data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "When two fair 12-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that both the two-digit number $ab$ (where $a$ and $b$ are digits) and each of $a$ and $b$ individually are divisible by 4?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 38800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When two fair 6-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 3?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac19"
} | {
"index": 38801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When two fair 8-sided dice (labeled from 1 to 8) are tossed, the numbers \\(a\\) and \\(b\\) are obtained. What is the probability that the two-digit number \\(ab\\) (where \\(a\\) and \\(b\\) are digits) and both \\(a\\) and \\(b\\) are divisible by 4?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 38802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When two fair dice are rolled once each, what is the probability that one of the upward-facing numbers is 2, given that the two numbers are not the same?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 38803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When two fair dice are rolled, the probability that the sum of the numbers facing up is $5$ equals \\_\\_\\_\\_\\_\\_．"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 38804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When two fair dice are thrown once each, what is the probability that the upward-facing numbers are different and that one of them shows a 3?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"index": 38805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When two fair dice are thrown, the numbers obtained are $a$ and $b$, respectively. Express the probability that the slope $k$ of the line $bx+ay=1$ is greater than or equal to $-\\dfrac{2}{5}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{6}"
} | {
"index": 38806,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When two numbers are sequentially and randomly picked from the set {1, 2, 3, 4}, what is the probability that the product of the two picked numbers is even?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 38807,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When two positive integers are multiplied, the result is 24. When these two integers are added, the result is 11. What is the result when the smaller integer is subtracted from the larger integer?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 38808,
"solution": "The positive divisor pairs of 24 are: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Of these, the pair whose sum is 11 is 3 and 8. The difference between these two integers is $8 - 3 = 5$."
} |
DeepScaleR | [
{
"role": "user",
"content": "When two standard dice and one 8-sided die (with faces showing numbers from 1 to 8) are tossed, the numbers \\(a, b, c\\) are obtained respectively where \\(a\\) and \\(b\\) are from the standard dice and \\(c\\) is from the 8-sided die. Find the probability that \\((a-1)(b-1)(... | math | {
"style": "rule",
"ground_truth": "\\frac{175}{288}"
} | {
"index": 38809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When we say that Ray is climbing up the stairs $m$ at a time, we mean that he starts on the floor (step $0$) then jumps to step $m$ and then to $2m$ and so on until the number of steps to the top is less than $m$. Ray climbs up a flight of stairs of $n$ steps in two ways. When ... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 38810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When written out in full, the number \\((10^{2020}+2020)^{2}\\) has 4041 digits. What is the sum of the digits of this 4041-digit number?\nA) 9 \nB) 17 \nC) 25 \nD) 2048 \nE) 4041"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 38811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When you simplify $\\left[ \\sqrt [3]{\\sqrt [6]{a^9}} \\right]^4\\left[ \\sqrt [6]{\\sqrt [3]{a^9}} \\right]^4$, the result is:"
}
] | math | {
"style": "rule",
"ground_truth": "a^4"
} | {
"index": 38812,
"solution": "1. **Simplify each term inside the brackets:**\n - For the first term $\\sqrt[3]{\\sqrt[6]{a^9}}$, we simplify the exponents:\n \\[\n \\sqrt[3]{\\sqrt[6]{a^9}} = a^{\\frac{9}{6} \\cdot \\frac{1}{3}} = a^{\\frac{9}{18}} = a^{\\frac{1}{2}}\n \\]\n - For the second term $... |
DeepScaleR | [
{
"role": "user",
"content": "When you simplify $\\sqrt[3]{24a^4b^6c^{11}}$, what is the sum of the exponents of the variables that are outside the radical?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 38813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When young fishermen were asked how many fish each of them caught, the first one replied, \"I caught half the number of fish that my friend caught, plus 10 fish.\" The second one said, \"And I caught as many as my friend, plus 20 fish.\" How many fish did the fishermen catch?"
... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 38814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Where is Tony's cheese? The bottom right corner of a 3x3 grid :0 We’re doing weekly posts of problems until the start of the in-person tournament on April 13th (Sign up **[here](https://www.stanfordmathtournament.com/competitions/smt-2024)**!) Registrations for our synchronous ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"index": 38815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity?"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 38816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which digit is represented by $\\Theta$ if $252/\\Theta=\\underline{3\\Theta}+\\Theta$, where $\\underline{3\\Theta}$ represents a two-digit number with $3$ in the tens digit and $\\Theta$ in the ones digit?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 38817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which digit of $.12345$, when changed to $9$, gives the largest number?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 38818,
"solution": "To determine which digit change in $.12345$ to $9$ results in the largest number, we need to consider the positional value of each digit in the decimal.\n\n1. **Understanding Decimal Places**:\n - The decimal $.12345$ can be expanded as:\n \\[\n 0.12345 = 0.1 + 0.02 + 0.003 ... |
DeepScaleR | [
{
"role": "user",
"content": "Which digit will appear in the 534th place after the decimal point in the decimal representation of $\\frac{5}{13}$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 38819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which five-digit numbers are greater in quantity: those not divisible by 5 or those whose first and second digits from the left are not a five?"
}
] | math | {
"style": "rule",
"ground_truth": "72000"
} | {
"index": 38820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which fractions $ \\dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\\sqrt{2}$ (without using any table)."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt2\\approx1+\\frac1{2+\\frac1{2+\\frac1{2+\\frac1{2+\\frac12}}}}=\\boxed{\\frac{99}{70}}"
} | {
"index": 38821,
"solution": "\nWe are tasked with finding the fraction \\(\\frac{p}{q}\\), where \\( p, q \\) are positive integers less than 100, that is closest to \\(\\sqrt{2}\\). Additionally, we aim to determine how many digits after the decimal point coincide between this fraction and \\(\\sqrt{2}\\).\n\n##... |
DeepScaleR | [
{
"role": "user",
"content": "Which integer is closest to the cube root of 100?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 38822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which integers from 1 to 60,000 (inclusive) are more numerous and by how much: those containing only even digits in their representation, or those containing only odd digits in their representation?"
}
] | math | {
"style": "rule",
"ground_truth": "780"
} | {
"index": 38823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which letter will go in the square marked with $*$ in the grid where each of the letters A, B, C, D, and E appears exactly once in each row and column?"
}
] | math | {
"style": "rule",
"ground_truth": "B"
} | {
"index": 38824,
"solution": "Each letter A, B, C, D, E appears exactly once in each column and each row. The entry in the first column, second row cannot be A or E or B (the entries already present in that column) and cannot be C or A (the entries already present in that row). Therefore, the entry in the first co... |
DeepScaleR | [
{
"role": "user",
"content": "Which number appears most frequently in the second position when listing the winning numbers of a lottery draw in ascending order?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 38825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which number from the set $\\{1,2,3,4,5,6,7,8,9,10,11\\}$ must be removed so that the mean (average) of the numbers remaining in the set is 6.1?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 38826,
"solution": "The original set contains 11 elements whose sum is 66. When one number is removed, there will be 10 elements in the set. For the average of these elements to be 6.1, their sum must be $10 \\times 6.1=61$. Since the sum of the original 11 elements is 66 and the sum of the remaining 10 ... |
DeepScaleR | [
{
"role": "user",
"content": "Which number has the property that if it is multiplied by $1, 2, 3, 4, 5$, or $6$, the resulting product contains only the digits that appear in the original number?"
}
] | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"index": 38827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which number in the array below is both the largest in its column and the smallest in its row? (Columns go up and down, rows go right and left.)\n\\[\\begin{tabular}{ccccc} 10 & 6 & 4 & 3 & 2 \\\\ 11 & 7 & 14 & 10 & 8 \\\\ 8 & 3 & 4 & 5 & 9 \\\\ 13 & 4 & 15 & 12 & 1 \\\\ 8 & 2... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 38828,
"solution": "To solve this problem, we need to identify the number that is both the largest in its column and the smallest in its row from the given array. We will proceed by first finding the largest number in each column and then checking if any of these numbers are also the smallest in their re... |
DeepScaleR | [
{
"role": "user",
"content": "Which number is closest to \\(-3.4\\) on a number line?"
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 38829,
"solution": "On a number line, \\(-3.4\\) is between \\(-4\\) and \\(-3\\). This means that \\(-3.4\\) is closer to \\(-3\\) than to \\(-4\\), and so the answer is \\(-3\\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "Which number is greater than 0.7?"
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"index": 38830,
"solution": "Each of \\( 0.07, -0.41, 0.35, \\) and \\(-0.9\\) is less than 0.7. The number 0.8 is greater than 0.7."
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the cones below can be formed from a $252^{\\circ}$ sector of a circle of radius 10 by aligning the two straight sides?\n\n[asy]\ndraw((5.8,8.1)..(-10,0)--(0,0)--(3.1,-9.5)..cycle);\nlabel(\"10\",(-5,0),S);\nlabel(\"$252^{\\circ}$\",(0,0),NE);\n[/asy]\n\nA. base radius... | math | {
"style": "rule",
"ground_truth": "C"
} | {
"index": 38831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following could NOT be the lengths of the external diagonals of a right regular prism [a \"box\"]? (An $\\textit{external diagonal}$ is a diagonal of one of the rectangular faces of the box.)\n$\\text{(A) }\\{4,5,6\\} \\quad \\text{(B) } \\{4,5,7\\} \\quad \\text{... | math | {
"style": "rule",
"ground_truth": "\\{4,5,7\\}"
} | {
"index": 38832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following divisions is not equal to a whole number: $\\frac{60}{12}$, $\\frac{60}{8}$, $\\frac{60}{5}$, $\\frac{60}{4}$, $\\frac{60}{3}$?"
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"index": 38833,
"solution": "Since $\\frac{60}{8}=60 \\div 8=7.5$, then this choice is not equal to a whole number. Note as well that $\\frac{60}{12}=5, \\frac{60}{5}=12, \\frac{60}{4}=15$, and $\\frac{60}{3}=20$ are all whole numbers."
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following expressions is equal to an odd integer for every integer $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "2017+2n"
} | {
"index": 38834,
"solution": "When $n=1$, the values of the five expressions are 2014, 2018, 2017, 2018, 2019. \n When $n=2$, the values of the five expressions are 2011, 2019, 4034, 2021, 2021. \n Only the fifth expression $(2017+2n)$ is odd for both of these choices of $n$, so this must be the correct answer. \n... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following expressions is not equivalent to $3x + 6$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}(3x) + \\frac{2}{3}(9)"
} | {
"index": 38835,
"solution": "We look at each of the five choices: (A) $3(x + 2) = 3x + 6$ (B) $\\frac{-9x - 18}{-3} = \\frac{-9x}{-3} + \\frac{-18}{-3} = 3x + 6$ (C) $\\frac{1}{3}(3x) + \\frac{2}{3}(9) = x + 6$ (D) $\\frac{1}{3}(9x + 18) = 3x + 6$ (E) $3x - 2(-3) = 3x + (-2)(-3) = 3x + 6$ The expression that is n... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following five values of \\( n \\) is a counterexample to the statement: For a positive integer \\( n \\), at least one of \\( 6n-1 \\) and \\( 6n+1 \\) is prime?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 38836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following fractions has the greatest value: $\\frac{3}{10}$, $\\frac{4}{7}$, $\\frac{5}{23}$, $\\frac{2}{3}$, $\\frac{1}{2}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 38837,
"solution": "The fractions $\\frac{3}{10}$ and $\\frac{5}{23}$ are each less than $\\frac{1}{2}$ (which is choice $(E)$) so cannot be the greatest among the choices. The fractions $\\frac{4}{7}$ and $\\frac{2}{3}$ are each greater than $\\frac{1}{2}$, so $\\frac{1}{2}$ cannot be the greatest among... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following integers cannot be written as a product of two integers, each greater than 1: 6, 27, 53, 39, 77?"
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 38838,
"solution": "We note that $6=2 \times 3$ and $27=3 \times 9$ and $39=3 \times 13$ and $77=7 \times 11$, which means that each of $6,27,39$, and 77 can be written as the product of two integers, each greater than 1. Thus, 53 must be the integer that cannot be written in this way. We can check that ... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following integers is equal to a perfect square: $2^{3}$, $3^{5}$, $4^{7}$, $5^{9}$, $6^{11}$?"
}
] | math | {
"style": "rule",
"ground_truth": "4^{7}"
} | {
"index": 38839,
"solution": "Since $4 = 2^{2}$, then $4^{7} = (2^{2})^{7} = 2^{14} = (2^{7})^{2}$, which means that $4^{7}$ is a perfect square. We can check, for example using a calculator, that the square root of each of the other four choices is not an integer, and so each of these four choices cannot be expre... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following is a possible value of $x$ if given two different numbers on a number line, the number to the right is greater than the number to the left, and the positions of $x, x^{3}$ and $x^{2}$ are marked on a number line?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"index": 38840,
"solution": "From the number line shown, we see that $x<x^{3}<x^{2}$. \n If $x>1$, then successive powers of $x$ are increasing (that is, $x<x^{2}<x^{3}$ ). \n Since this is not the case, then it is not true that $x>1$. \n If $x=0$ or $x=1$, then successive powers of $x$ are equal. This is not the... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following is closest in value to 7?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{50}"
} | {
"index": 38841,
"solution": "We note that $7=\\sqrt{49}$ and that $\\sqrt{40}<\\sqrt{49}<\\sqrt{50}<\\sqrt{60}<\\sqrt{70}<\\sqrt{80}$. \n This means that $\\sqrt{40}$ or $\\sqrt{50}$ is the closest to 7 of the given choices. \n Since $\\sqrt{40} \\approx 6.32$ and $\\sqrt{50} \\approx 7.07$, then $\\sqrt{50}$ is ... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following is equal to $110 \\%$ of 500?"
}
] | math | {
"style": "rule",
"ground_truth": "550"
} | {
"index": 38842,
"solution": "Solution 1: $10 \\%$ of 500 is $\\frac{1}{10}$ of 500, which equals 50. Thus, $110 \\%$ of 500 equals $500+50$, which equals 550. Solution 2: $110 \\%$ of 500 is equal to $\\frac{110}{100} \\times 500=110 \\times 5=550$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following is equal to $2017 - \\frac{1}{2017}$?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{2018 \\times 2016}{2017}$"
} | {
"index": 38843,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following is equal to $9^{4}$?"
}
] | math | {
"style": "rule",
"ground_truth": "3^{8}"
} | {
"index": 38844,
"solution": "Since $9=3 \\times 3$, then $9^{4}=(3 \\times 3)^{4}=3^{4} \\times 3^{4}=3^{8}$. Alternatively, we can note that $9^{4}=9 \\times 9 \\times 9 \\times 9=(3 \\times 3) \\times(3 \\times 3) \\times(3 \\times 3) \\times(3 \\times 3)=3^{8}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following is equal to \\(\\frac{4}{5}\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"index": 38845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following lines, when drawn together with the $x$-axis and the $y$-axis, encloses an isosceles triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "y=-x+4"
} | {
"index": 38846,
"solution": "Since the triangle will include the two axes, then the triangle will have a right angle. For the triangle to be isosceles, the other two angles must be $45^{\\circ}$. For a line to make an angle of $45^{\\circ}$ with both axes, it must have slope 1 or -1. Of the given possibilities, t... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following multiplication expressions has a product that is a multiple of 54? (Fill in the serial number). \n$261 \\times 345$ \n$234 \\times 345$ \n$256 \\times 345$ \n$562 \\times 345$"
}
] | math | {
"style": "rule",
"ground_truth": "$234 \\times 345$"
} | {
"index": 38847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \\(12^2\\) and \\(13^2\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"index": 38848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \\(12^{2}\\) and \\(13^{2}\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"index": 38849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following numbers is closest to 1: $\frac{11}{10}$, $\frac{111}{100}$, 1.101, $\frac{1111}{1000}$, 1.011?"
}
] | math | {
"style": "rule",
"ground_truth": "1.011"
} | {
"index": 38850,
"solution": "When we convert each of the possible answers to a decimal, we obtain 1.1, 1.11, 1.101, 1.111, and 1.011. Since the last of these is the only one greater than 1 and less than 1.1, it is closest to 1."
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following numbers is equal to 33 million?"
}
] | math | {
"style": "rule",
"ground_truth": "33000000"
} | {
"index": 38851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following numbers is less than $\\frac{1}{20}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{25}"
} | {
"index": 38852,
"solution": "If $0<a<20$, then $\\frac{1}{a}>\\frac{1}{20}$. Therefore, $\\frac{1}{15}>\\frac{1}{20}$ and $\\frac{1}{10}>\\frac{1}{20}$. Also, $\\frac{1}{20}=0.05$ which is less than both 0.5 and 0.055. Lastly, $\\frac{1}{20}>\\frac{1}{25}$ since $0<20<25$. Therefore, $\\frac{1}{25}$ is the only o... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following numbers is not an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{2014}{4}$"
} | {
"index": 38853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following points is farthest from the origin: $(0,5)$, $(1,2)$, $(3,-4)$, $(6,0)$, $(-1,-2)?$"
}
] | math | {
"style": "rule",
"ground_truth": "(6,0)"
} | {
"index": 38854,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following quantities is the largest? (Write $A$, $B$, or $C$.) \\[ A.\\ \\ \\frac{2006}{2005}+\\frac{2006}{2007} \\qquad B.\\ \\ \\frac{2006}{2007}+\\frac{2008}{2007} \\qquad C.\\ \\ \\frac{2007}{2006}+\\frac{2007}{2008} \\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\text{A}"
} | {
"index": 38855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following words has the largest value, given that the first five letters of the alphabet are assigned the values $A=1, B=2, C=3, D=4, E=5$?"
}
] | math | {
"style": "rule",
"ground_truth": "BEE"
} | {
"index": 38856,
"solution": "We calculate the value of each of the five words as follows: - The value of $B A D$ is $2+1+4=7$ - The value of $C A B$ is $3+1+2=6$ - The value of $D A D$ is $4+1+4=9$ - The value of $B E E$ is $2+5+5=12$ - The value of $B E D$ is $2+5+4=11$. Of these, the word with the largest value... |
DeepScaleR | [
{
"role": "user",
"content": "Which of the following, when rounded to the nearest hundredth, does not round to 65.14?\nA) 65.141\nB) 65.138\nC) 65.1339999\nD) 65.1401\nE) 65.14444\n\nYour answer should be a letter: A, B, C, D, or E."
}
] | math | {
"style": "rule",
"ground_truth": "C"
} | {
"index": 38857,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which of the followings gives the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 38858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which one of the following is not equivalent to $0.000000375$?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{3}{8} \\times 10^{-7}$"
} | {
"index": 38859,
"solution": "To find which option is not equivalent to $0.000000375$, we first convert $0.000000375$ into scientific notation:\n\n1. **Convert to Scientific Notation:**\n \\[\n 0.000000375 = 3.75 \\times 10^{-7}\n \\]\n\n2. **Evaluate Each Option:**\n - **Option (A) $3.75 \\times 10^{-7}$:... |
DeepScaleR | [
{
"role": "user",
"content": "Which one of the following points is not on the graph of $y=\\dfrac{x}{x+1}$?"
}
] | math | {
"style": "rule",
"ground_truth": "(-1,1)"
} | {
"index": 38860,
"solution": "To determine which point is not on the graph of the function $y = \\frac{x}{x+1}$, we need to substitute each point's $x$-coordinate into the function and check if the corresponding $y$-coordinate matches.\n\n1. **Point (A) $(0,0)$:**\n \\[\n y = \\frac{0}{0+1} = \\frac{0}{1} = 0\... |
DeepScaleR | [
{
"role": "user",
"content": "Which one satisfies $n^{29} \\equiv 7 \\pmod {65}$?"
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 38861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which pair of numbers does NOT have a product equal to $36$?"
}
] | math | {
"style": "rule",
"ground_truth": "{\\frac{1}{2},-72}"
} | {
"index": 38862,
"solution": "To solve this problem, we need to calculate the product of each pair of numbers given in the options and check which pair does not have a product equal to $36$.\n\n1. **Option A: $\\{-4,-9\\}$**\n \\[\n (-4) \\times (-9) = 36\n \\]\n Here, the product is $36$.\n\n2. **Option B... |
DeepScaleR | [
{
"role": "user",
"content": "Which terms must be removed from the sum\n$\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{6}+\\frac{1}{8}+\\frac{1}{10}+\\frac{1}{12}$\nif the sum of the remaining terms is to equal $1$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8} \\text{ and } \\frac{1}{10}"
} | {
"index": 38863,
"solution": "1. **Calculate the total sum of all terms**: We start by finding the sum of all the terms given in the problem:\n \\[\n \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{6} + \\frac{1}{8} + \\frac{1}{10} + \\frac{1}{12}\n \\]\n To add these fractions, we need a common denominator. The l... |
DeepScaleR | [
{
"role": "user",
"content": "Which triplet of numbers has a sum NOT equal to 1?"
}
] | math | {
"style": "rule",
"ground_truth": "1.1 + (-2.1) + 1.0"
} | {
"index": 38864,
"solution": "To find which triplet of numbers has a sum NOT equal to 1, we will add the numbers in each triplet and check the result.\n\n1. **Triplet (A):** $(1/2, 1/3, 1/6)$\n \\[\n \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{6} = \\frac{3}{6} + \\frac{2}{6} + \\frac{1}{6} = \\frac{6}{6} = 1\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Which type of conic section is described by the equation \\[(x+5)^2 = (4y-3)^2 - 140?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of the above."
}
] | math | {
"style": "rule",
"ground_truth": "(\\text{H})"
} | {
"index": 38865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which type of conic section is described by the equation \\[\\sqrt{x^2 + (y-1)^2} + \\sqrt{(x-5)^2 + (y+3)^2} = 10?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of the above."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(E)}"
} | {
"index": 38866,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Which type of conic section is described by the equation \\[|y+5| = \\sqrt{(x-2)^2 + y^2}?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of the above."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(P)}"
} | {
"index": 38867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles p... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 38868,
"solution": "1. **Calculate the time to reach the shore**: \n Steve rows at a speed of 4 miles per hour. Since they are 1 mile from shore, the time \\( t \\) it takes to reach the shore is given by:\n \\[\n t = \\frac{\\text{distance}}{\\text{speed}} = \\frac{1 \\text{ mile}}{4 \\text{ miles... |
DeepScaleR | [
{
"role": "user",
"content": "While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After five minutes, how likely is Sherry to be one edge away from where she started?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{16}"
} | {
"index": 38869,
"solution": "Let the starting vertex be the 'bottom' one. Then there is a 'top' vertex, and 4 'middle' ones. If $p(n)$ is the probability that Sherry is on a middle vertex after $n$ minutes, $p(0)=0$, $p(n+1)=(1-p(n))+p(n) \\cdot \\frac{1}{2}$. This recurrence gives us the following equations. $$\... |
DeepScaleR | [
{
"role": "user",
"content": "While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$ ."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 38870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While eating out, Mike and Joe each tipped their server $2$ dollars. Mike tipped $10\\%$ of his bill and Joe tipped $20\\%$ of his bill. What was the difference, in dollars between their bills?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 38871,
"solution": "1. **Define the variables:**\n Let $m$ represent Mike's bill and $j$ represent Joe's bill.\n\n2. **Set up the equations based on the given percentages:**\n - Mike tipped $10\\%$ of his bill, which is $2$ dollars. Therefore, we have the equation:\n \\[\n \\frac{10}{100}m = ... |
DeepScaleR | [
{
"role": "user",
"content": "While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\\lfloor 1000a\\rfloor$ ."
}
] | math | {
"style": "rule",
"ground_truth": "941"
} | {
"index": 38872,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $\\$14$ each, $4$-pound rocks worth $\\$11$ each, and $1$-pound rocks worth $\\$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 38873,
"solution": "To determine the maximum value of rocks Carl can carry, we first calculate the value per pound for each type of rock:\n- **5-pound rocks:** $\\frac{\\$14}{5} = \\$2.80$ per pound\n- **4-pound rocks:** $\\frac{\\$11}{4} = \\$2.75$ per pound\n- **1-pound rocks:** $\\frac{\\$2}{1} = \\$2... |
DeepScaleR | [
{
"role": "user",
"content": "While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the s... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"index": 38874,
"solution": "Note that $x$ degrees is equal to $\\frac{\\pi x}{180}$ radians. Also, for $\\alpha \\in \\left[0 , \\frac{\\pi}{2} \\right]$, the two least positive angles $\\theta > \\alpha$ such that $\\sin{\\theta} = \\sin{\\alpha}$ are $\\theta = \\pi-\\alpha$, and $\\theta = 2\\pi + \\alpha$.\n... |
DeepScaleR | [
{
"role": "user",
"content": "While hiking in a valley, a hiker first walks 15 miles north, then 8 miles east, then 9 miles south, and finally 2 miles east. How far is the hiker from the starting point after completing these movements?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{34}"
} | {
"index": 38875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While moving down an escalator, Walker counted 50 steps. Trotman, who moved three times faster than Walker, counted 75 steps. If the escalator were to stop, how many steps could be counted on its visible part? It is assumed that both individuals moved at a constant speed and th... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 38876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While one lion cub, who is 6 minutes away from the water hole, heads there, another, having already quenched its thirst, heads back along the same road 1.5 times faster than the first. At the same time, a turtle starts towards the water hole along the same road, being 32 minute... | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"index": 38877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While standing in line to buy concert tickets, Kit moved 60 feet closer to the ticket window over a period of 30 minutes. At this rate, how many minutes will it take her to move the remaining 70 yards to the ticket window?"
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 38878,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$ , there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$ . Given that $|a-b| = 2\\sqrt{3... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 38879,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While waiting at the post office, Lena moved 40 feet closer to the counter over a period of 20 minutes. At this rate, how many minutes will it take her to move the remaining 100 meters to the counter?"
}
] | math | {
"style": "rule",
"ground_truth": "164.042"
} | {
"index": 38880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw 3 squares $\\square_{1}, \\square_{2}, \\square_{3}$ on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set $\\{1,2,3,4,5\\}$ such that no ... | math | {
"style": "rule",
"ground_truth": "\\[ 541 \\]"
} | {
"index": 38881,
"solution": "Relabel $a_{1}, a_{2}, a_{3}$ as $a, b, c$. This is minimized at $x=\\frac{-b}{2 a}$, so $M=c-\\frac{b^{2}}{4 a}$. If in the end $a=5$ or $b \\in\\{1,2\\}$, then $\\frac{b^{2}}{4 a} \\leq 1$ and $M \\geq 0$. The only way for Ana to block this is to set $b=5$, which will be optimal if ... |
DeepScaleR | [
{
"role": "user",
"content": "While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a 2020-term strictly increasing geometric sequence with an integer common ratio $r$. Every second, Alesha erases the two smallest terms on ... | math | {
"style": "rule",
"ground_truth": "\\[\n\\boxed{2018}\n\\]"
} | {
"index": 38882,
"solution": "Because we only care about when the ratio of $A$ to $B$ is an integer, the value of the first term in $S$ does not matter. Let the initial term in $S$ be 1 . Then, we can write $S$ as $1, r, r^{2}, \\ldots, r^{2019}$. Because all terms are in terms of $r$, we can write $A=r^{a}$ and $... |
DeepScaleR | [
{
"role": "user",
"content": "While walking on a plane surface, a traveler first headed 18 miles north, then 11 miles west, then 6 miles south and finally 6 miles east. How many miles from the starting point was the traveler after these four legs of the journey?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 38883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While watching a circus show, I counted out the number of acrobats and elephants. I counted 40 legs and 15 heads. How many acrobats did I see in the show?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 38884,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other be... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 38885,
"solution": "We proceed with casework based on the person who sits first after the break.\n$\\textbf{Case 1:}$ A is first. Then the first three people in the row can be ACE, ACF, ADB, ADF, AEB, AEC, AFB, AFC, or AFD, which yield 2, 1, 2, 2, 1, 2, 0, 1, and 1 possible configurations, respectively, ... |
DeepScaleR | [
{
"role": "user",
"content": "Whole numbers that read the same from left to right and right to left are called symmetrical. For example, the number 513315 is symmetrical, whereas 513325 is not. How many six-digit symmetrical numbers exist such that adding 110 to them leaves them symmetrical?"
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 38886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Whole numbers whose decimal representation reads the same from left to right as from right to left are called symmetrical. For example, the number 5134315 is symmetrical, while 5134415 is not. How many seven-digit symmetrical numbers exist such that adding 1100 to them leaves t... | math | {
"style": "rule",
"ground_truth": "810"
} | {
"index": 38887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Will stands at a point \\(P\\) on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of \\(n^{\\circ}\\) and \\((n+1)^{\\circ}\\) with the tangent at \\(P\\), where \\(n\\) is a positive integer less than 90. ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 38888,
"solution": "Note that we want the path drawn out by the lasers to come back to \\(P\\) in as few steps as possible. Observe that if a laser is fired with an angle of \\(n\\) degrees from the tangent, then the number of points it creates on the circle is \\(\\frac{180}{\\operatorname{gcd}(180, n)}... |
DeepScaleR | [
{
"role": "user",
"content": "William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\\circ$ decline for $1$ kilometer. If William travels at $... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 38889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "William is taking the 25-question, multiple choice American Mathematics Competition. Each question has five answer choices. William guesses random answers for the last four questions. What is the probability that he will get at least one of these final four questions right?"
... | math | {
"style": "rule",
"ground_truth": "\\frac{369}{625}"
} | {
"index": 38890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Winnie has 17 red balloons, 33 white balloons, 65 green balloons, and 83 chartreuse balloons. With no regard to color, Winnie gives each of her 8 nearest and dearest friends some balloons. Each friend gets the same number of balloons and she hands out as many balloons as poss... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 38891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Winnie has 45 cherry lollipops, 116 wintergreen lollipops, 4 grape lollipops, and 229 shrimp cocktail lollipops. With no regard to flavor, Winnie gives each of her 11 nearest and dearest friends some lollipops. Each friend gets the same number of lollipops and she hands out a... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 38892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Winnie the Pooh decided to give Piglet a birthday cake in the shape of a regular hexagon. On his way, he got hungry and cut off 6 pieces from the cake, each containing one vertex and one-third of a side of the hexagon (see the illustration). As a result, he gave Piglet a cake w... | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"index": 38893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Winnie wrote all the integers from 1 to 2017 inclusive on a board. She then erased all the integers that are a multiple of 3. Next, she reinstated all those integers that are a multiple of 6. Finally, she erased all integers then on the board which are a multiple of 27. Of the ... | math | {
"style": "rule",
"ground_truth": "373"
} | {
"index": 38894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Winnie-the-Pooh eats 3 cans of condensed milk and a jar of honey in 25 minutes, while Piglet eats them in 55 minutes. Pooh eats one can of condensed milk and 3 jars of honey in 35 minutes, while Piglet eats them in 1 hour 25 minutes. How long will it take for them to eat 6 cans... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 38895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in the passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the revote was $\\frac{12}{11}$ of the... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 38896,
"solution": "Let $x$ be the number of members who voted for the bill the first time, and $y$ be the number of members who voted against it the first time. Since the total number of members voting is $400$, we have:\n\\[ x + y = 400 \\]\n\nThe bill was defeated the first time, so $y > x$. Let the m... |
DeepScaleR | [
{
"role": "user",
"content": "With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team.\n\nThere are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, o... | math | {
"style": "rule",
"ground_truth": "1881"
} | {
"index": 38897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "With all angles measured in degrees, calculate the product $\\prod_{k=1}^{30} \\csc^2(3k)^\\circ$, and express the result as $m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 38898,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "With all angles measured in degrees, consider the product $\\prod_{k=1}^{22} \\sec^2(4k)^\\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 38899,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.