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 | 3441 | 1e31fddae4604de7993ca6651aaf83a6 | [
"其它"
] | 1 | single_choice | If $A:B=3:5$, $B:C=3:2$, find $A:B:C$=~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$3:5:4$ "
}
],
[
{
"aoVal": "B",
"content": "$9:15:10$ "
}
],
[
{
"aoVal": "C",
"content": "$9:3:10$ "
}
],
[
{
"aoVal": "D",
"content": "$8:15:10$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions"
] | [
"$A:B=3:5$ $B:C=3:2$ $A:B:C=9:15:10$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3445 | 2b6382d9d4a4469e9ed9f8b673329bda | [] | 1 | single_choice | $$0.1\times 0.2\times 0.3=$$．($1979-1980$ Math League.com contest problem, $7$\textsuperscript{th~}Grade, Question \#$12$) | [
[
{
"aoVal": "A",
"content": "$$0.0006$$ "
}
],
[
{
"aoVal": "B",
"content": "$$0.006$$ "
}
],
[
{
"aoVal": "C",
"content": "$$0.06$$ "
}
],
[
{
"aoVal": "D",
"content": "$$0.6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Multiplication and Division of Decimals"
] | [
"$$0.1\\times 0.2\\times 0.3=(0.1\\times 0.2)\\times 0.3=0.02\\times 0.3=0.006$$. Therefore, the answer is $$\\rm B$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3446 | 624635a251b54314a8925a0ba454fb8b | [] | 1 | single_choice | If $$5$$ plates weigh as much as $$9$$ mugs, then $$45$$ mugs weigh as much as~\uline{~~~~~~~~~~}~plates. | [
[
{
"aoVal": "A",
"content": "$$18$$ "
}
],
[
{
"aoVal": "B",
"content": "$$25$$ "
}
],
[
{
"aoVal": "C",
"content": "$$72$$ "
}
],
[
{
"aoVal": "D",
"content": "$$81$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"Since $$9$$ mugs $$=5$$ plates $$45\\div9=5$$ times of the above equation. $$45$$ mugs $$=25$$ plates "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3454 | 1e3c3accf2c04b6598fb9a14a393487b | [] | 1 | single_choice | Find the missing number: $$512\times2 = 32\times $$． | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$8$$ "
}
],
[
{
"aoVal": "C",
"content": "$$16$$ "
}
],
[
{
"aoVal": "D",
"content": "$$32$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"$$512\\times2=1024=32\\times32$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3459 | 127cb31e10214df495a74adbe7416b58 | [] | 1 | single_choice | Calculate: $$\left( \frac{5}{8}+\frac{1}{17} \right)\times 8+\frac{9}{17}=$$． | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$$\\left( \\frac{5}{8}+\\frac{1}{17} \\right)\\times 8+\\frac{9}{17}$$ $$=\\frac{5}{8}\\times 8+\\frac{1}{17}\\times 8+\\frac{9}{17}$$ $$=5+\\frac{8}{17}+\\frac{9}{17}$$ $$=5+1$$ $$=6$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3460 | 1e3f0fa891f045519e1133da38909027 | [
"其它"
] | 1 | single_choice | A container had $27$ ℓ of longan drink. The drink is made up of three $2$-ℓ\textbf{~}bottles of longan syrup and some water. What is the volume of water used to make the drink? | [
[
{
"aoVal": "A",
"content": "6ℓ "
}
],
[
{
"aoVal": "B",
"content": "18ℓ "
}
],
[
{
"aoVal": "C",
"content": "21ℓ "
}
],
[
{
"aoVal": "D",
"content": "27ℓ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Unit Conversion->Converting between Units of Volume and Capacity"
] | [
"$$2 \\times 3 = 6$$ $$27 - 6 = 21$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3463 | 7026cf6b608a4f818f5ff7ac650a5eb1 | [] | 1 | single_choice | Express $108:9$ in the 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 | 3465 | 1e454f8eab3f444aa67bf46e71ab21fb | [] | 1 | single_choice | $$5 + 50 + 500 = 5 \times$$． | [
[
{
"aoVal": "A",
"content": "$$10$$ "
}
],
[
{
"aoVal": "B",
"content": "$$100$$ "
}
],
[
{
"aoVal": "C",
"content": "$$111$$ "
}
],
[
{
"aoVal": "D",
"content": "$$550$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"$$5 + 50 + 500 =555= 5 \\times111$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3466 | 128812c80a2c488e974b72d57cf184fe | [] | 1 | single_choice | If $$x+2 = y$$ and $$y+4x = 12$$, then $$x=$$． | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"$$x+2=y$$, then $$x=y-2$$. We can get $$y+4(y-2)=12$$, $$y=4$$, $$x=y-2=2$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3470 | 5da8438ecb1d4cbab0f3b729ef6d0c0c | [
"其它"
] | 2 | single_choice | In the following Figures $\left (a\right )$ and $\left (b\right )$, each number inside a small triangle is the sum of the numbers inside in the neighbouring small circles. The number inside each circle is either $1$, $2$, $3$, $4$, $5$, $6$, $7$ or $8$. The sum of whole numbers inside the circles in Figure $\left (a\right )$ is $1+8+8+2+3+5=27$. What is the largest possible sum of whole numbers inside the circles in Figure $\left (b\right )$? | [
[
{
"aoVal": "A",
"content": "$$48$$ "
}
],
[
{
"aoVal": "B",
"content": "$$52$$ "
}
],
[
{
"aoVal": "C",
"content": "$$56$$ "
}
],
[
{
"aoVal": "D",
"content": "$$64$$ "
}
],
[
{
"aoVal": "E",
"content": "$$72$$ "
}
],
[
{
"aoVal": "F",
"content": "$$74$$ "
}
],
[
{
"aoVal": "G",
"content": "$$80$$ "
}
],
[
{
"aoVal": "H",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"B "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3472 | 874b263665b34a9da8ad6b6c590d4f59 | [] | 3 | single_choice | Known $$A=\frac{{{3}^{2019}}+1}{{{3}^{2020}}+1}$$，$$B=\frac{{{3}^{2020}}+1}{{{3}^{2021}}+1}$$, compare between $$A$$ and $$B$$． | [
[
{
"aoVal": "A",
"content": "$A=B$ "
}
],
[
{
"aoVal": "B",
"content": "$A\\textgreater B$ "
}
],
[
{
"aoVal": "C",
"content": "$A\\textless B$ "
}
],
[
{
"aoVal": "D",
"content": "All of the above "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$${{3}^{2019}}=a$$，$$A=\\frac{a+1}{3a+1}$$，$$B=\\frac{3a+1}{9a+1}$$， $$\\frac{1}{A}=\\frac{3a+1}{a+1}=\\frac{3(a+1)-2}{a+1}=3-\\frac{2}{a+1}$$， $$\\frac{1}{B}=\\frac{9a+1}{3a+1}=\\frac{3(3a+1)-2}{3a+1}=3-\\frac{2}{3a+1}$$， ∴$$\\frac{1}{A} ~\\textless{} ~\\frac{1}{B}$$， ∴$$A\\textgreater B$$． "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3474 | 98edf6e63399407aa73543c2149a6bf1 | [] | 1 | single_choice | The fraction $$\frac23$$ keeps the same value when both its numerator and denominator are． | [
[
{
"aoVal": "A",
"content": "multiplied by $$2~ $$ "
}
],
[
{
"aoVal": "B",
"content": "increased by $$2$$ "
}
],
[
{
"aoVal": "C",
"content": "decreased by $$2$$ "
}
],
[
{
"aoVal": "D",
"content": "$$ $$squared$$ $$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Basic Understanding of Fractions"
] | [
"The problem is about the basic property of fractions, namely, multiplying or dividing the numerator and denominator by the same number (except $$0$$), the value of the fraction is unchanged. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3476 | 1a1cc88de0e64f09812019e4b1172b02 | [
"其它"
] | 1 | single_choice | The original price of a product was $$80$$ dollars, and it\textquotesingle s on sale for 30\% off, this product isdollars cheaper than before． | [
[
{
"aoVal": "A",
"content": "$$7$$ "
}
],
[
{
"aoVal": "B",
"content": "$$24$$ "
}
],
[
{
"aoVal": "C",
"content": "$$30$$ "
}
],
[
{
"aoVal": "D",
"content": "$$56$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"the new price is $$70\\textbackslash\\%$$ of the original price，$$so$$ the new price is $$80\\times 70\\textbackslash\\%=56$$； $$80-56=24$$． so choose $$\\text{B}$$． "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3479 | ab80fe0a39244700a0e526d7f0f2d8af | [
"其它"
] | 2 | single_choice | What is the product of five fractions, whose denominators are $5-1$ in descending order, and the numerators are $1-5$ in ascending order？ | [
[
{
"aoVal": "A",
"content": "$\\frac1{25}$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac1{2}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac1{5}$ "
}
],
[
{
"aoVal": "D",
"content": "$$0$$ "
}
],
[
{
"aoVal": "E",
"content": "$$1$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"$\\frac{1\\times2\\times3\\times4\\times5}{5\\times4\\times3\\times2\\times1}=1$ "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3480 | 3d92e164087d42fa931c54b418b1e175 | [] | 1 | single_choice | Tony and Mary are waiting in line to go shopping. Tony is fifth in line, while Mary is 365th. How many people are standing between Tony and Mary in the line? | [
[
{
"aoVal": "A",
"content": "$$359$$ "
}
],
[
{
"aoVal": "B",
"content": "$$360$$ "
}
],
[
{
"aoVal": "C",
"content": "$$361$$ "
}
],
[
{
"aoVal": "D",
"content": "$$315$$ "
}
],
[
{
"aoVal": "E",
"content": "$$314$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating->Comparing and Ordering"
] | [
"The question is asking for the number of people between Tony and Mary, which is $$365-5-1=359$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3483 | 46b440005be84fd18e846d3efe10b2bd | [] | 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 |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3486 | 22a0b3a190324c37ab626509b1732d76 | [] | 1 | single_choice | Find the value of $$ 6\times \frac{5}{3}$$． | [
[
{
"aoVal": "A",
"content": "$5$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac{33}{3}$ "
}
],
[
{
"aoVal": "C",
"content": "$10$ "
}
],
[
{
"aoVal": "D",
"content": "$6\\frac{12}{3}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Operations of Fractions"
] | [
"omitted "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3490 | be08771eef3c49ddb9ba3db7db86ac3b | [] | 1 | single_choice | The $$40^{}\text{th}$$ number in the sequence $$2$$, $$6$$, $$10$$, $$14$$, $$\cdots \cdots $$ is~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$171$$ "
}
],
[
{
"aoVal": "B",
"content": "$$158$$ "
}
],
[
{
"aoVal": "C",
"content": "$$164$$ "
}
],
[
{
"aoVal": "D",
"content": "$$160$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Arithmetic Sequences"
] | [
"4$\\times$40-2=158 "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3506 | 3003b148278e495c8e5fe00eb84c1489 | [] | 1 | single_choice | Teacher Nicole bought some sweets and divided it equally among $9$ children. If everyone got $6$ sweets, there would still be some sweets remaining. What is the most number of sweets Teacher Nicole could have bought? What is the least number of sweets Teacher Nicole could have bought? | [
[
{
"aoVal": "A",
"content": "$$63$$，$$54$$ "
}
],
[
{
"aoVal": "B",
"content": "$$63$$，$$55$$ "
}
],
[
{
"aoVal": "C",
"content": "$$62$$，$$54$$ "
}
],
[
{
"aoVal": "D",
"content": "$$62$$，$$55$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"~\\uline{~~~~~~~~~~}~$\\div 9=6$ $\\text{R}$~\\uline{~~~~~~~~~~}~ Greatest possible remainder is $8$ while smallest possible remainder is $1$. Greatest possible number of sweets is $$9\\times 6+8=62$$, while the least possible number of sweets is $$9\\times 6+1=55$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3513 | 46c0be14bee14856837f2bb3bbba9e91 | [] | 1 | single_choice | What is the correct ordering of the three numbers $$\dfrac{5}{19}$$, $$\dfrac{7}{21}$$, and $$\dfrac{9}{23}$$, in increasing order? ($$2012$$ AMC $$8$$ Problem, Question \#$$4$$) | [
[
{
"aoVal": "A",
"content": "$$\\dfrac{9}{23}\\textless{} \\dfrac{7}{21}\\textless\\dfrac{5}{19}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\dfrac{5}{19}\\textless{} \\dfrac{7}{21}\\textless{} \\dfrac{9}{23}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\dfrac{9}{23}\\textless{} \\dfrac{5}{19}\\textless{} \\dfrac{7}{21}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\dfrac{5}{19}\\textless{} \\dfrac{9}{23}\\textless{} \\dfrac{7}{21}$$ "
}
],
[
{
"aoVal": "E",
"content": "$$\\dfrac{7}{21}\\textless{} \\dfrac{5}{19}\\textless{} \\dfrac{9}{23}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Comparing, Ordering and Estimating->Comparing and Ordering"
] | [
"Method $$1$$: The value of $$\\dfrac{7}{21}$$ is $$\\dfrac{1}{3}$$. Now we give all the fractions a common denominator. $$\\dfrac{5}{19} \\Rightarrow \\dfrac{345}{1311}$$, $$\\dfrac{1}{3} \\Rightarrow \\dfrac{437}{1311}$$, $$\\dfrac{9}{23} \\Rightarrow \\dfrac{513}{1311}$$. Ordering the fractions from least to greatest, we find that they are in the order listed. Therefore, $$\\frac{5}{19}\\textless{} \\frac{7}{21}\\textless{} \\frac{9}{23}$$. Method $$2$$: Instead of finding the LCD, we can subtract each fraction from $$1$$ to get a common numerator. Thus, $$1- \\dfrac{5}{19}= \\dfrac{14}{19}$$, $$1- \\dfrac{7}{21}= \\dfrac{14}{21}$$, $$1- \\dfrac{9}{23}= \\dfrac{14}{23}$$. All three fraction have the common numerator $$14$$. Now the order of the fractions is obvious. $$\\dfrac{14}{19}\\textgreater\\dfrac{14}{21}\\textgreater\\dfrac{14}{23}\\Rightarrow\\dfrac{5}{19}\\textless\\dfrac{7}{21}\\textless\\dfrac{9}{23}$$. Therefore, $$\\dfrac{5}{19}\\textless\\dfrac{7}{21}\\textless\\dfrac{9}{23}$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3516 | 74cc1b1392844f10a8296d9ca636f246 | [
"其它"
] | 1 | single_choice | Which of the following is a linear equation written in function form? | [
[
{
"aoVal": "A",
"content": "$x=15$ "
}
],
[
{
"aoVal": "B",
"content": "$y=5x+7b-120c$ "
}
],
[
{
"aoVal": "C",
"content": "$y^{2}=4$ "
}
],
[
{
"aoVal": "D",
"content": "$y=-3x^{2}-1$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Linear Equations with one Variable"
] | [
"An equation is in function form when it is solved for $y$. A linear equation is also an equation in which the highest power of the variable is always $1$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3524 | 22c982cc58dd48bfa1c40c681be0cfe2 | [] | 1 | single_choice | The school has $$11200$$ books. The school puts them into 70 equal piles. How many books are in each pile? | [
[
{
"aoVal": "A",
"content": "$$16$$ "
}
],
[
{
"aoVal": "B",
"content": "$$160$$ "
}
],
[
{
"aoVal": "C",
"content": "$$1600$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division->Division of Whole Numbers->Division within the Multiplication Tables"
] | [
"omitted "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3528 | 1672bda384924e28854176bf2fb85dfd | [
"其它"
] | 2 | single_choice | \textbf{The height and age of each child in a random sample of children was recorded. The value of the correlation coefficient between height and age for the children in the sample was 0.8. Based on the least-squares regression line created from the data to predict the height of a child based on age, which of the following is a correct statement?} | [
[
{
"aoVal": "A",
"content": "\\textbf{~On average, the height of a child is 80\\% of the age of the child.} "
}
],
[
{
"aoVal": "B",
"content": "\\textbf{The least-squares regression line of height versus age will have a slope of 0.8.} "
}
],
[
{
"aoVal": "C",
"content": "\\textbf{The proportion of the variation in height that is explained by a regression on age is 0.64.~} "
}
],
[
{
"aoVal": "D",
"content": "\\textbf{The least-squares regression line will correctly predict height based on age 80\\% of the time.} "
}
],
[
{
"aoVal": "E",
"content": "\\textbf{The least-squares regression line will correctly predict height based on age 64\\% of the time.} "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$$R^{2} = r^{2} = 0.8^{2} = 0.64$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3530 | 3dabab5b96624787a245fc062dcceaac | [
"其它"
] | 1 | single_choice | Karen works part-time at a local convenience store and earns 10 dollars per hour. She wants to spend next Saturday afternoon attending a music concert. The full price of a concert ticket is 75 dollars, but Karen was able to get a discounted price of 50 dollars from a friend who purchased the ticket but has become unable to attend. If Karen took 4 hours off from her job to attend the concert, what was her opportunity cost of attending the concert? | [
[
{
"aoVal": "A",
"content": "$40 "
}
],
[
{
"aoVal": "B",
"content": "$50 "
}
],
[
{
"aoVal": "C",
"content": "$75 "
}
],
[
{
"aoVal": "D",
"content": "$90 "
}
],
[
{
"aoVal": "E",
"content": "$115 "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"Economic costs are the sum of explicit and implicit costs. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3548 | 22e045f94d734c70ae620acf85abf427 | [
"其它"
] | 1 | single_choice | The measures of angles $A, B$ and $C$ are $32^{\circ}, 68^{\circ},90^{\circ}$. The triangle formed by the three angles will be~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "right triangle "
}
],
[
{
"aoVal": "B",
"content": "acute triangle "
}
],
[
{
"aoVal": "C",
"content": "obtuse triangle "
}
],
[
{
"aoVal": "D",
"content": "no triangle "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns of Number Tables->Triangle Number Table"
] | [
"$32^{\\circ} + 68^{\\circ} + 90^{\\circ}=190^{\\circ}$ The sum of a triangle\\textquotesingle s interior angles is $180^{\\circ}$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3550 | 3931982941c345f0a8d76daf21fc40c2 | [] | 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 |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3551 | a6e5ead356d54fac99546aabe7f17d76 | [
"其它"
] | 4 | single_choice | The zeroes of the function $f(x)=x^{2}-a x+a$ are integers. What is the sum of the possible values of $a$? (Adapted From 2015 AMC 10A Problems, Question \#23) | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
],
[
{
"aoVal": "E",
"content": "$$7$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"By Vieta\\textquotesingle s Formula, $a$ is the sum of the integral zeros of the function, and so $a$ is integral. Because the zeros are integral, the discriminant of the function, $a^{2}-4a$, is a perfect square, say $k^{2}$. Then adding $4$ to both sides and completing the square yields $$ (a-2)^{2}=k^{2}+4$$. Therefore, $(a-2)^{2}-k^{2}=4$ and $$((a-2)-k)((a-2)+k)=4$$. Let $(a-2)-k=u$ and $(a-2)+k=v$; then, $a-2=\\frac{u+v}{2}$ and so $a=\\frac{u+v}{2}+2$. Listing all possible $(u, v)$ pairs such that $a$ is integral: $(2,2), (-2,-2)$ (not counting transpositions because this does not affect $u+v$). Then, $a=4,0$. These $a$ sum to $4$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3559 | 2bb183f57c874e5a9ccdb202e9c715a0 | [
"其它"
] | 1 | single_choice | Consider these two operations: $a ~♦ ~b = a^{2}~−~b^{2}$ $a✭b = (a~−~b)^{2}$ What is the value of $(5♦3)✭6$? | [
[
{
"aoVal": "A",
"content": "$$-20$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$16$$ "
}
],
[
{
"aoVal": "D",
"content": "$$100$$ "
}
],
[
{
"aoVal": "E",
"content": "$$220$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"NA "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3560 | 66fee83f732e43809b1ace2cd72208e3 | [
"其它"
] | 2 | single_choice | What is the least possible value of $(x y-1)^{2}+(x+y)^{2}$ for real numbers $x$ and $y$? | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac{1}{4}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac{1}{2}$ "
}
],
[
{
"aoVal": "D",
"content": "$$1$$ "
}
],
[
{
"aoVal": "E",
"content": "$$2$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Indefinite Equations"
] | [
"We expand the original expression, then factor the result by grouping: $$ \\begin{aligned} (x y-1)^{2}+(x+y)^{2} \\&=\\left(x^{2} y^{2}-2 x y+1\\right)+\\left(x^{2}+2 x y+y^{2}\\right) \\textbackslash\\textbackslash{} \\&=x^{2} y^{2}+x^{2}+y^{2}+1 \\textbackslash\\textbackslash{} \\&=x^{2}\\left(y^{2}+1\\right)+\\left(y^{2}+1\\right) \\textbackslash\\textbackslash{} \\&=\\left(x^{2}+1\\right)\\left(y^{2}+1\\right) . \\end{aligned} $$ Clearly, both factors are positive. By the Trivial Inequality, we have $$ \\left(x^{2}+1\\right)\\left(y^{2}+1\\right) \\geq(0+1)(0+1)= 1 . $$ Note that the least possible value of $(x y-1)^{2}+(x+y)^{2}$ occurs at $x=y=0$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3565 | 66ff863edd3e4713acb053b95ba5a6bf | [
"其它"
] | 1 | single_choice | How many even numbers are there? 1, 3, 4, 6, 7, 9, 5, 8. | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers"
] | [
"$$Omitted.$$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3568 | c756417bfada4d5aadd889f639895861 | [] | 1 | single_choice | Find the value of $$26+(12-9\div3)\times4-2$$. | [
[
{
"aoVal": "A",
"content": "$$28$$ "
}
],
[
{
"aoVal": "B",
"content": "$$54$$ "
}
],
[
{
"aoVal": "C",
"content": "$$60$$ "
}
],
[
{
"aoVal": "D",
"content": "$$138$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations"
] | [
"$$26+(12-9\\div3)\\times4-2$$ $$=26+9\\times4-2$$ $$=26+36-2$$ $$=62-2$$ $$=60$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3573 | 500024a621c14e5cbf54686eaa478cfa | [
"其它"
] | 0 | single_choice | Avril has a card. The number on the card is a neighbouring number of 10, but is not a neighbouring number of 12. What is the number on Avril\textquotesingle s card?~\uline{~~~~~~~~~~}~ | [
[
{
"aoVal": "A",
"content": "$$11$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers"
] | [
"$$Omitted.$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3580 | 3dbed9270c2c4257a6404f9aa19147e9 | [] | 1 | single_choice | If $$☆-10=△-5$$, then $$☆$$~\uline{~~~~~~~~~~}~$$△$$. | [
[
{
"aoVal": "A",
"content": "$$\\textgreater$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\textless$$ "
}
],
[
{
"aoVal": "C",
"content": "$$=$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation"
] | [
"If $$☆=10$$, then $$△=5$$，$$☆\\textgreater△$$． So, the answer is $$\\text{A}$$． "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3584 | 7e1e900890fc425881271983d9edc4cf | [
"其它"
] | 2 | single_choice | \textbf{In a population of university students, 20\% of the students have experienced feelings of math anxiety. If we select 8 students, what is the probability that exactly three have experienced math anxiety?} | [
[
{
"aoVal": "A",
"content": "$${8 \\choose 3} (0.20)^{3} (0.80)^{8}$$ "
}
],
[
{
"aoVal": "B",
"content": "$${8 \\choose 3} (0.20)^{3} (0.80)^{5}$$ "
}
],
[
{
"aoVal": "C",
"content": "$${5 \\choose 3} (0.20)^{3} (0.80)^{5}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$(0.20)^{3} (0.80)^{5}$$ "
}
],
[
{
"aoVal": "E",
"content": "$$(0.20)^{5} (0.80)^{3}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"\\textbf{Binomial distribution with parameters n = 8 and p = 0.20.} \\textbf{$$P(X = 3) ={8 \\choose 3}(0.20)^{3}(0.80)^{5}$$} "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3593 | 592e9f729f374ddeaea3da4042ae046a | [
"其它"
] | 3 | single_choice | Points $A$ and $B$ are 10 units apart. Points $B$ and $C$ are 4 units apart. Points $C$ and $D$ are 3 units apart. If $A$ and $D$ are as close as possible, then the number of units between them is ．(1996 AJHSME, Question 8) | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$11$$ "
}
],
[
{
"aoVal": "E",
"content": "$$17$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Inequalities"
] | [
"If $A B=10$ and $B C=4$, then $(10-4) \\leq A C \\leq(10+4)$ by the triangle inequality. In the triangle inequality, the equality is only reached when the \"triangle\" $A B C$ is really a degenerate triangle, and $A B C$ are collinear. Simplifying, this means the smallest value $A C$ can be is 6 . Applying the triangle inequality on $A C D$ with $A C=6$ and $C D=3$, we know that $6-3 \\leq A D \\leq 6+3$ when $A C$ is minimized. If $A C$ were larger, then $A D$ could be larger, but we want the smallest $A D$ possible, and not the largest. Thus, $A D$ must be at least 3 , but cannot be smaller than 3 . Therefore, $B$ is the answer. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3606 | 70472ba92bb8462fbf1483f09bfeb716 | [
"其它"
] | 2 | single_choice | There are $20$ balls of the same size in a box. Lucas says: "Half of them are red." Peter says: "The number of red balls is $5$ times that of green balls." Claire says: "There are three colors of balls in the box: red, light blue, and green." How many dark blue balls are there in the box? | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
],
[
{
"aoVal": "E",
"content": "$$10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"There is no dark blue ball in the box. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3607 | 3dc9f5b6faef4b578bfe9618979f1c5d | [] | 1 | single_choice | $$9.25\times 0.8+9\frac{1}{4}\times 0.2=$$~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$9.25$$ "
}
],
[
{
"aoVal": "B",
"content": "$$92.5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$925$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$$9.25\\times 0.8+9\\frac{1}{4}\\times 0.2$$ $$=9.25\\times 0.8+9.25\\times 0.2$$ $$=9.25\\times (0.8+0.2)$$ $$=9.25\\times 1$$ $$=9.25$$ So, $$\\text{A}$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3609 | 7047692906134bddacffba16390027ae | [
"其它"
] | 1 | single_choice | Suppose that $x$ and $y$ are nonzero real numbers such that $\frac{x+y}{x}=2$, what is the value of $\frac{x}{y}$? | [
[
{
"aoVal": "A",
"content": "$$\\frac{1}{3}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$-1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$1$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"Rearranging, we find $x+y=2x, x=y, \\frac xy = 1$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3610 | 1ecadfb2e16b439a87c6394a9e8deb72 | [
"其它"
] | 1 | single_choice | Cathy wants to cut a wooden stick. In how many places does she need to break a wooden stick in order to get $7$ pieces? | [
[
{
"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"
] | [
"$7 - 1 = 6$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3611 | 46e4a87e1ac24d54b0fd3d02a48231fc | [] | 1 | single_choice | If $a◆b=a\times 2+b$, then $2◆3=$． | [
[
{
"aoVal": "A",
"content": "$$7$$ "
}
],
[
{
"aoVal": "B",
"content": "$$12$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Operating Directly"
] | [
"$2◆3=2\\times2+3=7$ So the answer is $\\rm A$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3625 | b02d4ef1fe1543afb5f20cb92c75f843 | [
"其它"
] | 2 | single_choice | Starting with some gold coins and some empty treasure chests, I tried to put $9$ gold coins in each treasure chest, but that left $2$ treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins did I have? ( 2017 AMC8, Questions \#17) | [
[
{
"aoVal": "A",
"content": "$$9$$ "
}
],
[
{
"aoVal": "B",
"content": "$$27$$ "
}
],
[
{
"aoVal": "C",
"content": "$$45$$ "
}
],
[
{
"aoVal": "D",
"content": "$$63$$ "
}
],
[
{
"aoVal": "E",
"content": "$$81$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Linear Equations with Multiple Variables"
] | [
"We can represent the amount of gold with $g$ and the amount of chests with $c$. We can use the problem to make the following equations: $$ \\begin{gathered} 9 c-18=g \\textbackslash\\textbackslash{} 6 c+3=g \\end{gathered} $$ We do this because for $9$ chests there are $2$ empty and if $9$ were in each $9$ multiplied by $2$ is $18$ left. Therefore, $6 c+3=9 c-18$. This implies that $c=7$. We therefore have $g=45$. So, our answer is (C) 45 "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3626 | b96f1d1fa1b84755ae4eb58c3a5d8c40 | [
"其它"
] | 2 | single_choice | \textbf{A stationery store owner calculated the mean and median price of all types of pen: 5 and 3, respectively. He plans to have a 20\% off sale. What are the new mean and median of pens in that store?} | [
[
{
"aoVal": "A",
"content": "\\textbf{mean: 4; median: 2.4} "
}
],
[
{
"aoVal": "B",
"content": "\\textbf{mean: 5; median: 2.4} "
}
],
[
{
"aoVal": "C",
"content": "\\textbf{~mean: 4; median: 3} "
}
],
[
{
"aoVal": "D",
"content": "\\textbf{~mean: 5; median: 3} "
}
],
[
{
"aoVal": "E",
"content": "\\textbf{mean: 2.4; median: 4} "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"\\textbf{If you times some number to the data set,~ the mean or median needs to be times with the same number. So for here, everything times with 0.8.} "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3630 | 1ede330fdd6049e8918a4285c09e4e07 | [] | 1 | single_choice | Which of the following is the same as~$44\times25$? (~\uline{~~~~~~~~~~}~) | [
[
{
"aoVal": "A",
"content": "$$4\\times 25+4\\times 25$$ "
}
],
[
{
"aoVal": "B",
"content": "$$44\\times 2+44\\times 5$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4\\times 2+4\\times 5$$ "
}
],
[
{
"aoVal": "D",
"content": "$$40\\times 25+4\\times 25$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Distributive Law of Whole Numbers"
] | [
"$44\\times25=\\left( 40+4\\right)\\times25=40\\times25+4\\times25$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3631 | ec6eed200b684d00bab77b36dd2527ad | [] | 1 | single_choice | $$50\textbackslash\%$$ of $$30\textbackslash\%=15\textbackslash\%$$of ． | [
[
{
"aoVal": "A",
"content": "$$10\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "B",
"content": "$$25\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "C",
"content": "$$100\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "D",
"content": "$$150\\textbackslash\\%$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Operations of Fractions"
] | [
"$$50\\textbackslash\\%\\times 30\\textbackslash\\%=$$ half of $$30\\textbackslash\\%=15\\textbackslash\\%=15\\textbackslash\\%$$ of $$100\\textbackslash\\%$$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3634 | 6276aba0474f460c92cb2640a2ad7487 | [] | 1 | single_choice | $$2001 + (2000 - 1999 + 1998 - 1997 + 1996 - \cdots + 2 - 1) =$$． | [
[
{
"aoVal": "A",
"content": "$$2001$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3001$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4001$$ "
}
],
[
{
"aoVal": "D",
"content": "$$4002$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction "
] | [
"$$2001+(2000-1999)+(1998-1997)+\\cdots +(2-1)=2001+(1000$$ones$$)$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3647 | 1ad556b4e9064276bfd8751dfa17f620 | [] | 1 | single_choice | How many polynomials are there in the following expressions: $$-4{{x}^{2}}+2$$，$$-\frac{1}{3}mn$$，$$ \pi $$，$$\frac{{{(2x-y)}^{2}}}{3}$$，$$32\frac{1}{4}$$~ ~． | [
[
{
"aoVal": "A",
"content": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$3$$ "
}
],
[
{
"aoVal": "D",
"content": "$$4$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"$$-4{{x}^{2}}+2$$，$$\\frac{{{(2x-y)}^{2}}}{3}$$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3655 | b4d4e937f058442fad371a5f852a74af | [
"其它"
] | 0 | single_choice | What value does the $4$ represent in the number $55.431$? | [
[
{
"aoVal": "A",
"content": "$$4$$ "
}
],
[
{
"aoVal": "B",
"content": "$$0.4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$0.04$$ "
}
],
[
{
"aoVal": "D",
"content": "$$0.004$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals"
] | [
"The number $$4$$ is located on the tenth place, thus representing $$0.04$$. Check Lesson 4 Concept 1 on textbook "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3659 | 1715d2fdc80f4e09a84ab204d79438ca | [
"其它"
] | 1 | single_choice | Assuming $a \neq 3, b \neq 4$, and $c \neq 5$, what is the value in simplest form of the following expression? (Adapted From 2020 AMC 10A Problems, Question \#3) $$ \frac{2a-6}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{2c-10}{4-b} $$ | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$-4$$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac{a b c}{15}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\frac{4}{a b c}-\\frac{1}{15}$ "
}
],
[
{
"aoVal": "E",
"content": "$\\frac{1}{15}-\\frac{1}{a b c}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"If $x \\neq y$, then $\\frac{x-y}{y-x}=-1$. We use this fact to simplify the original expression: $$ \\frac{2a-6}{5-c} \\cdot \\frac{b-4}{3-a} \\cdot \\frac{2c-10}{4-b} = -2\\times 2= 4$$. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3670 | 54b649894b934e0cba3206221b716f49 | [
"其它"
] | 4 | single_choice | The zeroes of the function $f(x)=x^{2}-a x+2 a$ are integers. What is the sum of the possible values of $a$? (2015 AMC 10A Problems, Question \#23) | [
[
{
"aoVal": "A",
"content": "$$7$$ "
}
],
[
{
"aoVal": "B",
"content": "$$8$$ "
}
],
[
{
"aoVal": "C",
"content": "$$16$$ "
}
],
[
{
"aoVal": "D",
"content": "$$18$$ "
}
],
[
{
"aoVal": "E",
"content": "$$18$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"By Vieta\\textquotesingle s Formula, $a$ is the sum of the integral zeros of the function, and so $a$ is integral. Because the zeros are integral, the discriminant of the function, $a^{2}-8 a$, is a perfect square, say $k^{2}$. Then adding $16$ to both sides and completing the square yields $$ (a-4)^{2}=k^{2}+16 $$. Therefore, $(a-4)^{2}-k^{2}=16$ and $$((a-4)-k)((a-4)+k)=16$$. Let $(a-4)-k=u$ and $(a-4)+k=v$; then, $a-4=\\frac{u+v}{2}$ and so $a=\\frac{u+v}{2}+4$. Listing all possible $(u, v)$ pairs such that $a$ is integral: $(2,8),(4,4),(-2,-8),(-4,-4)$ (not counting transpositions because this does not affect $u+v$). Then, $a=9,8,-1,0$. These $a$ sum to $16$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3672 | 6bb72cbcde7d487a91260b749950d349 | [] | 1 | single_choice | Alice, Tom, Chloe and Susan ate chicken nuggets together. Everyone ate at least 2 pieces. The person who ate the least ate 4 pieces less than the person who ate the most. How many pieces of chicken nuggets did they each eat? (adapted from $$2021$$ Math kangaroo Problems, Level $$1-2$$, Question \#$$8$$） | [
[
{
"aoVal": "A",
"content": "$8, 2, 4, 3$ "
}
],
[
{
"aoVal": "B",
"content": "$4, 1, 5,9$ "
}
],
[
{
"aoVal": "C",
"content": "$6,5,0,10$ "
}
],
[
{
"aoVal": "D",
"content": "$2,7,4,1$ "
}
],
[
{
"aoVal": "E",
"content": "$2,6,4,8$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Subtraction of Whole Numbers->Subtraction in Horizontal Form"
] | [
"In B, C, D, the smallest number less than $2$. In the E, the difference between the largest number and the smallest number is $6$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3674 | a2555ac4be44460ba407c88b485ea2f7 | [] | 1 | single_choice | Which of the following quotients is $$1$$ more than $$162\div18$$? | [
[
{
"aoVal": "A",
"content": "$$128\\div 16$$ "
}
],
[
{
"aoVal": "B",
"content": "$$120\\div 15$$ "
}
],
[
{
"aoVal": "C",
"content": "$$132\\div 12$$ "
}
],
[
{
"aoVal": "D",
"content": "$$110\\div 11$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"Since $$162 \\div18 =9$$, we want a quotient of $$10$$. That\\textquotesingle s $$\\text{D}$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3675 | a255be4b4c0f4b48ac0525419e8e37ec | [
"其它"
] | 1 | single_choice | A box contains five cards, numbered $1,2,3,4$, and $5$. Three cards are selected randomly without replacement from the box. What is the probability that $4$ is the largest value selected? (2017 AMC 8 Problems, Question \#10) | [
[
{
"aoVal": "A",
"content": "$\\frac{1}{10}$ "
}
],
[
{
"aoVal": "B",
"content": "$\\frac{1}{5}$ "
}
],
[
{
"aoVal": "C",
"content": "$\\frac{3}{10}$ "
}
],
[
{
"aoVal": "D",
"content": "$\\frac{2}{5}$ "
}
],
[
{
"aoVal": "E",
"content": "$\\frac{1}{2}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"There are $\\_5C\\_3 = 10$ possible groups of cards that can be selected. If $4$ is the largest card selected, then the other two cards must be either $1$,$2$, or $3$, for a total $\\_3C\\_2 = 3$ groups of cards. Then, the probability is just $(\\text{C}) \\frac{3}{10}$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3680 | 279dc43f53c444dfa63bc701e36fbba4 | [
"其它"
] | 1 | single_choice | The distance between A and B is $$350$$ $\text{km}$. Kin and Mary drive away from A and B respectively at $8$ a.m. and go towards each other at same time. Kin drives $$40$$ $\text{km/h}$, and Mary drives $$50$$ $\text{km/h}$. Mary rested for $$2$$ hours on her way and then continues driving. They will meet at~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$10$$ a.m. "
}
],
[
{
"aoVal": "B",
"content": "$$11$$ a.m. "
}
],
[
{
"aoVal": "C",
"content": "$$12$$ p.m. "
}
],
[
{
"aoVal": "D",
"content": "$$1$$ p.m. "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"$$(350-80)$$$\\div$$$(40+50)=3$$hr $$8+2+3=13$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3688 | a6f758ab251c4224b7ee884e09e3eaa8 | [
"其它"
] | 1 | single_choice | Bob sees that the license plate number of the car consists of $5$ digits: $1$, $2$, $6$, $7$, $9$. 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": "$$1$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6$$ "
}
],
[
{
"aoVal": "D",
"content": "$$7$$ "
}
],
[
{
"aoVal": "E",
"content": "$$9$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Addition of Whole Numbers->Addition in Horizontal Form"
] | [
"$1+7=2+6$ "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3691 | 705d7653e5f647f5bfa886bc80e7d4e1 | [] | 0 | single_choice | Tiantian drank $$6\frac{3}{8}$$ litres of water. Matthew drank $$4\frac{5}{6}$$ litres of water. How many litres of water did the both of them drink altogether? Give your answer as a mixed number in the simplest form. | [
[
{
"aoVal": "A",
"content": "$$10\\frac{4}{7}\\ell$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10\\frac{5}{8}\\ell$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10\\frac{15}{24}\\ell$$ "
}
],
[
{
"aoVal": "D",
"content": "$$11\\frac{5}{24}\\ell$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions->Operations of Fractions"
] | [
"$$6\\frac{3}{8}+4\\frac{5}{6}=6\\frac{9}{24}+4\\frac{20}{24}=11\\frac{5}{24}\\ell$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3696 | 235f41c1516f4254a787441b51f58255 | [
"其它"
] | 0 | single_choice | $$3\times2016 + 0\times2016 + 3\times2016=$$? | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2016$$ "
}
],
[
{
"aoVal": "C",
"content": "$$6048$$ "
}
],
[
{
"aoVal": "D",
"content": "$$12096$$ "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$$(3+0+3)\\times2016=6\\times2016=12096$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3697 | 1b23ae71e3c64ad78ea448d4579e6f20 | [
"其它"
] | 2 | single_choice | In the diagram below, a circle centered at $O$ has radius $4 \text{cm}$. It is divided into $4$ regions by two chords that are perpendicular to each other at point $N$. It is known that $OM=1 \text{cm}$, $MN=2\text{cm}$. Find the value of: Area of $(I + III) -$ Area of $(II+IV)$ | [
[
{
"aoVal": "A",
"content": "$$0$$ "
}
],
[
{
"aoVal": "B",
"content": "$$2$$ "
}
],
[
{
"aoVal": "C",
"content": "$$4$$ "
}
],
[
{
"aoVal": "D",
"content": "$$6$$ "
}
],
[
{
"aoVal": "E",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Unit Conversion->Converting between Units of Area"
] | [
"D "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3699 | 4288c25670dd4ab69ce3b78675fb4028 | [
"其它"
] | 3 | single_choice | Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump $5$ pads to the right or $3$ pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located $2023$ pads to the right of her starting position? | [
[
{
"aoVal": "A",
"content": "$$405$$ "
}
],
[
{
"aoVal": "B",
"content": "$$407$$ "
}
],
[
{
"aoVal": "C",
"content": "$$409$$ "
}
],
[
{
"aoVal": "D",
"content": "$$411$$ "
}
],
[
{
"aoVal": "E",
"content": "$$413$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"D "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3703 | 5def908615b74af682ba4d7aa495b56b | [] | 1 | single_choice | $$3^{336}\times 9^{336}\times 27^{336}=$$． | [
[
{
"aoVal": "A",
"content": "$$3^{1008}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$3^{1344}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$3^{1680}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$3^{2016}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers"
] | [
"$$3^{336}\\times 9^{336}\\times 27^{336}=3^{336}\\times 3^{672}\\times 3^{1008}=3^{336+672+1008}=3^{2016}$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3704 | 5defb5e9e14240dda7a12fa7ca76c198 | [] | 2 | single_choice | What is the value of $$1+3+5+\cdots +2017+2019-2-4-6-\cdots -2016-2018$$? | [
[
{
"aoVal": "A",
"content": "$$1010$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1009$$ "
}
],
[
{
"aoVal": "C",
"content": "$$1008$$ "
}
],
[
{
"aoVal": "D",
"content": "$$-1009$$ "
}
],
[
{
"aoVal": "E",
"content": "$$-1010$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Addition and Subtraction ->Grouping in Fast Addition and Subtraction of Whole Numbers"
] | [
"Solution 1 Rearranging the terms, we get $$(1-2)+(3-4)+(5-6)+\\cdots (2017-2018)+2019$$, and our answer is $$-1009+2019=1010$$. Solution 2 We can rewrite the given expression as $$1+(3-2)+(5-4)+\\cdots +(2017-2016)+(2019-2018)=1+1+1+\\cdots +1$$. The number of $$1$$s is the same as the number of terms in $$1$$, $$3$$, $$5$$, $$7\\cdots $$, $$2017$$, $$2019$$. Thus the answer is $$1010$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3710 | 1b2cd5850dee4ee8b3d7a37ad88c73e5 | [] | 1 | single_choice | $$2\times \left( 3+1 \right)\times \left( {{3}^{2}}+1 \right)\times \left( {{3}^{4}}+1 \right)\times \left( {{3}^{8}}+1 \right)\times \left( {{3}^{16}}+1 \right)\times \left( {{3}^{32}}+1 \right)=$$． | [
[
{
"aoVal": "A",
"content": "$${{3}^{64}}+1$$． "
}
],
[
{
"aoVal": "B",
"content": "$${{3}^{128}}-1$$． "
}
],
[
{
"aoVal": "C",
"content": "$${{3}^{32}}-1$$． "
}
],
[
{
"aoVal": "D",
"content": "$${{3}^{64}}-1$$． "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations through Formulas-> Difference of Two Squares Formula"
] | [
"$$=\\left( 3-1 \\right)\\times \\left( 3+1 \\right)\\times \\left( {{3}^{2}}+1 \\right)\\times \\left( {{3}^{4}}+1 \\right)\\times \\left( {{3}^{8}}+1 \\right)\\times \\left( {{3}^{16}}+1 \\right)\\times \\left( {{3}^{32}}+1 \\right)$$ $$=\\left( {{3}^{2}}-1 \\right)\\times \\left( {{3}^{2}}+1 \\right)\\times \\left( {{3}^{4}}+1 \\right)\\times \\left( {{3}^{8}}+1 \\right)\\times \\left( {{3}^{16}}+1 \\right)\\times \\left( {{3}^{32}}+1 \\right)$$ $$=\\left( {{3}^{4}}-1 \\right)\\times \\left( {{3}^{4}}+1 \\right)\\times \\left( {{3}^{8}}+1 \\right)\\times \\left( {{3}^{16}}+1 \\right)\\times \\left( {{3}^{32}}+1 \\right)$$ $$=\\cdots $$ $$=\\left( {{3}^{32}}-1 \\right)\\times \\left( {{3}^{32}}+1 \\right)$$ $$={{3}^{64}}-1$$． "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3720 | 2c1b705ff8804097b2500148e418d4f5 | [
"其它"
] | 2 | single_choice | Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5 cups. What percent of the total capacity of the pitcher did each cup receive?~ (2020 AMC 8, Question \#5) | [
[
{
"aoVal": "A",
"content": "$$5$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$15$$ "
}
],
[
{
"aoVal": "D",
"content": "$$20$$ "
}
],
[
{
"aoVal": "E",
"content": "$$25$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Percentage Calculation"
] | [
"The pitcher is $\\frac{3}{4}$ full, i.e. $75 \\textbackslash\\%$ full. Therefore each cup receives $\\frac{75}{5}=(\\mathbf{C}) 15$ percent of the total capacity. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3724 | 1f435fb31dc24633add9a3cee6e3c71d | [
"其它"
] | 1 | single_choice | The solution to the rational equation $$\frac{16}{4-{{x}^{2}}}+\frac{x-2}{x+2}=\frac{x+2}{x-2}$$ is~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$x=2$$ "
}
],
[
{
"aoVal": "B",
"content": "no solution "
}
],
[
{
"aoVal": "C",
"content": "$$x=2$$ or $$x=3$$ "
}
],
[
{
"aoVal": "D",
"content": "$$x=3$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$$\\begin{eqnarray}\\frac{16}{4-{{x}^{2}}}+\\frac{x-2}{x+2}\\&=\\&\\frac{x+2}{x-2}\\textbackslash\\textbackslash{} -16+{{(x-2)}^{2}}\\&=\\&{{(x+2)}^{2}}\\textbackslash\\textbackslash{} x\\&=\\&-2\\end{eqnarray}$$ After verification, $$x=-2$$ is not a solution. There is no solution. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3727 | 7065c569679d41bc99f202a7fecccd64 | [
"其它"
] | 0 | single_choice | The sum of $$5$$ consecutive numbers is $$500$$. Among all five numbers, what is the smallest value? | [
[
{
"aoVal": "A",
"content": "$$95$$ "
}
],
[
{
"aoVal": "B",
"content": "$$96$$ "
}
],
[
{
"aoVal": "C",
"content": "$$97$$ "
}
],
[
{
"aoVal": "D",
"content": "$$98$$ "
}
],
[
{
"aoVal": "E",
"content": "$$99$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Arithmetic Sequences"
] | [
"$$500\\div5=100$$(middle number is $$100$$) $$98, 99, 100, 101. 102$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3747 | 5960412e9af14db390229664072b88fb | [] | 1 | single_choice | What is the product of $63$ and $4$? | [
[
{
"aoVal": "A",
"content": "$$242$$ "
}
],
[
{
"aoVal": "B",
"content": "$$252$$ "
}
],
[
{
"aoVal": "C",
"content": "$$262$$ "
}
],
[
{
"aoVal": "D",
"content": "$$272$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"omitted "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3749 | 351782cab5df433daa5997cd2a28ed7a | [
"其它"
] | 1 | single_choice | What is the last number in Row $12$? | [
[
{
"aoVal": "A",
"content": "$$121$$ "
}
],
[
{
"aoVal": "B",
"content": "$$144$$ "
}
],
[
{
"aoVal": "C",
"content": "$$169$$ "
}
],
[
{
"aoVal": "D",
"content": "$$196$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns of Number Tables->Triangle Number Table"
] | [
"B "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3757 | aba9ad9073b54015953fc5b9ed974c18 | [] | 1 | single_choice | In multiplying a number by $\dfrac{1}{100}$, the result may be obtained by moving the decimal point of that number． | [
[
{
"aoVal": "A",
"content": "two places to the left "
}
],
[
{
"aoVal": "B",
"content": "one place to the left "
}
],
[
{
"aoVal": "C",
"content": "two places to the right "
}
],
[
{
"aoVal": "D",
"content": "one place to the right "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals"
] | [
"Multiplying a number by $\\dfrac{1}{100}$, is the same as dividing by $100$. When dividing by $100$, move the decimal point $2$ places left. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3760 | 8785f07bb64f4d199fce316435209f46 | [] | 1 | single_choice | If $$3$$ apples weigh as much as $$4$$ pears, and $$2$$ pears weigh as much as $$5$$ plums, then $$9$$ apples weigh as much as~ \uline{?~} plums. | [
[
{
"aoVal": "A",
"content": "$$10$$ "
}
],
[
{
"aoVal": "B",
"content": "$$19$$ "
}
],
[
{
"aoVal": "C",
"content": "$$27$$ "
}
],
[
{
"aoVal": "D",
"content": "$$30$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"If $$9$$ apples weigh as much as $$12$$ pears, and $$12$$ pears weigh as much as $$30$$ plums, then $$9$$ apples weigh as much as $$30$$ plums. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3765 | 23a03c2f1a3644c98e89d5dc863788f1 | [
"其它"
] | 3 | single_choice | Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-19$, the remainder is 99 , and when $P(x)$ is divided by $x-99$, the remainder is 19. What is the remainder when $P(x)$ is divided by $(x-19)(x-99)$? (1999 AHSME Problems, Question \#17) | [
[
{
"aoVal": "A",
"content": "$-x+80$ "
}
],
[
{
"aoVal": "B",
"content": "$x+80$ "
}
],
[
{
"aoVal": "C",
"content": "$-x+118$ "
}
],
[
{
"aoVal": "D",
"content": "$x+118$ "
}
],
[
{
"aoVal": "E",
"content": "$$0$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"Since the divisor $(x-19)(x-99)$ is a quadratic, the degree of the remainder is at most linear. We can write $P(x)$ in the form $$ P(x)=Q(x)(x-19)(x-99)+c x+d $$ where $c x+d$ is the remainder. By the Remainder Theorem, plugging in 19 and 99 gives us a system of equations: $$ \\begin{aligned} \\& 99 c+d=19 \\textbackslash\\textbackslash{} \\& 19 c+d=99\\end{aligned} $$ Solving gives us $c=-1$ and $d=118$, thus, our answer is $(\\text{C})-x+118$. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3766 | 472a5c4cdb29469796258393af1a531d | [
"其它"
] | 1 | single_choice | $1-2-3+4+5-6-7+8-9=$ | [
[
{
"aoVal": "A",
"content": "$$-6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$-7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$-8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$-9$$ "
}
],
[
{
"aoVal": "E",
"content": "$$-10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Mixed Operations"
] | [
"$(1-2-3+4)+(5-6-7+8)-9=-9$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3768 | 1f7714f9e97147aa9e001efc8baf6915 | [
"其它"
] | 1 | single_choice | Given that $$a\Delta b=a+b-4$$, for example, $$3\Delta 2 = 3 +2-4$$, what is $$3\Delta4$$? | [
[
{
"aoVal": "A",
"content": "$$3$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$7$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition"
] | [
"Nil "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3771 | 504b32b250fb49a9bbc10bee97fee12e | [
"其它"
] | 2 | single_choice | \textbf{Which of the following is/are categorical variable(s)?} \textbf{I. The mean of Grade 1 boys' heights} \textbf{II. The colors of jackets in the class} \textbf{III. The types of pens in a stationary store} | [
[
{
"aoVal": "A",
"content": "I "
}
],
[
{
"aoVal": "B",
"content": "II "
}
],
[
{
"aoVal": "C",
"content": "III "
}
],
[
{
"aoVal": "D",
"content": "I, II "
}
],
[
{
"aoVal": "E",
"content": "II, III "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"\\textbf{A categorical variable is defined by the set of groups or categories (qualitative values) that individuals are placed into; it is not a numerical value. The mean of~ Grade 1 boys' heights is a number. The colors of jackets in the class and the types of pens in a stationary store cannot be described by numbers.~} "
] | E |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3772 | 1f7eb4fb982d4bf5982c64129f97004a | [
"其它"
] | 1 | single_choice | A movie ticket costs $17$ dollars and the tax is $a$ dollars. George purchases a movie ticket and writes on his ledgar : " the ticket cost $17+a$ dollars". Is this description accurate? | [
[
{
"aoVal": "A",
"content": "yes "
}
],
[
{
"aoVal": "B",
"content": "no "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Basic Concepts of Equation->Equivalent Substitution->Algebraic Expressions"
] | [
"When writing an algebraic expression with units, we need to write it in parentheses. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3778 | 4731d852550249d9b9a7309f4966d347 | [] | 1 | single_choice | $$\frac{5}{12}\div \frac{25}{24}=$$． | [
[
{
"aoVal": "A",
"content": "$$\\frac{2}{5}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{5}{2}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{35}{24}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{5}{12}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"omitted "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3787 | 948bded92bbd4133890ba60f6f56897c | [] | 1 | single_choice | Calculate: $${{2}^{2}}\div {{2}^{3}}\times {{2}^{4}}\div {{2}^{5}}\times {{2}^{6}}\div \cdots \div {{2}^{99}}\times {{2}^{100}}=$$~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$${{2}^{48}}$$ "
}
],
[
{
"aoVal": "B",
"content": "$${{2}^{49}}$$ "
}
],
[
{
"aoVal": "C",
"content": "$${{2}^{50}}$$ "
}
],
[
{
"aoVal": "D",
"content": "$${{2}^{51}}$$ "
}
],
[
{
"aoVal": "E",
"content": "$${{2}^{1}}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers->Exponentiation of Powers"
] | [
"$$({{2}^{100}}\\div {{2}^{99}})\\times ({{2}^{98}}\\div {{2}^{97}})\\times \\cdot \\cdot \\cdot \\times ({{2}^{6}}\\div {{2}^{5}})\\times ({{2}^{4}}\\div {{2}^{3}})\\times {{2}^{2}}={{2}^{51}}$$ "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3790 | 1b9ac48c167d41bab1b4c01a21ed3f16 | [] | 0 | single_choice | The next number in the sequence $$1, 3, 7, 13, 21, \cdots$$ is． | [
[
{
"aoVal": "A",
"content": "$$37$$ "
}
],
[
{
"aoVal": "B",
"content": "$$35$$ "
}
],
[
{
"aoVal": "C",
"content": "$$33$$ "
}
],
[
{
"aoVal": "D",
"content": "$$31$$ "
}
],
[
{
"aoVal": "E",
"content": "$$29$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns in Number Sequences"
] | [
"Difference between the numbers is $$2$$, $$4$$, $$6$$, $$8$$. Next one is $$21+10$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3796 | 597787bbc80f439298db867b86cc5a09 | [
"其它"
] | 1 | single_choice | Given that the points $A(-2, y\_1)$, $B(0, y\_2)$ and $C(3, y\_3)$ are all on the graph of $f(x)=-2x^{2}+4x+m$, then the relationship among $y\_1$, $y\_2$ and $y\_3$ is: | [
[
{
"aoVal": "A",
"content": "$y\\_2 \\textgreater{} y\\_3\\textgreater y\\_1$ "
}
],
[
{
"aoVal": "B",
"content": "$y\\_1~\\textgreater{} y\\_3\\textgreater y\\_2$ "
}
],
[
{
"aoVal": "C",
"content": "$y\\_2 \\textgreater{} y\\_1\\textgreater y\\_3$ "
}
],
[
{
"aoVal": "D",
"content": "$y\\_3~\\textgreater{} y\\_2\\textgreater y\\_1$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"The parabola opens down with axis of symmetry $x=1$. Therefore, point $B$ is closer the axis of symmetry than point $C$, and point $C$ is closer the axis of symmetry than point $A$ "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3812 | b98b1cb0032b4092b2dd0921cab69e1a | [
"其它"
] | 1 | single_choice | Which of the following fraction is smaller than $$\frac{1}{9}$$? | [
[
{
"aoVal": "A",
"content": "$$\\frac{1}{6}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$\\frac{1}{7}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac{1}{8}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac{1}{10}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"With same numerator, the one we split into more portion is smaller "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3813 | c773de52af524fe88c804f2ec4603b7b | [
"其它"
] | 2 | single_choice | \textbf{A statistical test involves the following null and alternative hypotheses.~} \textbf{H0: $\mu$ = 64} \textbf{Ha: $\mu$ \textgreater{} 64} \textbf{Which of the following describes a Type II error?} | [
[
{
"aoVal": "A",
"content": "\\textbf{Failing to reject the null hypothesis when the population mean is 64} "
}
],
[
{
"aoVal": "B",
"content": "\\textbf{Failing to reject the null hypothesis when the population mean is greater than 64} "
}
],
[
{
"aoVal": "C",
"content": "\\textbf{Rejecting the null hypothesis when the population mean is 64} "
}
],
[
{
"aoVal": "D",
"content": "\\textbf{Rejecting the null hypothesis when the population mean is greater than 64} "
}
],
[
{
"aoVal": "E",
"content": "\\textbf{Failing to reject the null hypothesis when the p-value is less than the significance level} "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"\\textbf{Type II error ($\\beta$): the error of failing to reject the null hypothesis when it is false.~} "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3815 | 54ee05569ec14bff8a720b098a19699b | [] | 1 | single_choice | The sum of the recurring decimals $$0.\overline {62}$$ and $$0.\overline {16}$$ is~\uline{~~~~~~~~~~}~. | [
[
{
"aoVal": "A",
"content": "$$0.\\overline{78}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$78\\textbackslash\\%$$ "
}
],
[
{
"aoVal": "C",
"content": "$$\\frac {78}{100}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$0.788$$ "
}
],
[
{
"aoVal": "E",
"content": "$0.\\overline{46}$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals"
] | [
"$0.\\overline{62}+0.\\overline{16}=0.\\overline{78}$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3820 | 30cf4704ddfc457da143a5ef52f6feb8 | [] | 1 | single_choice | Three members of the Euclid Middle School girls\textquotesingle~softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $$2-$$digit primes. Bethany: And the sum of your two uniform numbers is the day of my birthday earlier this month. Caitlin: That\textquotesingle s funny. The sum of your two uniform numbers is the day of my birthday later this month. Ashley: And the sum of your two uniform numbers is today\textquotesingle s date. What number does Caitlin wear? ($$2014$$ AMC $$8$$ Problem, Question \#$$23$$) | [
[
{
"aoVal": "A",
"content": "$$11$$ "
}
],
[
{
"aoVal": "B",
"content": "$$13$$ "
}
],
[
{
"aoVal": "C",
"content": "$$17$$ "
}
],
[
{
"aoVal": "D",
"content": "$$19$$ "
}
],
[
{
"aoVal": "E",
"content": "$$23$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Sequences and Number Tables->Patterns in Number Sequences"
] | [
"The maximum amount of days any given month can have is $$31$$, and the smallest, two-digit primes are $$11$$, $$13$$, and $$17$$. There are a few different sums that can be deduced from the following numbers, which are $$24, 30$$, and $$28$$, all of which represent the three days. Therefore, since Brittany says that the other two people\\textquotesingle s uniform numbers add up to her birthday eadier in the month, that means Caitlin and Ashley\\textquotesingle s numbers must add up to $$24$$. Similarly, Caitlin says that the other two people\\textquotesingle s uniform numbers add up to her birhday later in the month, so the sum must add up to $$30$$. This leaves $$28$$ as today\\textquotesingle s date. From this, Caitlin was referring to the uniform numbers $$13$$ and $$17$$ telling us that her number is $$11$$, giving our solution as $$(\\text{A})=11$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3821 | b04cecab072c4865b55344aeec46b9d2 | [
"其它"
] | 2 | single_choice | Given that $$23+25+27 +\ldots+(2k-1)=m^{2}$$, where $$k$$ and $$m$$ are whole numbers, $$k\textgreater30$$, find the value of $$m$$. | [
[
{
"aoVal": "A",
"content": "$$11$$ "
}
],
[
{
"aoVal": "B",
"content": "$$35$$ "
}
],
[
{
"aoVal": "C",
"content": "$$41$$ "
}
],
[
{
"aoVal": "D",
"content": "$$60$$ "
}
],
[
{
"aoVal": "E",
"content": "None of the above "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"D "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3823 | 4bd0a22a8ad94e79ac55754732d4fee7 | [] | 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 |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3838 | 39c9189f56a94f608b795b4483116ad7 | [] | 1 | single_choice | Which of the following is not equal to $$10$$? | [
[
{
"aoVal": "A",
"content": "$$100\\div 10$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10 \\div1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10\\times1$$ "
}
],
[
{
"aoVal": "D",
"content": "$$100\\times 10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"Since $$100 \\times 10 = 1000$$, choice $$\\text{D}$$ is not equal to $$10$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3839 | b05030fbabff417ca02c25cde2103cbf | [] | 1 | single_choice | What is the remainder when $$16+16+16 +16$$ is divided by $$4$$? | [
[
{
"aoVal": "A",
"content": "$$16$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2$$ "
}
],
[
{
"aoVal": "D",
"content": "$$0$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"Since $$16\\div4$$ has remainder $$0$$, the remainder is $$0 + 0+0+0 = 0$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3846 | 42ccfae50dbc4187ba6af0433ee5c4be | [
"其它"
] | 1 | single_choice | An amusement park has a collection of scale models, with ratio $1: 45$, of buildings and other sights from around the country. If the height of the One World Trade Center is $1770$ feet. What is the height in feet of its replica to the nearest whole number? (adapted from 2018 AMC 8, Question 1) | [
[
{
"aoVal": "A",
"content": "$$36$$ "
}
],
[
{
"aoVal": "B",
"content": "$$37$$ "
}
],
[
{
"aoVal": "C",
"content": "$$38$$ "
}
],
[
{
"aoVal": "D",
"content": "$$39$$ "
}
],
[
{
"aoVal": "E",
"content": "$$41$$ "
}
]
] | [
"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 $1: 45$. If the height of the center is $1770$ feet, to find the height of the model, we divide by $45$ . That gives us $39.3$ which rounds to $39$ . Therefore, to the nearest whole number, the duplicate is (D) $39$ feet. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3853 | 1fd00e7f29c7486198a68ed3e6f15128 | [] | 1 | single_choice | Which choice is correct: $$3.5:2.55=$$~\uline{~~~~~~~~~~}~．$$4.6:1.15=$$~\uline{~~~~~~~~~~}~． | [
[
{
"aoVal": "A",
"content": "$$70:51$$，$$5:2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7:5$$，$$4:1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$16:11$$，$$4:1$$ "
}
],
[
{
"aoVal": "D",
"content": "$$70:51$$，$$4:1$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Ratios and Proportions"
] | [
"$$70:51$$，$$4:1$$． "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3858 | 2c867a7f68914e3cafd1dbf9b5f78978 | [] | 1 | single_choice | $$24\times 26\times 28\times 30\times 32=48\times 52\times 56\times 60\times $$． | [
[
{
"aoVal": "A",
"content": "$$2$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4$$ "
}
],
[
{
"aoVal": "C",
"content": "$$16$$ "
}
],
[
{
"aoVal": "D",
"content": "$$64$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"$$24\\times 26\\times 28\\times 30\\times 32=\\left(24\\times 26\\times 28\\times 30\\right)\\times \\left(2\\times 2\\times 2\\times 2\\right)\\times \\underline{2}$$. "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3859 | 8b530aa0896648939109881f8cf2d117 | [
"其它"
] | 1 | single_choice | How many positive integers can fill the blank in the sentence below? ``One positive integer is~\uline{~~~~~~~~~~}~more than twice another, and the sum of the two numbers is $28$.'' | [
[
{
"aoVal": "A",
"content": "$$6$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$8$$ "
}
],
[
{
"aoVal": "D",
"content": "$$9$$ "
}
],
[
{
"aoVal": "E",
"content": "$$10$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"NA "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3867 | b4f4df45f5ea410f9e2d89520ccb9904 | [] | 1 | single_choice | What is the value of $$5^{4}+5^{4}+5^{4}+5^{4}+5^{4}$$? | [
[
{
"aoVal": "A",
"content": "$$5^{4}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$5^{5}$$ "
}
],
[
{
"aoVal": "C",
"content": "$$5^{6}$$ "
}
],
[
{
"aoVal": "D",
"content": "$$5^{10}$$ "
}
],
[
{
"aoVal": "E",
"content": "$$5^{20}$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Power->Computing Powers"
] | [
"$$5^{1}\\cdot5^{4}=5^{1+4}=5^{5}$$, \\uline{Teacher should introduce the formula of $$a^{m}a^{n}=a^{m+n}$$}. "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3881 | fa75fd0dfe124ac7873a1559941601c8 | [
"其它"
] | 1 | single_choice | A soccer player computes his win ratio by dividing the number of matches he has won by the total number of matches he has played. At the start of a weekend, his win ratio is exactly~$0.5$. During the weekend, he plays five games, winning three and losing two. At the end of the weekend, his win ratio is greater than~$0.505$. What\textquotesingle s the largest number of matches he could\textquotesingle ve won before the weekend began? | [
[
{
"aoVal": "A",
"content": "$$46$$ "
}
],
[
{
"aoVal": "B",
"content": "$$47$$ "
}
],
[
{
"aoVal": "C",
"content": "$$48$$ "
}
],
[
{
"aoVal": "D",
"content": "$$92$$ "
}
],
[
{
"aoVal": "E",
"content": "$$96$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"Assume she won~$x$~games before the weekend, we obtain the inequality~$\\frac{x+3}{2x+5}\\gt0.505$~. Solve the inequality, we get~$x\\lt47.5$ "
] | B |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3884 | 59929c594f1a463fb610507dad2eb6a3 | [
"其它"
] | 1 | single_choice | Three positive integers are equally spaced on a number line. The middle number is $15$, and the largest number is $4$ times the smallest number. What is the smallest of these three numbers? | [
[
{
"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"
] | [
"NA "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3890 | 1ff4ecabfcef440cb4eaa0bddd64eddc | [
"其它"
] | 0 | single_choice | Eddie has a card. The number on the card is a neighbouring number of 10, but is not a neighbouring number of 12. What is the number on Eddie\textquotesingle s card?~\uline{~~~~~~~~~~}~ | [
[
{
"aoVal": "A",
"content": "$$11$$ "
}
],
[
{
"aoVal": "B",
"content": "$$10$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$8$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers"
] | [
"$$Omitted.$$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3899 | 507b57bec3784290a4eab6f868a40544 | [] | 1 | single_choice | If $$20\times 30$$ is divided by $$40$$, the remainder is． | [
[
{
"aoVal": "A",
"content": "$$30$$ "
}
],
[
{
"aoVal": "B",
"content": "$$15$$ "
}
],
[
{
"aoVal": "C",
"content": "$$10$$ "
}
],
[
{
"aoVal": "D",
"content": "$$0$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"$$(20\\times 30)\\div 40=600\\div 40=15$$; remainder $$=0$$. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3901 | 709513e96abd4ded852505b21b2787a8 | [
"其它"
] | 0 | single_choice | Which of the following numbers\textquotesingle{} value does not change after removing all ``$$0$$''s ． | [
[
{
"aoVal": "A",
"content": "$$90.221$$ "
}
],
[
{
"aoVal": "B",
"content": "$$4.106$$ "
}
],
[
{
"aoVal": "C",
"content": "$$22.990$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Decimals->Basic Understanding of Decimals"
] | [
"Only $$22.990$$\\textquotesingle s $$\"0\"$$ can be removed without causing other digits to change their places from the original number. "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3909 | 7e6813975d7945cd8406d39daaca3a63 | [
"其它"
] | 1 | single_choice | Mary cooked $9$ cakes. She cut two of them into $3$ pieces. How many cakes does she have now? | [
[
{
"aoVal": "A",
"content": "$$11$$ "
}
],
[
{
"aoVal": "B",
"content": "$$12$$ "
}
],
[
{
"aoVal": "C",
"content": "$$13$$ "
}
],
[
{
"aoVal": "D",
"content": "$$14$$ "
}
],
[
{
"aoVal": "E",
"content": "$$15$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"$9 - 2 + 2 \\times 3 = 13$ "
] | C |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3916 | 20107026968540f2b243a533c84d1cb2 | [
"其它"
] | 2 | single_choice | Camila writes down five positive integers. The unique mode of these integers is 2 greater than their median, and the median is 2 greater than their arithmetic mean. What is the least possible value for the mode? | [
[
{
"aoVal": "A",
"content": "$$5$$ "
}
],
[
{
"aoVal": "B",
"content": "$$7$$ "
}
],
[
{
"aoVal": "C",
"content": "$$9$$ "
}
],
[
{
"aoVal": "D",
"content": "$$11$$ "
}
],
[
{
"aoVal": "E",
"content": "$$13$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Fractions"
] | [
"1，$$3$$，$$9$$，$$11$$，11 "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3917 | 476c3cb23fbc4074b1cda64d7e13446b | [] | 2 | single_choice | If $$a$$@$$b=\frac {a\times b}{a+b}$$ for positive integers $$a$$ and $$b$$ , what is $$5$$@$$10$$? | [
[
{
"aoVal": "A",
"content": "$$\\frac 3{10}$$ "
}
],
[
{
"aoVal": "B",
"content": "$$1$$ "
}
],
[
{
"aoVal": "C",
"content": "$$2$$ "
}
],
[
{
"aoVal": "D",
"content": "$$\\frac {10}3$$ "
}
],
[
{
"aoVal": "E",
"content": "$$50$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Operations with New Definition->Operating Directly->Ordinary Type"
] | [
"Substitute $$a=5$$ and $$b=10$$ into the expression for $$a$$@$$b$$ to get: $$5$$@$$10=\\frac {5\\times 10}{5+10}=\\frac {50}{15}=\\frac {10}3$$. Thus, the answer choice $$\\frac {10}3$$ is correct. "
] | D |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3921 | a27f5d3e5c0249bca175128c973b0b88 | [
"其它"
] | 2 | single_choice | Is it true that If an odd function has zero in its domain, then it must pass through the origin. (Hint: $0$ is in the domain, we can apply $f(-x) = -f(x)$ at $x=0$) | [
[
{
"aoVal": "A",
"content": "True "
}
],
[
{
"aoVal": "B",
"content": "False "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules"
] | [
"By definition of an odd function, $f(-x) = -f(x)$ for all $x$ in the domain. Since $x = 0$ is in the domain, we have $f(-0) = -f(0)$, which means $f(0) = -f(0)$, or $2f(0) = 0$, and thus $f(0) = 0$ (this represents the origin). "
] | A |
prime_math_competition_en_single_choice_8K_dev | 2023-07-07T00:00:00 | 3923 | 8b613d7ee3634ec9aae24ecdf9b9768d | [] | 1 | single_choice | Dividing a certain two-digit number by $$10$$ leaves a remainder of $$9$$, and dividing it by $$9$$ leaves a remainder of $$8$$. The sum of the number\textquotesingle s two digits is． | [
[
{
"aoVal": "A",
"content": "$$7$$ "
}
],
[
{
"aoVal": "B",
"content": "$$9$$ "
}
],
[
{
"aoVal": "C",
"content": "$$13$$ "
}
],
[
{
"aoVal": "D",
"content": "$$17$$ "
}
]
] | [
"Overseas Competition->Knowledge Point->Calculation Modules->Whole Numbers->Whole Numbers Multiplication and Division"
] | [
"Dividing a certain two-digit number by $$10$$ leaves a remainder of $$9$$, so it is $$19$$, $$29$$, $$39$$, $$49$$, $$59$$, $$69$$, $$79$$, $$89$$, or $$99$$. The only number listed with remainder $$8$$ when divided by $$9$$ is $$89$$, so the number is $$89$$ and $$8+9=17$$. "
] | D |
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