$: When $$\\frac{x}{y}\\textgreater1$$, $$\\frac{x+z}{y+z}\\textless\\frac{x}{y}$$. Hence, $$\\dfrac{19}{15}\\textless{} \\dfrac{17}{13}\\textless{} \\dfrac{15}{11}$$. "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5842 | 976b3d1a1b144a04a900bb2230e6ae0a | [] | 1 | single_choice | $$1+3\frac{1}{6}+5\frac{1}{12}+7\frac{1}{20}+9\frac{1}{30}+11\frac{1}{42}=$$． | [
[
{
"aoVal": "A",
"content": "$$36\\frac{5}{14}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$25\\frac{5}{14}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$36\\frac{1}{3}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$25\\frac{1}{3}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"$$\\begin{eqnarray}\\&\\&1+3\\frac{1}{6}+5\\frac{1}{12}+7\\frac{1}{20}+9\\frac{1}{30}+11\\frac{1}{42}\\textbackslash\\textbackslash{} \\&=\\&1+3+5+7+9+11+\\frac{1}{6}+\\frac{1}{12}+\\frac{1}{20}+\\frac{1}{30}+\\frac{1}{42}\\textbackslash\\textbackslash{} \\&=\\&\\left[ (1+9)+(3+7)+(5+11) \\right]+\\left( \\frac{1}{2}-\\frac{1}{3} \\right)+\\left( \\frac{1}{3}-\\frac{1}{4} \\right)+\\left( \\frac{1}{4}-\\frac{1}{5} \\right)+\\left( \\frac{1}{5}-\\frac{1}{6} \\right)+\\left( \\frac{1}{6}-\\frac{1}{7} \\right)\\textbackslash\\textbackslash{} \\&=\\&36+\\left( \\frac{1}{2}-\\frac{1}{7} \\right)\\textbackslash\\textbackslash{} \\&=\\&36\\frac{5}{14}\\end{eqnarray}$$ $$\\text{A}$$． "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5844 | 9bee9ed41d56456e806f5b644156f880 | [
"其它"
] | 1 | single_choice | For real numbers $w$ and $z$, $$ \frac{\frac{1}{w}+\frac{1}{z}}{\frac{1}{w}-\frac{1}{z}}=2014 $$ What is $\frac{w+z}{w-z} ?$ | [
[
{
"aoVal": "A",
"content": "- 2014 "
}
],
[
{
"aoVal": "B",
"content": "$\\frac{-1}{2014}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac{1}{2014}$ "
}
],
[
{
"aoVal": "D",
"content": "$$1$$ "
}
],
[
{
"aoVal": "E",
"content": "$$2014$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Proportional Equations"
] | [
"Muliply both sides by $\\left(\\frac{1}{w}-\\frac{1}{z}\\right)$ to get $\\frac{1}{w}+\\frac{1}{z}=2014\\left(\\frac{1}{w}-\\frac{1}{z}\\right)$. Then, add $2014 \\cdot \\frac{1}{z}$ to both sides and subtract $\\frac{1}{w}$ from both sides to get $2015 \\cdot \\frac{1}{z}=2013 \\cdot \\frac{1}{w}$. Then, we can plug in the most simple values for $z$ and $w(2015$ and 2013 , respectively), and find $\\frac{2013+2015}{2013-2015}=\\frac{2(2014)}{-2}=-2014$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5847 | c03024067ed4459fb0da177563799922 | [] | 1 | single_choice | In a dining room, there are $$15$$ chairs, $$5$$ tables, and $$20$$ cups. What is the ratio of chairs to cups in the simplest form? | [
[
{
"aoVal": "A",
"content": "$$1:4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$15:20$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4:3$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3:4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions->Ratio"
] | [
"There are $$15$$ chairs and $$20$$ cups. So the ratio of chairs to cups is $$15:20$$. The simplest form is $$3:4$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5848 | 8669095855174076aa52cd6a6d6a6403 | [
"其它"
] | 2 | single_choice | A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^{\text {th }}$-graders to $6^{\text {th }}$-graders is $7: 6$, and the the ratio of $8^{\text {th }}$-graders to $7^{\text {th }}$ graders is $4: 7$. What is the smallest number of students that could be participating in the project? (2013 AMC 8, Question 16) | [
[
{
"aoVal": "A",
"content": "$$21$$ "
}
],
[
{
"aoVal": "B",
"content": "$$47$$ "
}
],
[
{
"aoVal": "C",
"content": "$$56$$ "
}
],
[
{
"aoVal": "D",
"content": "$$94$$ "
}
],
[
{
"aoVal": "E",
"content": "$$100$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions"
] | [
"We multiply the first ratio by 2 on both sides, and the second ratio by 3 to get the same number for 8 th graders, in order that we can put the two ratios together: $$ \\begin{aligned} \\&7: 6=7(2): 6(2)=14: 12 \\textbackslash\\textbackslash{} \\&4: 7=4(3): 7(3)=12: 21\\end{aligned} $$ Therefore, the ratio of 8th graders to 7th graders to 6th graders is $14: 12: 21$. Since the ratio is in lowest terms, the smallest number of students participating in the project is $$ 14+12+21=\\text { (B) } 47 $$ Notice if you get $$28:24:42$$, you have to simplfy to $14: 12: 21$ for the smallest number of students. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5853 | a98c08c7398f41c58acf73281169a666 | [
"其它"
] | 1 | single_choice | There is a tournament at the pool. First, $$13$$ children signed up and then another $$19$$ children signed up. Six teams with an equal number of members each are needed for the tournament. At least how many more children need to sign up so that the six teams can be formed?~ | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$3$$ "
}
],
[
{
"aoVal": "D",
"content": "$$4$$ "
}
],
[
{
"aoVal": "E",
"content": "$$5$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division->Division of Whole Numbers->Division with Remainders"
] | [
"$$13 + 19 = 32$$; $$32 \\div 6 = 5R2$$. There are $$2$$ students left, which means that $$6 - 2 = 4$$ more students are needed to form $6$ teams. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5856 | 820601bdad7a4229a6e69c7bbb945720 | [
"其它"
] | 1 | single_choice | A data set consists of $6$ (not distinct) positive integers: $1,7,5,2,5$, and $X$. The average (arithmetic mean) of the 6 numbers equals a value in the data set. What is the sum of all positive values of $X$? (2022 AMC 10A Problems, Question \#8) | [
[
{
"aoVal": "A",
"content": "$$10$$ "
}
],
[
{
"aoVal": "B",
"content": "$$26$$ "
}
],
[
{
"aoVal": "C",
"content": "$$32$$ "
}
],
[
{
"aoVal": "D",
"content": "$$36$$ "
}
],
[
{
"aoVal": "E",
"content": "$$40$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"First, note that $1+7+5+2+5=20$. The mean cannot be $1$ or $2$. There are $3$ possible cases: Case 1: the mean is $5$ . $$ X=5 \\cdot 6-20=10 \\text {. } $$ Case 2: the mean is $7$. $$ X=7 \\cdot 6-20=22 \\text {. } $$ Case 3: the mean is $X$. $$ \\frac{20+X}{6}=X \\Rightarrow X=4 \\text {. } $$ Hence, the answer is $10+22+4=$ (D) $36$ . "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5859 | bba22136ee2e41d497870149fd53535d | [] | 1 | single_choice | The average value of ＄$$2$$, ＄$$4$$, ＄$$6$$, ＄$$8$$, and ＄$$10$$ is~\uline{~~~~~~~~~~}~pennies. | [
[
{
"aoVal": "A",
"content": "$$3000$$ "
}
],
[
{
"aoVal": "B",
"content": "$$600$$ "
}
],
[
{
"aoVal": "C",
"content": "$$550$$ "
}
],
[
{
"aoVal": "D",
"content": "$$500$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Counting Modules->Statistics and Probability"
] | [
"Average $$=$$(＄$$2+$$＄$$4+$$＄$$6+$$＄$$8+$$＄$$10$$)$$\\div 5=$$＄$$6=600$$ pennies. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5863 | 9305e091cd7a464797336270905727df | [
"其它"
] | 1 | single_choice | Which of the following number is different from the others． | [
[
{
"aoVal": "A",
"content": "$$0.08$$ "
}
],
[
{
"aoVal": "B",
"content": "$$8\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{8}{10}$$ "
}
],
[
{
"aoVal": "D",
"content": "Eight percents "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Basic Understanding of Fractions->Converting Between Fractions, Percentage and Decimals"
] | [
"$$8\\textbackslash\\%=0.08$$， But $$\\frac{8}{10}=0.8$$． So, the answer is $$\\text{C}$$． "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5866 | c4c0ccc675fa45f08fb4b1cb6eb30735 | [
"其它"
] | 1 | single_choice | An operator $$\star$$ acts on two numbers to give the following outcomes: $$3\star2=51$$ $$5\star3=82$$ $$6\star1=75$$ $$9\star4=135$$ What is $$7\star5$$ equal to? | [
[
{
"aoVal": "A",
"content": "$$112$$ "
}
],
[
{
"aoVal": "B",
"content": "$$121$$ "
}
],
[
{
"aoVal": "C",
"content": "$$122$$ "
}
],
[
{
"aoVal": "D",
"content": "$$212$$ "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition"
] | [
"$$a\\star b=\\underbrace{(a+b)}\\_{\\rm{tens}} \\underbrace{(a-b)}\\_{\\rm{ones}}$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5869 | c03d4f5effc148258bd033ce97615339 | [
"其它"
] | 0 | single_choice | What value does the hundredth place represent in the number $12.345$? | [
[
{
"aoVal": "A",
"content": "$$0.005$$ "
}
],
[
{
"aoVal": "B",
"content": "$$0.04$$ "
}
],
[
{
"aoVal": "C",
"content": "$$0.3$$ "
}
],
[
{
"aoVal": "D",
"content": "$$2$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals"
] | [
"The hundredth place of a number is the second digit after the decimal point, which is $$4$$. Since the number $$4$$ is on the hundredth place, it represents $$0.04$$. Check Lesson 4 Concept 1 on textbook "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5877 | c0438245cc614aa0b3a08a58ac9b2025 | [
"其它"
] | 2 | single_choice | \textbf{Suppose a certain scale is not calibrated correctly, and as a result, the mass of any object is displayed as 0.75 kilogram less than its actual mass. What is the correlation between the actual masses of a set of objects and the respective masses of the same set of objects displayed by the scale?} | [
[
{
"aoVal": "A",
"content": "$$-1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$-0.75$$ "
}
],
[
{
"aoVal": "C",
"content": "$$0$$ "
}
],
[
{
"aoVal": "D",
"content": "$$0.75$$ "
}
],
[
{
"aoVal": "E",
"content": "$$1$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"\\textbf{Since the actual mass is always 0.75kg more than the displayed mass, they are perfectly correlated.~} "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5879 | 931614bbd6f647378d43fdd495c4e8dd | [
"其它"
] | 2 | single_choice | \textbf{High school students from track teams in the state participated in a training program to improve running times. Before the training, the mean running time for the students to run a mile was 402 seconds with standard deviation 40 seconds. After completing the program, the mean running time for the students to run a mile was 368 seconds with standard deviation 30 seconds. Let X represent the running time of a randomly selected student before training, and let Y represent the running time of the same student after training. Which of the following is true about the distribution of X - Y ?} | [
[
{
"aoVal": "A",
"content": "\\textbf{The variables X and Y are independent; therefore, the mean is 34 seconds and the standard deviation is 10 seconds.} "
}
],
[
{
"aoVal": "B",
"content": "\\textbf{~The variables X and Y are independent; therefore, the mean is 34 seconds and the standard deviation is 50 seconds.} "
}
],
[
{
"aoVal": "C",
"content": "\\textbf{The variables X and Y are not independent; therefore, the standard deviation is 50 seconds and the mean cannot be determined with the information given.} "
}
],
[
{
"aoVal": "D",
"content": "\\textbf{The variables X and Y are not independent; therefore, the mean is 34 seconds and the standard deviation cannot be determined with the information given.} "
}
],
[
{
"aoVal": "E",
"content": "\\textbf{The variables X and Y are not independent; therefore, neither the mean nor the standard deviation can be determined with the information given.} "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"\\textbf{Mean: 402 - 368 = 34} \\textbf{SD: X and Y are not independent, correlation is unknown → SD unknown} "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5882 | b2a3c9afc7a3409ea6fdceeb0d342cc5 | [
"其它"
] | 1 | single_choice | There is a number $X.$ Three times the sum of $4$ and number $X$ is $36$. What is the value of number $X$? | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
],
[
{
"aoVal": "E",
"content": "$$10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation"
] | [
"$3(4+x)=36, x=8$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5886 | 97923c13be2345a7bc3a4a76bf73fc55 | [
"其它"
] | 1 | single_choice | insert pic Which number is hidden behind the square? | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4$$ "
}
],
[
{
"aoVal": "D",
"content": "$$5$$ "
}
],
[
{
"aoVal": "E",
"content": "$$6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Finding Patterns->Encryption and Decryption"
] | [
"triangle: 7-4=3 square: 9-3=6 "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5891 | fbcb884a3c254dc1a780ac7ece18fea3 | [] | 1 | single_choice | $$\left( {3+2+1} \right)\times 10=30+20+$$~\uline{~~~~~~~~~~}~ | [
[
{
"aoVal": "A",
"content": "$$10$$ "
}
],
[
{
"aoVal": "B",
"content": "$$11$$ "
}
],
[
{
"aoVal": "C",
"content": "$$33$$ "
}
],
[
{
"aoVal": "D",
"content": "$$44$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Distributive Law of Whole Numbers"
] | [
"$$\\left( {3+2+1} \\right)\\times 10=60=30+20+10$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5893 | a098d006651a45ff941c9cbd6f4663a2 | [
"其它"
] | 2 | single_choice | Vivian creates a new operation: $m$@$n=m\times m-n\times n$. What is the value of $11$@$9$? | [
[
{
"aoVal": "A",
"content": "$$99$$ "
}
],
[
{
"aoVal": "B",
"content": "$$121$$ "
}
],
[
{
"aoVal": "C",
"content": "$$81$$ "
}
],
[
{
"aoVal": "D",
"content": "$$40$$ "
}
],
[
{
"aoVal": "E",
"content": "$$30$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition"
] | [
"$11^{2}-9^{2}=40$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5894 | 97967926c768457d8d2fda04f71b43aa | [
"其它"
] | 1 | single_choice | How many two-digits positive integers has at least one $8$ as its digit? | [
[
{
"aoVal": "A",
"content": "$$9$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$11$$ "
}
],
[
{
"aoVal": "D",
"content": "$$18$$ "
}
],
[
{
"aoVal": "E",
"content": "$$19$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"D "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5897 | 86983524dd3c40168f4432922b147c6c | [
"其它"
] | 1 | single_choice | When Koko and Koala does not sleep, he eats 50 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": "$$50$$ "
}
],
[
{
"aoVal": "C",
"content": "$$100$$ "
}
],
[
{
"aoVal": "D",
"content": "$$200$$ "
}
],
[
{
"aoVal": "E",
"content": "$$400$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Unit Conversion"
] | [
"(24 - 20) x 5 = 200. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5904 | a9ac7f3b0b0d4e3ab730b9d73120a87e | [] | 1 | single_choice | Calculate：$$\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}=$$． | [
[
{
"aoVal": "A",
"content": "$$\\frac{4}{5}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{5}{6}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{11}{12}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{19}{20}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"$$\\textasciitilde\\textasciitilde\\textasciitilde\\textasciitilde\\frac{1}{2}+\\frac{1}{6}+\\frac{1}{12}+\\frac{1}{20}$$ $$=\\frac{30}{60}+\\frac{10}{60}+\\frac{5}{60}+\\frac{3}{60}$$ $$=\\frac{48}{60}$$ $$=\\frac{4}{5}$$． "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5905 | f731a9e56db84d72b7d84506bd3d61fb | [
"其它"
] | 1 | single_choice | Two non-zero real numbers, $a$ and $b$, satisfy $a b=a-b$. Which of the following is a possible value of $\frac{a}{b}+\frac{b}{a}-a b$? (2000 AMC 12 Problems, Question \#11) | [
[
{
"aoVal": "A",
"content": "$-2$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac{-1}{2}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac{1}{3}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\frac{1}{2}$ "
}
],
[
{
"aoVal": "E",
"content": "$$2$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$$\\frac{a}{b}+\\frac{b}{a}-a b=\\frac{a^{2}+b^{2}}{a b}-(a-b)=\\frac{a^{2}+b^{2}}{a-b}-\\frac{(a-b)^{2}}{(a-b)}=\\frac{2 a}{a-b}=\\frac{2(a-b)}{a-b}=2 \\Rightarrow \\text{E}.$$ "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5923 | ffbda26415564c15978cf0e993f0c3c3 | [] | 1 | single_choice | Which of these is closest to $$4$$? | [
[
{
"aoVal": "A",
"content": "$$4.07 $$ "
}
],
[
{
"aoVal": "B",
"content": "$$ 3.925 $$ "
}
],
[
{
"aoVal": "C",
"content": "$$3.979 $$ "
}
],
[
{
"aoVal": "D",
"content": "$$4.12 $$ "
}
],
[
{
"aoVal": "E",
"content": "$$4.024$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating->Comparing and Ordering"
] | [
"Note that $$4.07-4 = 0.07$$; $$4-3.925 = 0.075$$; $$4-3.979 =0.021$$; $$4.12-4 = 0.12$$ and $$4.024-4 = 0.024$$. So, of the five options, $$3.979$$ is nearest to $$4$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5927 | fbd7a8f1a94d491aaead04db07f4c11c | [
"其它"
] | 2 | single_choice | What is the unit digit of $23^{2023}$? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
],
[
{
"aoVal": "E",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"D "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5928 | 8ec0c46149f64ad69d70a63e9a712178 | [
"其它"
] | 2 | single_choice | Nancy bought 17 cones of ice-cream for her three children. Misha ate twice as many cones as Ana. Dan ate more ice-cream than Ana but less than Misha. How many cones of ice-cream did Dan eat? | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
],
[
{
"aoVal": "E",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities->Solving Inequalities"
] | [
"Let x be the number of ice-cream cones that Ana ate. Then, 2x is the number of cones that Misha ate, and the number of cones that Dan ate is: 17 - x - 2x = 17 - 3x. It is given that Dan ate more than Misha but less than Ana; therefore, x \\textless{} 17 - 3x \\textless{} 2x 4x \\textless{} 17 \\textless{} 5x only x=4 satisfies this inequality. The number of cones that Dan ate is thus 17- 3 x 4 = 5. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5933 | 933c936488a14a869d9d0df5c3cf1f62 | [
"其它"
] | 2 | single_choice | For $\triangle ABC$, all its side lengths are integer. The primeter of $\triangle ABC$ with a side of length $14$ and a side length of $8$ is at least ． | [
[
{
"aoVal": "A",
"content": "$$25$$ "
}
],
[
{
"aoVal": "B",
"content": "$$26$$ "
}
],
[
{
"aoVal": "C",
"content": "$$27$$ "
}
],
[
{
"aoVal": "D",
"content": "$$28$$ "
}
],
[
{
"aoVal": "E",
"content": "$$29$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities"
] | [
"We know from the triangle inequality that the last side, $s$, fulfills $s+8\\textgreater14$. Therefore, $P\\textgreater14+14$. The least integer value of $P$ is $29$ "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5938 | 8ec66856efb5485296a8605b07abb64a | [
"其它"
] | 1 | single_choice | Gilda has a bag of marbles. She gives $30 \textbackslash\%$ of them to her friend Pedro. Then Gilda gives $25 \textbackslash\%$ of what is left to another friend, Ebony. Finally, Gilda gives $10 \textbackslash\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself? (adapted 2019 AMC 8, Question \#8) | [
[
{
"aoVal": "A",
"content": "$$47.25\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "B",
"content": "$50.25\\textbackslash\\%$ "
}
],
[
{
"aoVal": "C",
"content": "$$65\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "D",
"content": "$$70\\textbackslash\\%$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Percentage Calculation"
] | [
"After Gilda gives $30 \\textbackslash\\%$ of the marbles to Pedro, she has $70 \\textbackslash\\%$ of the marbles left. If she then gives $25 \\textbackslash\\%$ of what\\textquotesingle s left to Ebony, she has $(0.75 * 0.70)=52.5 \\textbackslash\\%$ of what she had at the beginning. Finally, she gives $10 \\textbackslash\\%$ of what\\textquotesingle s left to her brother, so she has $(0.9 * 0.525) =$ (A) $47.25\\textbackslash\\%$ of what she had in the beginning left. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5940 | cdfc906b992b4e32909143593c16003c | [
"其它"
] | 1 | single_choice | If $$a=2$$, $$b=4$$, $${{a}^{2}}+3b=$$． | [
[
{
"aoVal": "A",
"content": "$$8$$ "
}
],
[
{
"aoVal": "B",
"content": "$$9$$ "
}
],
[
{
"aoVal": "C",
"content": "$$12$$ "
}
],
[
{
"aoVal": "D",
"content": "$$14$$ "
}
],
[
{
"aoVal": "E",
"content": "$$16$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"$$3^{2}+2\\times1.5=12$$． "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5950 | 86c964220e704651a6a96c84896574e0 | [
"其它"
] | 2 | single_choice | If $$\left\textbar{} 2x-5 \right\textbar=7$$, $x=$~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$-1$$ or $$6$$ "
}
],
[
{
"aoVal": "C",
"content": "$$1$$ or $$-6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$1$$ or $$6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities->Solving Inequalities"
] | [
"$$2x-5=\\pm 7$$ $$x=6$$ or $$x=-1$$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5951 | f2a03f7cfe02484b9790140f0ff89d06 | [
"其它"
] | 1 | single_choice | A circle is centered at $O$, $AB$ is a diameter and $C$ is a point on the circle with $\angle COB=50^{\circ}$. What is the degree measure of $\angle CAB$? | [
[
{
"aoVal": "A",
"content": "$$20$$ "
}
],
[
{
"aoVal": "B",
"content": "$$25$$ "
}
],
[
{
"aoVal": "C",
"content": "$$45$$ "
}
],
[
{
"aoVal": "D",
"content": "$$50$$ "
}
],
[
{
"aoVal": "E",
"content": "$$65$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"B "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5954 | e040aefe934845ce8e79ec8dc8c79414 | [
"其它"
] | 2 | single_choice | For how many values of $a$ is it true that the line $y=x+a^{2}-6$ passes through the vertex of the parabola $y=4x^{2}-8x+a^{2}$? (Adapted From 2005 AMC 12B Problem, Question \#8) | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2$$ "
}
],
[
{
"aoVal": "D",
"content": "$$10$$ "
}
],
[
{
"aoVal": "E",
"content": "infinitely many "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"We see that the vertex of the quadratic function $y=4x^{2}-8x+a^{2}$ is $\\left(1, a^{2}-4\\right)$. If $\\left(1, a^{2}-4\\right)$ will be on the line $y=x+a^{2}-6$, $a^{2} -4=1+a^{2}-6$. Solve for $a$, there is no solution. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5961 | a9d2c15a7f7a4b8d8279195b6dd1492e | [
"其它"
] | 1 | single_choice | Tiara\textquotesingle s toll pass account has a value of $$$24$$. Each time she uses the toll road, $$$1.4$$ is deducted from the account. When the value drops below $$$10$$, she must add value to the toll pass. Assume there are $$x$$ times Tiara can use the toll road without having to add value to the toll pass. Which inequality can represent the situation? | [
[
{
"aoVal": "A",
"content": "$24-1.4x\\geq10$ "
}
],
[
{
"aoVal": "B",
"content": "$24-1.4x\\textgreater10$ "
}
],
[
{
"aoVal": "C",
"content": "$24+1.4x\\geq10$ "
}
],
[
{
"aoVal": "D",
"content": "$24-1.4x\\leq10$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities"
] | [
"$24-1.4x\\geq10$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5965 | e0486418bf7d44f3b2e83a41aef2dbf2 | [
"其它"
] | 2 | single_choice | How many $0$s are there in the end of the result of the factorial of $25$? | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
],
[
{
"aoVal": "E",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations through Formulas->Factorization"
] | [
"We can get $1+1+1+1+2=6$ factor $5$s from $5, 10, 15, 20, 25$ in total, which can make six $10$s after multiplying $2$. Six $10$s means that there should be $6$ $0$s in the end. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5966 | ae5616138e414d46ad65462e58a02b0d | [
"其它"
] | 0 | single_choice | From a starting number, Olivia was supposed to subtract $3$, and then divide by $9$, but instead, Olivia subtracted $9$, then divided by $3$, getting $43$. If the correct instructions were followed, what would the result be? | [
[
{
"aoVal": "A",
"content": "$$15$$ "
}
],
[
{
"aoVal": "B",
"content": "$$34$$ "
}
],
[
{
"aoVal": "C",
"content": "$$43$$ "
}
],
[
{
"aoVal": "D",
"content": "$$51$$ "
}
],
[
{
"aoVal": "E",
"content": "$$138$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"A "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5977 | b7621c0ba71c42c3a75a5239a7c542c9 | [] | 1 | single_choice | What is the product of $$5$$ and $$830$$? | [
[
{
"aoVal": "A",
"content": "$$166$$ "
}
],
[
{
"aoVal": "B",
"content": "$$825$$ "
}
],
[
{
"aoVal": "C",
"content": "$$835$$ "
}
],
[
{
"aoVal": "D",
"content": "$$4150$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"omitted "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5979 | ae5f03bb32424af0a391e9574cd0d4e4 | [
"其它"
] | 1 | single_choice | In how many ways can the letters in \textbf{BEEKEEPER} be rearranged so that two or more \textbf{E}s do not appear together? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$12$$ "
}
],
[
{
"aoVal": "D",
"content": "$$24$$ "
}
],
[
{
"aoVal": "E",
"content": "$$120$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"NA "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5983 | 9c624e6926624477bfc6e5dbd9c275d1 | [
"其它"
] | 1 | single_choice | There is some water in a bowl, which is on a electronic scales. Mike pours water into the bowl. After pouring $2$ glasses of water, the sacle shows $300$ grams. After pouring a total of $6$ glasses of water, the scale shows $700$ grams. What is the quality of the water at the start? | [
[
{
"aoVal": "A",
"content": "$$100$$ g "
}
],
[
{
"aoVal": "B",
"content": "$$150$$ g "
}
],
[
{
"aoVal": "C",
"content": "$$200$$ g "
}
],
[
{
"aoVal": "D",
"content": "$$250$$ g "
}
],
[
{
"aoVal": "E",
"content": "$$275$$ g "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"The quality of each glass of water: $(700 - 300) \\div (6 - 2) = 100$ g At first: $300 - 100 \\times 2 = 100$ g "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5987 | 9c64b65e33764d92b463680bc6f57fce | [] | 1 | single_choice | $\left (876\times543\right ) -\left (543\times876\right )=$． | [
[
{
"aoVal": "A",
"content": "$0$ "
}
],
[
{
"aoVal": "B",
"content": "$1$ "
}
],
[
{
"aoVal": "C",
"content": "$237834$ "
}
],
[
{
"aoVal": "D",
"content": "$475668$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations"
] | [
"$\\left (876\\times543\\right ) - \\left (543\\times876\\right ) = \\left (876\\times543\\right ) - (876\\times543) = 0$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5989 | f752a5714d1849c1aea67db0ee0736f6 | [] | 1 | single_choice | $ 0.33 =$． | [
[
{
"aoVal": "A",
"content": "$$\\frac{3}{10}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{33}{100}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{1}{3}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{3}{8}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals"
] | [
"$0.33 = 33\\div100 = \\dfrac{33}{100}$, so choice $\\text{B}$ is correct. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5990 | fbf096c3bbba4cf699e239e70ae477c0 | [
"其它"
] | 1 | single_choice | $$26+24-22+20-18+16-14+12-10+8-6+4-2=$$？． | [
[
{
"aoVal": "A",
"content": "$$48$$ "
}
],
[
{
"aoVal": "B",
"content": "$$38$$ "
}
],
[
{
"aoVal": "C",
"content": "$$36$$ "
}
],
[
{
"aoVal": "D",
"content": "$$10$$ "
}
],
[
{
"aoVal": "E",
"content": "$$0$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Addition of Whole Numbers"
] | [
"$$=26+(24-22)+(20-18)+(16-14)+(12-10)+(8-6)+(4-2)$$ $$=26+2+2+2+2+2+2$$ $$=38$$． "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 5992 | dbcf0a189a0c4d04ad5e92bc7ff8610b | [
"其它"
] | 1 | single_choice | Ace and Sam were painting a house. Ace painted $$\frac{3}{10}$$ of the house. Sam painted $$\frac{2}{10}$$ more than Ace. What fraction of the house still needs to be painted? | [
[
{
"aoVal": "A",
"content": "$$\\frac{2}{10}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{3}{10}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{5}{10}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{8}{10}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"Sam: $$\\frac{2}{10}+\\frac{3}{10}=\\frac{5}{10}$$ Total: $$\\frac{5}{10}+\\frac{3}{10}=\\frac{8}{10}$$ Left: $$\\frac{10}{10}-\\frac{8}{10}=\\frac{2}{10}$$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6005 | ffddeae52b284e15bab7cfc255bf60fd | [] | 1 | single_choice | Which of the following is not an expression? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$a$$ "
}
],
[
{
"aoVal": "C",
"content": "$$a+b=a+b$$ "
}
],
[
{
"aoVal": "D",
"content": "$$b-3$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"Equations are not expressions. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6007 | ce24b20353324308beb9c5816e64e785 | [] | 1 | single_choice | Which of these pairs of fractions are equivalent? | [
[
{
"aoVal": "A",
"content": "$$3\\frac{1}{2}$$ and $$2\\frac{1}{4}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1\\frac{1}{3}$$ and $$\\frac{8}{6}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$1\\frac{1}{4}$$ and $$\\frac{3}{4}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3\\frac{3}{9}$$ and $$\\frac{9}{9}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Operations of Fractions"
] | [
"$\\frac{8}{6}=\\frac{4}{3}=1\\frac{1}{3}$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6016 | a57ddb37d6ed43c5870c0eaf89f138b4 | [
"其它"
] | 0 | single_choice | Cassandra is helping her mother to pack $$75$$ cupcakes. The boxes that her mother prepare can only fit $$7$$ cupcakes. She must ensure the box is full before she can use the next box. How many boxes she can fill up? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10$$ "
}
],
[
{
"aoVal": "D",
"content": "$$68$$ "
}
],
[
{
"aoVal": "E",
"content": "$$70$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division->Division of Whole Numbers"
] | [
"$$75\\div7=10R5$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6017 | a57df383beac40338b217b1165af8c31 | [
"其它"
] | 1 | single_choice | Judy has less than $40$ game coins. The game coins can be divided evenly between $3, 4$ or $6$ students. However, they cannot be divided evenly between $7$ students because $4$ more game coins would be needed. How many game coins does Judy have? | [
[
{
"aoVal": "A",
"content": "$$12$$ "
}
],
[
{
"aoVal": "B",
"content": "$$18$$ "
}
],
[
{
"aoVal": "C",
"content": "$$48$$ "
}
],
[
{
"aoVal": "D",
"content": "$$24$$ "
}
],
[
{
"aoVal": "E",
"content": "$$35$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"The game coins can be divided evenly between $3, 4$ or $6$ students, which means the number of game coins should be divisible by $3, 4,$ or $6$, so we can eliminate the options $E$ . The game coins cannot be divided evenly between $7$ students because $4$ more game coins would be needed, so after the number of game coins plus $4$, the result can be divisible by $7$. Thus, the answer is $24$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6018 | 93995029076142a185c7e58b04684ffb | [
"其它"
] | 1 | single_choice | Calculate $$0.3\dot{8}\div 0.\dot{5}1\dot{8}$$ | [
[
{
"aoVal": "A",
"content": "$$\\frac12$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac23$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac34$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac45$$ "
}
],
[
{
"aoVal": "E",
"content": "$$\\frac56$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"C "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6030 | 8f2f2fce4e684914b6babbb9c5fc8cb6 | [
"其它"
] | 2 | single_choice | How many odd numbers are there? 5, 12, 3, 6, 7, 9, 10, 4 | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers"
] | [
"$$Omitted.$$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6031 | a58ad6ba591241df99dd6133a72663a2 | [
"其它"
] | 1 | single_choice | Which of the followings is not an algebraic expression? | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$ax+32$$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac{2}{y}$+88 "
}
],
[
{
"aoVal": "D",
"content": "$$x=33y$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"$D$ is an equation "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6032 | 8f32dec7ba814dbe96c5aa84535509aa | [
"其它"
] | 2 | single_choice | For $\triangle ABC$, all its side lengths are integers. The perimeter of $\triangle ABC$ with a side of length $25$ and a side length of $18$ is at least ． | [
[
{
"aoVal": "A",
"content": "$$25$$ "
}
],
[
{
"aoVal": "B",
"content": "$$18$$ "
}
],
[
{
"aoVal": "C",
"content": "$$52$$ "
}
],
[
{
"aoVal": "D",
"content": "$$51$$ "
}
],
[
{
"aoVal": "E",
"content": "$$50$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities"
] | [
"We know from the triangle inequality that the last side, $s$, fulfills $s+18\\textgreater25$. Therefore, $P\\textgreater25+25$. The least integer value of $P$ is $51$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6035 | a5906eaa101a4069a4b553bcdf768e6b | [
"其它"
] | 3 | single_choice | Alina writes the numbers $1,2,\cdots ,9$ on a separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done? | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3$$ "
}
],
[
{
"aoVal": "E",
"content": "$$4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"C "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6037 | dbe86917a4d24e928d025f19e56f0202 | [] | 1 | single_choice | Find the least dividend and divisor in each division equation below. $\textasciitilde$ ~\uline{~~~~~~~~~~}~$$\div$$5$$=4R4$$ $\textasciitilde$ $$22\div$$~\uline{~~~~~~~~~~}~$$=7R1$$ | [
[
{
"aoVal": "A",
"content": "24,3 "
}
],
[
{
"aoVal": "B",
"content": "23,3 "
}
],
[
{
"aoVal": "C",
"content": "24,4 "
}
],
[
{
"aoVal": "D",
"content": "23,5 "
}
],
[
{
"aoVal": "E",
"content": "$$23.4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division->Division of Whole Numbers->Division with Remainders"
] | [
"The remainder should be less than the divisor. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6040 | ae95adebc9624fe99f9304387342c0c6 | [
"其它"
] | 1 | single_choice | At equilibrium in the market of apartments, the consumer surplus is $\textbackslash$ 120,000$ and the producer surplus is $\textbackslash$ 180,000$. The government decide to set a price ceiling. As a result, the consumer surplus becomes $ \textbackslash$ 140,000$ and the producer surplus becomes $\textbackslash$ 140,000$. What is the deadweight loss from the price ceiling? | [
[
{
"aoVal": "A",
"content": "$\\textbackslash$140,000$ "
}
],
[
{
"aoVal": "B",
"content": "$\\textbackslash$60,000$ "
}
],
[
{
"aoVal": "C",
"content": "$\\textbackslash$40,000$ "
}
],
[
{
"aoVal": "D",
"content": "$\\textbackslash$20,000$ "
}
],
[
{
"aoVal": "E",
"content": "$\\textbackslash$0$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"At the equilibrium, the total surplus is the sum of consumer and producer surplus, which is $ \\textbackslash$120,000 + \\textbackslash$180,000 = \\textbackslash$300,000$. After the price ceiling is imposed, the consumer surplus increases to $\\textbackslash$140,000$, while the producer surplus decreases to $\\textbackslash$140,000$. The new total surplus is $ \\textbackslash$140,000 + \\textbackslash$140,000 = \\textbackslash$280,000$. Therefore, the deadweight loss is the difference between the total surplus before and after the price ceiling, which is $\\textbackslash$300,000 - \\textbackslash$280,000 = \\textbackslash$20,000$ . "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6043 | e074ac0e89a5481f8502879b30e3db43 | [] | 1 | single_choice | $ 0.33 =$． | [
[
{
"aoVal": "A",
"content": "$$\\frac{3}{10}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{33}{100}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{1}{3}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{3}{8}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals"
] | [
"$0.33 = 33\\div100 = \\dfrac{33}{100}$, so choice $\\text{B}$ is correct. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6064 | d762e7cbc3404964bea5f74cd4c3c300 | [
"其它"
] | 1 | single_choice | Simplify: $$b\left( a+1 \right)-a\left( b-1 \right)$$． | [
[
{
"aoVal": "A",
"content": "$$b+a$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2ab+b-a$$ "
}
],
[
{
"aoVal": "C",
"content": "$$b-a$$ "
}
],
[
{
"aoVal": "D",
"content": "$$ab-1$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"$$b\\left( a+1 \\right)-a\\left( b-1 \\right)=ab+b-ab+a=b+a$$, so the answer is$\\text{A}$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6066 | 983a7dbf7dc04c58ac9c2e7a2774641d | [
"其它"
] | 1 | single_choice | Stefani is ordering fertilizer for her garden. A local garden supply store sells fertilizer by the yard (that really means cubic yard, but that is what they call it).For every yard she orders there is a $\textbackslash$$15charge and ,in addition ,there is a $\textbackslash$$40 delivery charge. ( Total Cost =~$\textbackslash$$15(yard)+~$\textbackslash$$40.) $\textasciitilde$ If the average order is 4.3 yards with standard deviation of 2.9 yards, what is the mean and standard deviation of the Total Cost of a typical order? | [
[
{
"aoVal": "A",
"content": "mean = $\\textbackslash$$4.30; standard deviation =~$\\textbackslash$$2.90 "
}
],
[
{
"aoVal": "B",
"content": "mean = $\\textbackslash$$64.50; standard deviation = $\\textbackslash$$43.50 "
}
],
[
{
"aoVal": "C",
"content": "mean = $\\textbackslash$$105.50; standard deviation =~$\\textbackslash$$43.50 "
}
],
[
{
"aoVal": "D",
"content": "mean = $\\textbackslash$$105.50; standard deviation = $\\textbackslash$$83.50 "
}
],
[
{
"aoVal": "E",
"content": "Cannot be determined because each order may not be independent~ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"The correct answer is (c). When using a linear transformation (Total cost 15 (yard)+40), the mean is changed by both the multiplication and addition. But the standard deviation is only changed by the multiplication. Mean = 15(4.3) +40=$\\textbackslash$$ 104.50; =15(2.9) =~$\\textbackslash$$43.50. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6067 | b7a593d9d8e64821a5647aded84bfe94 | [] | 1 | single_choice | $$40 \times 50 \times 60 = 4 \times 5\times 6 \times$$． | [
[
{
"aoVal": "A",
"content": "$$10$$ "
}
],
[
{
"aoVal": "B",
"content": "$$100$$ "
}
],
[
{
"aoVal": "C",
"content": "$$456$$ "
}
],
[
{
"aoVal": "D",
"content": "$$1000$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"$$4\\times10\\times5\\times10\\times6\\times10=4\\times5\\times6\\times10\\times10\\times10=4\\times5\\times6\\times1000$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6068 | aeab27cc64c148aabc871312158023fd | [] | 1 | single_choice | The average of $$4000$$ fours is four times the average of $$2000$$． | [
[
{
"aoVal": "A",
"content": "ones "
}
],
[
{
"aoVal": "B",
"content": "twos "
}
],
[
{
"aoVal": "C",
"content": "fours "
}
],
[
{
"aoVal": "D",
"content": "eights "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"The average of $$4000$$ fours is $$4$$; this is $$4$$ times the average of $$2000$$ ones. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 6078 | dbff2710027845f39384ecf984724aea | [] | 1 | single_choice | Evaluate $$\frac{1}{2+\dfrac{1}{2+\dfrac{1}{2}}}$$ | [
[
{
"aoVal": "A",
"content": "$\\dfrac{1}{3}$ "
}
],
[
{
"aoVal": "B",
"content": "$\\dfrac{2}{5}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\dfrac{2}{9}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\dfrac{5}{12}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Complex Fractions"
] | [
"$$\\frac{1}{2+\\dfrac{1}{2+\\dfrac{1}{2}}}=\\frac{1}{2+ \\dfrac{1}{ \\dfrac{5}{2}}}= \\frac{1}{2+ \\dfrac{2}{5}}= \\frac{1}{ \\dfrac{12}{5}}= \\frac{5}{12}$$. "
] | D |
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