data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\frac{2021}{2020} - \\frac{2020}{2021}$ is equal to the fraction $\\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is 1. What is $p?$"
}
] | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"index": 32300,
"solution": "1. Let $a = 2020$. We start by rewriting the given expression using $a$:\n \\[\n \\frac{2021}{2020} - \\frac{2020}{2021} = \\frac{a+1}{a} - \\frac{a}{a+1}\n \\]\n\n2. To combine these fractions, find a common denominator, which is $a(a+1)$:\n \\[\n \\frac{(a+1)^2 - a^2}{a(a+... |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\frac{4k+8}{4}$ simplifies to an expression of the form $ak+b$ where $a$ and $b$ are integers. Find $\\frac{a}{b}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 32301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\frac{\\cos 85^{\\circ} + \\sin 25^{\\circ} \\cos 30^{\\circ}}{\\cos 25^{\\circ}}$ simplify to a specific value."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 32302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\frac{\\sqrt{3}\\tan 12^{\\circ} - 3}{(4\\cos^2 12^{\\circ} - 2)\\sin 12^{\\circ}}$ equals \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-4\\sqrt{3}"
} | {
"index": 32303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\frac{k^{2}}{1.001^{k}}$ reaches its maximum value with which natural number $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "2001"
} | {
"index": 32304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\frac{x-3}{4x}$ is equal to zero for what value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 32305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\frac{x^2-3x+2}{x^2-5x+6} \\div \\frac{x^2-5x+4}{x^2-7x+12}$, when simplified is:"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 32306,
"solution": "1. **Factorize each quadratic expression** in the given complex fraction:\n - Numerator of the first fraction: $x^2 - 3x + 2 = (x-2)(x-1)$.\n - Denominator of the first fraction: $x^2 - 5x + 6 = (x-3)(x-2)$.\n - Numerator of the second fraction: $x^2 - 5x + 4 = (x-4)(x-1)$.\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\left(1+\\frac{1}{2}\\right)\\left(1+\\frac{1}{3}\\right)\\left(1+\\frac{1}{4}\\right)\\left(1+\\frac{1}{5}\\right)\\left(1+\\frac{1}{6}\\right)\\left(1+\\frac{1}{7}\\right)\\left(1+\\frac{1}{8}\\right)\\left(1+\\frac{1}{9}\\right)$ is equal to what?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 32307,
"solution": "The expression is equal to $\\left(\\frac{3}{2}\\right)\\left(\\frac{4}{3}\\right)\\left(\\frac{5}{4}\\right)\\left(\\frac{6}{5}\\right)\\left(\\frac{7}{6}\\right)\\left(\\frac{8}{7}\\right)\\left(\\frac{9}{8}\\right)\\left(\\frac{10}{9}\\right)$ which equals $\\frac{3 \\cdot 4 \\cdot... |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x$. Find the value of $$\\left\\lfloor\\frac{2002!}{2001!+2000!+1999!+\\cdots+1!}\\right\\rfloor.$$"
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 32308,
"solution": "2000 We break up 2002! = 2002(2001)! as $$2000(2001!)+2 \\cdot 2001(2000!)=2000(2001!)+2000(2000!)+2002 \\cdot 2000(1999!) >2000(2001!+2000!+1999!+\\cdots+1!)$$ On the other hand, $$2001(2001!+2000!+\\cdots+1!)>2001(2001!+2000!)=2001(2001!)+2001!=2002!$$ Thus we have $2000<2002!/(2001... |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\log_{y^6}{x}\\cdot\\log_{x^5}{y^2}\\cdot\\log_{y^4}{x^3}\\cdot\\log_{x^3}{y^4}\\cdot\\log_{y^2}{x^5}$ can be written as $a\\log_y{x}$ for what constant $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac16"
} | {
"index": 32309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $\\sqrt{\\frac{4}{3}} - \\sqrt{\\frac{3}{4}}$ is equal to:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{6}"
} | {
"index": 32310,
"solution": "1. **Simplify $\\sqrt{\\frac{4}{3}}$:**\n \\[\n \\sqrt{\\frac{4}{3}} = \\frac{\\sqrt{4}}{\\sqrt{3}} = \\frac{2}{\\sqrt{3}}\n \\]\n Rationalizing the denominator:\n \\[\n \\frac{2}{\\sqrt{3}} \\cdot \\frac{\\sqrt{3}}{\\sqrt{3}} = \\frac{2\\sqrt{3}}{3}\n \\]\n\n2. **Simpli... |
DeepScaleR | [
{
"role": "user",
"content": "The expression $a^3-a^{-3}$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "\\left(a-\\frac{1}{a}\\right)\\left(a^2+1+\\frac{1}{a^2}\\right)"
} | {
"index": 32311,
"solution": "1. **Identify the form of the expression**: We are given the expression $a^3 - a^{-3}$ and need to simplify it. We recognize that this expression can be rewritten using the difference of cubes formula.\n\n2. **Recall the difference of cubes formula**: The formula for the difference of... |
DeepScaleR | [
{
"role": "user",
"content": "The expression $x^2 + 13x + 30$ can be written as $(x + a)(x + b),$ and the expression $x^2 + 5x - 50$ written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 32312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $x^2 + 15x + 54$ can be written as $(x + a)(x + b),$ and the expression $x^2 - 17x + 72$ written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 32313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $x^2 + 17x + 70$ can be rewritten as $(x + a)(x + b)$, and the expression $x^2 - 18x + 80$ written as $(x - b)(x - c)$, where a, b, and c are integers. Calculate the value of $a + b + c$."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 32314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $x^2 - 16x + 60$ can be written in the form $(x - a)(x - b)$, where $a$ and $b$ are both nonnegative integers and $a > b$. What is the value of $3b - a$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 32315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression $y^2+10y+33$ can be written as a combination of a square of a binomial and an integer. Find the integer."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 32316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression \\( 29 \\cdot 13 \\) is a product of two integers greater than 1. When this product is divided by any integer \\( k \\) where \\( 2 \\leqslant k \\leqslant 11 \\), the remainder is 1. What is the difference between the two smallest such integers?\n\n(29th Annual ... | math | {
"style": "rule",
"ground_truth": "27720"
} | {
"index": 32317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression \\(\\frac{3}{10}+\\frac{3}{100}+\\frac{3}{1000}\\) can be simplified by converting each fraction to a decimal, and then calculating the sum."
}
] | math | {
"style": "rule",
"ground_truth": "0.333"
} | {
"index": 32318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression \\(\\frac{3}{10}+\\frac{3}{100}+\\frac{3}{1000}\\) is equal to:"
}
] | math | {
"style": "rule",
"ground_truth": "0.333"
} | {
"index": 32319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression \\[(x+y+z)^{2006}+(x-y-z)^{2006}\\]is simplified by expanding it and combining like terms. How many terms are in the simplified expression?"
}
] | math | {
"style": "rule",
"ground_truth": "1{,}008{,}016"
} | {
"index": 32320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expression below has six empty boxes. Each box is to be fi\flled in with a number from $1$ to $6$ , where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible \ffinal result that can be achieved? $$ \\dfrac{\\frac{\\s... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 32321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The expressions $A$ = $1 \\times 2 + 3 \\times 4 + 5 \\times 6 + \\cdots + 37 \\times 38 + 39$ and $B$ = $1 + 2 \\times 3 + 4 \\times 5 + \\cdots + 36 \\times 37 + 38 \\times 39$ are obtained by writing multiplication and addition operators in an alternating pattern between suc... | math | {
"style": "rule",
"ground_truth": "722"
} | {
"index": 32322,
"solution": "We have \\[|A-B|=|1+3(4-2)+5(6-4)+ \\cdots + 37(38-36)-39(1-38)|\\]\\[\\implies |2(1+3+5+7+ \\cdots +37)-1-39(37)|\\]\\[\\implies |361(2)-1-39(37)|=|722-1-1443|=|-722|\\implies \\boxed{722}\\]"
} |
DeepScaleR | [
{
"role": "user",
"content": "The expressions \\[A=1 \\times 2 + 3 \\times 4 + 5 \\times 6 + \\cdots + 37 \\times 38 + 39\\]and \\[B = 1 + 2 \\times 3 + 4 \\times 5 + \\cdots + 36 \\times 37 + 38 \\times 39\\]are obtained by writing multiplication and addition operators in an alternating pattern between suc... | math | {
"style": "rule",
"ground_truth": "722"
} | {
"index": 32323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The extension of the altitude \\( BH \\) of triangle \\( ABC \\) intersects the circumcircle at point \\( D \\) (points \\( B \\) and \\( D \\) lie on opposite sides of line \\( AC \\)). The measures of arcs \\( AD \\) and \\( CD \\) that do not contain point \\( B \\) are \\( ... | math | {
"style": "rule",
"ground_truth": "1: \\sqrt{3}"
} | {
"index": 32324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The extensions of a telephone exchange have only 2 digits, from 00 to 99. Not all extensions are in use. By swapping the order of two digits of an extension in use, you either get the same number or the number of an extension not in use. What is the highest possible number of e... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 32325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The extensions of sides \\(AD\\) and \\(BC\\) of a convex quadrilateral \\(ABCD\\) intersect at point \\(M\\), and the extensions of sides \\(AB\\) and \\(CD\\) intersect at point \\(O\\). Segment \\(MO\\) is perpendicular to the angle bisector of \\(\\angle AOD\\). Find the ra... | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"index": 32326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The exterior angles of a triangle are proportional to the numbers 5: 7: 8. Find the angle between the altitudes of this triangle drawn from the vertices of its smaller angles."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 32327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The faces of a 12-sided die are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 such that the sum of the numbers on opposite faces is 13. The die is meticulously carved so that it is biased: the probability of obtaining a particular face \\( F \\) is greater than \\( \\frac{... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{48}"
} | {
"index": 32328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $4$. Another die's faces are marked with $2$, $2$, $5$, $6$, $7$, and $8$. Find the probability that the sum of the top two numbers will be $7$, $9$, or $11$.\nA) $\\frac{1}{18}$\nB) $\\frac{1}{... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 32329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $5$. Another die is marked with $2$, $3$, $4$, $6$, $7$, and $9$. What is the probability that the sum of the top two numbers will be $6$, $8$, or $10$?\nA) $\\frac{8}{36}$\nB) $\\frac{11}{36}$\... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{36}"
} | {
"index": 32330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The faces of an octahedral die are labeled with digits $1$ through $8$. What is the probability, expressed as a common fraction, of rolling a sum of $15$ with a pair of such octahedral dice?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{32}"
} | {
"index": 32331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probabilities that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 32332,
"solution": "To solve this problem, we need to understand the symmetry in the sum of the numbers on the faces of the dice when rolled. The minimum sum occurs when all dice show a $1$, which is $7 \\times 1 = 7$. The maximum sum occurs when all dice show a $6$, which is $7 \\times 6 = 42$. \n\nThe ... |
DeepScaleR | [
{
"role": "user",
"content": "The factors of $x^4+64$ are:"
}
] | math | {
"style": "rule",
"ground_truth": "(x^2-4x+8)(x^2+4x+8)"
} | {
"index": 32333,
"solution": "1. **Identify the Expression**: We start with the expression \\(x^4 + 64\\).\n\n2. **Recognize the Form**: Notice that \\(64\\) can be rewritten as \\(2^6\\), so we have:\n \\[\n x^4 + 64 = x^4 + 2^6\n \\]\n\n3. **Apply the Sophie Germain Identity**: The Sophie Germain Identity ... |
DeepScaleR | [
{
"role": "user",
"content": "The factory's planned output value for this year is $a$ million yuan, which is a 10% increase from last year. If the actual output value this year can exceed the plan by 1%, calculate the increase in the actual output value compared to last year."
}
] | math | {
"style": "rule",
"ground_truth": "11.1\\%"
} | {
"index": 32334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The famous German mathematician Dirichlet made significant achievements in the field of mathematics. He was the first person in the history of mathematics to pay attention to concepts and consciously \"replace intuition with concepts.\" The function named after him, $D\\left(x\... | math | {
"style": "rule",
"ground_truth": "(1)(4)"
} | {
"index": 32335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The famous Italian mathematician Fibonacci, while studying the problem of rabbit population growth, discovered a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, ..., where starting from the third number, each number is equal to the sum of the two numbers preceding it. This sequence ... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 32336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The famous skater Tony Hawk is riding a skateboard (segment $A B$) in a ramp, which is a semicircle with a diameter $P Q$. Point $M$ is the midpoint of the skateboard, and $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $P Q$. What values can the ang... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The famous skater Tony Hawk rides a skateboard (segment \\( AB \\)) on a ramp, which is a semicircle with diameter \\( PQ \\). Point \\( M \\) is the midpoint of the skateboard, and \\( C \\) is the foot of the perpendicular dropped from point \\( A \\) to the diameter \\( PQ \... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fare in Moscow with the \"Troika\" card in 2016 is 32 rubles for one trip on the metro and 31 rubles for one trip on ground transportation. What is the minimum total number of trips that can be made at these rates, spending exactly 5000 rubles?"
}
] | math | {
"style": "rule",
"ground_truth": "157"
} | {
"index": 32339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fenced area of a yard is a 15-foot by 12-foot rectangular region with a 3-foot by 3-foot square cut out, as shown. What is the area of the region within the fence, in square feet?\n\n[asy]draw((0,0)--(16,0)--(16,12)--(28,12)--(28,0)--(60,0)--(60,48)--(0,48)--cycle);\nlabel(... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"index": 32340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fenced area of a yard is an 18.5-foot by 14-foot rectangular region with a 3.5-foot by 3.5-foot square cutout. Calculate the area of the region within the fence, in square feet."
}
] | math | {
"style": "rule",
"ground_truth": "246.75"
} | {
"index": 32341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ferry \"Yi Rong\" travels at a speed of 40 kilometers per hour. On odd days, it travels downstream from point $A$ to point $B$, while on even days, it travels upstream from point $B$ to point $A$ (with the water current speed being 24 kilometers per hour). On one odd day, w... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 32342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ferry boat begins transporting tourists to an island every hour starting at 9 AM until its last trip, which starts at 4 PM. On the first trip at 9 AM, there were 120 tourists, and on each successive trip, there were 2 fewer tourists than on the previous trip. Determine the ... | math | {
"style": "rule",
"ground_truth": "904"
} | {
"index": 32343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fictional country of Isoland uses a 6-letter license plate system using the same 12-letter alphabet as the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, T, U, V). Design a license plate that starts with a vowel (A, E, I, O, U), ends with a consonant (G, K, P, R, T, V... | math | {
"style": "rule",
"ground_truth": "151200"
} | {
"index": 32344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fictional country of Isoland uses an alphabet with ten unique letters: A, B, D, E, I, L, N, O, R, U. License plates in Isoland are structured with five letters. How many different license plates are possible if they must begin with A or I, end with R, cannot include the let... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 32345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The field shown has been planted uniformly with wheat. [asy]\ndraw((0,0)--(1/2,sqrt(3)/2)--(3/2,sqrt(3)/2)--(2,0)--(0,0),linewidth(0.8));\nlabel(\"$60^\\circ$\",(0.06,0.1),E);\nlabel(\"$120^\\circ$\",(1/2-0.05,sqrt(3)/2-0.1),E);\nlabel(\"$120^\\circ$\",(3/2+0.05,sqrt(3)/2-0.1),... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 32346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fifth and eighth terms of a geometric sequence of real numbers are $7!$ and $8!$ respectively. What is the first term?"
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 32347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fifth grade has 120 teachers and students going to visit the Natural History Museum. A transportation company offers two types of vehicles to choose from:\n(1) A bus with a capacity of 40 people, with a ticket price of 5 yuan per person. If the bus is full, the ticket price... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 32348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fifth term of a geometric sequence of positive numbers is $11$ and the eleventh term is $5$. What is the eighth term of the sequence? Express your answer in simplest radical form. [asy]\nsize(150); defaultpen(linewidth(2));\nreal loc = 0;\nfor(int i = 0; i < 11; ++i) {\n\n... | math | {
"style": "rule",
"ground_truth": "\\sqrt{55}"
} | {
"index": 32349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fifth term of an arithmetic sequence is $11$. If the difference between two consecutive terms is $1$, what is the product of the first two terms?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 32350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fifth, sixth and seventh terms of an arithmetic sequence are 5, 9 and 13, respectively. What is the sum of the first four terms of the sequence? [asy]\nunitsize(8mm);\ndefaultpen(linewidth(.7pt)+fontsize(12pt));\n\nlabel(\"\\underline{\\quad}, \\underline{\\quad}, \\underli... | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"index": 32351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L$, without traveling along any portion of a road more than once. (Paula is allowe... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 32352,
"solution": "1. **Understanding the Problem**: We need to find the number of different routes Paula can take from city $A$ to city $L$, traversing exactly 13 roads, without repeating any road segment.\n\n2. **Vertex Visitation Constraints**: For any vertex $V$ (except $A$ and $L$), the maximum num... |
DeepScaleR | [
{
"role": "user",
"content": "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections ca... | math | {
"style": "rule",
"ground_truth": "732"
} | {
"index": 32353,
"solution": "This is equivalent to a node coloring of a cycle with 6 nodes. After repeatedly using deletion-contraction, the solution comes out to be $\\boxed{732}$"
} |
DeepScaleR | [
{
"role": "user",
"content": "The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.\n\nHow many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transfo... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 32354,
"solution": "To determine which transformations will map the figure onto itself, we analyze each type of transformation given in the problem:\n\n1. **Some rotation around a point on line $\\ell$:**\n - Consider a rotation of $180^\\circ$ around a point exactly halfway between an up-facing square... |
DeepScaleR | [
{
"role": "user",
"content": "The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.\n[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 32355,
"solution": "We analyze each of the four statements to determine which transformations will map the figure onto itself.\n\n1. **Some rotation around a point on line $\\ell$:**\n - Consider a rotation of $180^\\circ$ around a point that is exactly halfway between an up-facing square and a down-fa... |
DeepScaleR | [
{
"role": "user",
"content": "The figure drawn is not to scale. Which of the five segments shown is the longest? [asy]\npair A = (-3,0), B=(0,2), C=(3,0), D=(0,-1);\ndraw(D(MP(\"A\", A, W))--D(MP(\"B\", B, N))--D(MP(\"C\", C, E))--D(MP(\"D\", D, S))--A);\ndraw(B--D);\nMP(\"55^\\circ\", (0,-0.75), NW);\nMP(\... | math | {
"style": "rule",
"ground_truth": "CD"
} | {
"index": 32356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\\sqrt{m} + \\sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 32357,
"solution": "1. **Identify Key Triangles and Midpoint**: Let $M$ be the midpoint of $CD$. Since each side of the pentagon and the internal segments are of length $2$, and given the symmetry and equal lengths, we can infer that triangles $AED$ and $ABC$ are isosceles with a vertex angle of $120^\\c... |
DeepScaleR | [
{
"role": "user",
"content": "The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 32358,
"solution": "To solve this problem, we need to understand how a cube is formed and how the numbers on the faces are arranged. In a standard die, opposite faces sum up to 7. This means:\n- If one face shows 1, the opposite face shows 6.\n- If one face shows 2, the opposite face shows 5.\n- If one f... |
DeepScaleR | [
{
"role": "user",
"content": "The figure shown consists of a right triangle and two squares. If the figure's total area equals 850 square inches, what is the value of $x$ in inches? [asy]\nunitsize(5mm);\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\n\ndraw((0,5)--(0,-2)--(-2,-2)--(-2,0)--(5,0)--(5,5)--cycle... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 32359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The figure shown is a cube. The distance between vertices $B$ and $G$ is $5\\sqrt{2}$ units. What is the volume of the cube, in cubic units?\n\n[asy]\n\nsize(3cm,3cm);\n\npair A,B,C,D,a,b,c,d;\n\nA=(0,0);\nB=(1,0);\nC=(1,1);\nD=(0,1);\n\ndraw(A--B--C--D--A);\n\na=(-0.25,0.1);\n... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 32360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The figure shown represents a plot of land and is drawn using a scale in which 1 cm equals 2 miles. One square mile is 640 acres. How large is the actual plot of land, in acres? [asy]\ndraw((0,0)--(15,0)--(10,10)--(0,10)--cycle);\ndraw((0,1)--(1,1)--(1,0));\ndraw((0,9)--(1,9)--... | math | {
"style": "rule",
"ground_truth": "320000 \\mbox{ acres}"
} | {
"index": 32361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The figure shows a square in the interior of a regular hexagon. The square and regular hexagon share a common side. What is the degree measure of $\\angle ABC$? [asy]\nsize(150);\npair A, B, C, D, E, F, G, H;\nA=(0,.866);\nB=(.5,1.732);\nC=(1.5,1.732);\nD=(2,.866);\nE=(1.5,0);... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 32362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The figure shows a square of side $y$ units divided into a square of side $x$ units and four congruent rectangles. What is the perimeter, in units, of one of the four congruent rectangles? Express your answer in terms of $y$. [asy]\nsize(4cm);\ndefaultpen(linewidth(1pt)+fontsiz... | math | {
"style": "rule",
"ground_truth": "2y"
} | {
"index": 32363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The figure shows three squares with non-overlapping interiors. The area of the shaded square is 1 square inch. What is the area of rectangle $ABCD$, in square inches?\n\n[asy]size(100);\npair A = (0,0), D = (3,0),C = (3,2),B = (0,2);\ndraw(A--B--C--D--cycle);\ndraw(A--(1,0)--(1... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The figure shows two concentric circles. If the length of chord AB is 80 units and chord AB is tangent to the smaller circle, what is the area of the shaded region? Express your answer in terms of $\\pi$.\n\n[asy]\ndefaultpen(linewidth(.8pt));\ndotfactor=4;\nfilldraw(circle((0,... | math | {
"style": "rule",
"ground_truth": "1600\\pi"
} | {
"index": 32365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outs... | math | {
"style": "rule",
"ground_truth": "761"
} | {
"index": 32366,
"solution": "To solve the problem, we need to understand the pattern of how diamonds are added in each figure $F_n$.\n\n1. **Observing the pattern:**\n - $F_1$ has 1 diamond.\n - $F_2$ surrounds $F_1$ with a square and adds 4 diamonds (one on each side).\n - $F_3$ surrounds $F_2$ with a larg... |
DeepScaleR | [
{
"role": "user",
"content": "The first $20$ numbers of an arrangement are shown below. What would be the value of the $40^{\\mathrm{th}}$ number if the arrangement were continued?\n\n$\\bullet$ Row 1: $2,$ $2$\n\n$\\bullet$ Row 2: $4,$ $4,$ $4,$ $4$\n\n$\\bullet$ Row 3: $6,$ $6,$ $6,$ $6,$ $6,$ $6$\n\n$\\b... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first $30$ numbers of a similar arrangement are shown below. What would be the value of the $50^{\\mathrm{th}}$ number if the arrangement were continued?\n\n$\\bullet$ Row 1: $3,$ $3$\n\n$\\bullet$ Row 2: $6,$ $6,$ $6,$ $6$\n\n$\\bullet$ Row 3: $9,$ $9,$ $9,$ $9,$ $9,$ $9$\... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 32368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first 14 terms of the sequence $\\left\\{a_{n}\\right\\}$ are $4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, \\ldots$. Following this pattern, what is $a_{18}$?"
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 32369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first 14 terms of the sequence $\\{a_n\\}$ are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38. According to this pattern, find $a_{16}$."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 32370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$ .\n\n\\[ \\begin{tabular}{|c|c|c|}\\hline A & 1 & B \\hline 5 & C & 13 \\hline D & E & 3 \\hline\\end{tabular... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 32371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born?"
}
] | math | {
"style": "rule",
"ground_truth": "1979"
} | {
"index": 32372,
"solution": "1. **Identify the year of the seventh AMC 8**: \n The first AMC 8 was given in 1985. Since the AMC 8 is held annually, the subsequent AMC 8s would be in the years 1986, 1987, 1988, 1989, 1990, and the seventh AMC 8 would be in 1991.\n\n2. **Determine Samantha's age during the sevent... |
DeepScaleR | [
{
"role": "user",
"content": "The first and thirteenth terms of an arithmetic sequence are $\\frac79$ and $\\frac45$, respectively. What is the seventh term?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{71}{90}"
} | {
"index": 32373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first and thirteenth terms of an arithmetic sequence are 5 and 29, respectively. What is the fiftieth term?"
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 32374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first and twentieth terms of an arithmetic sequence are 3 and 63, respectively. What is the fortieth term?"
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 32375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first digit of a string of 2002 digits is a 1. Any two-digit number formed by consecutive digits within this string is divisible by 19 or 31. What is the largest possible last digit in this string?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 32376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first digit on the left of a six-digit number is 1. If this digit is moved to the last place, the resulting number is 64 times greater than the original number. Find the original number."
}
] | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"index": 32377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first few rows of a new sequence are given as follows:\n- Row 1: $3$\n- Row 2: $6, 6, 6, 6$\n- Row 3: $9, 9, 9, 9, 9, 9$\n- Row 4: $12, 12, 12, 12, 12, 12, 12, 12$\n\nWhat is the value of the $40^{\\mathrm{th}}$ number if this arrangement were continued?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 32378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first four terms in an arithmetic sequence are $x + y, x - y, xy,$ and $x/y,$ in that order. What is the fifth term?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{123}{40}"
} | {
"index": 32379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first four terms in an arithmetic sequence are $x+y$, $x-y$, $xy$, and $x/y$, in that order. What is the fifth term? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{123}{40}"
} | {
"index": 32380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first four terms of a sequence are $1,4,2$, and 3. Beginning with the fifth term in the sequence, each term is the sum of the previous four terms. What is the eighth term?"
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 32381,
"solution": "The first four terms of the sequence are $1,4,2,3$. Since each term starting with the fifth is the sum of the previous four terms, then the fifth term is $1+4+2+3=10$. Also, the sixth term is $4+2+3+10=19$, the seventh term is $2+3+10+19=34$, and the eighth term is $3+10+19+34=66$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 32382,
"solution": "1. **Identify the common difference**: Given an arithmetic sequence, the common difference $d$ can be calculated using any two consecutive terms. From the terms $x$ and $2x$, we have:\n \\[\n d = 2x - x = x\n \\]\n However, since $2x$ is the fourth term and $x$ is the second t... |
DeepScaleR | [
{
"role": "user",
"content": "The first four terms of an arithmetic sequence are $p$, $9$, $3p-q$, and $3p+q$. What is the $2010^{\\text{th}}$ term of this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "8041"
} | {
"index": 32383,
"solution": "1. **Identify the common difference**: \n Given the terms of the sequence are $p$, $9$, $3p-q$, and $3p+q$, we know that the common difference $d$ between consecutive terms can be calculated as:\n \\[\n d = (3p+q) - (3p-q) = 2q\n \\]\n\n2. **Set up equations for $p$ and $q$**:... |
DeepScaleR | [
{
"role": "user",
"content": "The first operation divides the bottom-left square of diagram $\\mathrm{a}$ into four smaller squares, as shown in diagram b. The second operation further divides the bottom-left smaller square of diagram b into four even smaller squares, as shown in diagram c; continuing this ... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 32384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first row of a triangle is given as:\n$$\n1, \\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}, \\ldots, \\frac{1}{1993}\n$$\n\nEach element of the following rows is calculated as the difference between two elements that are above it. The 1993rd row contains only one element. Find t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1993}"
} | {
"index": 32385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the $1$ at each end, row $4$ consists of only even numbers, as does row $2.$ How many of the first $20$ rows have this property? (Don't include row $0$ or row $1$). \\begin{tabular}{ccc... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 32386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a geometric sequence is 1024, and the 6th term is 125. What is the positive, real value for the 4th term?"
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 32387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a geometric sequence is 250. If the sum of the first 50 terms is 625 and the sum of the first 100 terms is 1225, find the sum of the first 150 terms."
}
] | math | {
"style": "rule",
"ground_truth": "1801"
} | {
"index": 32388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a geometric sequence is 729, and the 7th term is 64. What is the positive, real value for the 5th term?"
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 32389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence plus the square of the first term. What is the value of the first term which exceeds 10000?"
}
] | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"index": 32390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000?"
}
] | math | {
"style": "rule",
"ground_truth": "8192"
} | {
"index": 32391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a given sequence is 2, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 10000?"
}
] | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"index": 32392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a sequence is $2005$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the ${2005}^{\\text{th}}$ term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 32393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a sequence is $3107$. Each succeeding term is the sum of the squares of the digits of the previous term. What is the $614^{\\text{th}}$ term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a sequence is 1. Each subsequent term is 4 times the square root of the sum of all preceding terms plus 4.\n\nWhat is the sum of the first 1995 terms of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "15912121"
} | {
"index": 32395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a sequence is 1. Each subsequent term is 4 times the square root of the sum of all preceding terms plus 4. What is the sum of the first 1971 terms in the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "15531481"
} | {
"index": 32396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a sequence is 2, the second term is 3, and each subsequent term is formed such that each term is 1 less than the product of its two neighbors. What is the sum of the first 1095 terms of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "1971"
} | {
"index": 32397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a sequence is 934. Each following term is equal to the sum of the digits of the previous term, multiplied by 13.\n\nFind the 2013th term of the sequence."
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"index": 32398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of a sequence is 934. Each subsequent term is equal to the sum of the digits of the previous term multiplied by 13. Find the 2019th term of the sequence."
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"index": 32399,
"solution": ""
} |
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