data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working $20$... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 36349,
"solution": "\n1. **Identify the pattern of widgets sold each day**: Janabel's sales pattern starts with 1 widget on the first day and increases by 2 widgets each subsequent day. This forms an arithmetic sequence where the first term \\(a_1 = 1\\) and the common difference \\(d... |
deepscaler | [
{
"role": "user",
"content": "If the product $(3x^2 - 5x + 4)(7 - 2x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a,b,c,d$ are real numbers, then find $8a + 4b + 2c + d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 36350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system, the polar coordinate equation of circle $C$ is $ρ^{2}-8ρ\\sin (θ- \\dfrac {π}{3})+13=0$. Given points $A(1, \\dfrac {3π}{2})$ and $B(3, \\dfrac {3π}{2})$, where $P$ is a point on circle $C$, find the minimum value of the area of $\\triangle PAB$.... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 36351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $ABCD$ is a trapezoid with an area of $18$. $CD$ is three times the length of $AB$. What is the area of $\\triangle ABC$?\n\n[asy]\ndraw((0,0)--(1,3)--(10,3)--(15,0)--cycle);\ndraw((10,3)--(0,0));\nlabel(\"$D$\",(0,0),W);\nlabel(\"$A$\",(1,3),NW);\nlabel(\"$B$\"... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 36352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$, $CD$ is three times the length of $AB$. If the area of trapezoid $ABCD$ is $18$ square units, what is the area of $\\triangle ABC?$\n\n[asy]\ndraw((0,0)--(1,6)--(10,6)--(15,0)--cycle);\ndraw((10,6)--(0,0));\nlabel(\"$C$\",(0,0),W);\nlabel(\"$A$\",(1,6),NW);... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 36353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many points on the parabola \\( y = x^2 \\) (other than the origin) have the property that the tangent at these points intersects both coordinate axes at points with integer coordinates whose absolute values do not exceed 2020? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 36354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cutting equilateral triangle $BFC$ out of square $ABCD$ and translating it to the left of the square creates the shaded figure $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. What is the perimeter, in inches, of $ABFCDE$?\n\n[asy]\nfilldraw((0,0)--(sqrt(3),1)--(sqrt(3)+... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 36355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(2\\sin x,-\\cos x)$ and $\\overrightarrow{b}=(\\sqrt{3}\\cos x,2\\cos x)$, and function $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}+1$.\n\n(I) Find the smallest positive period of function $f(x)$, and find the range of $f(x)$ when $x\\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 36356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\\angle QRS=90^{\\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\\angle PUQ=\\angle RVT=y^{\\circ}$, the value of $y$ is Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "67.5"
} | {
"split": "train",
"index": 36357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a tetrahedron P-ABC, if PA, PB, and PC are mutually perpendicular, and PA=2, PB=PC=1, then the radius of the inscribed sphere of the tetrahedron P-ABC is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{4}"
} | {
"split": "train",
"index": 36358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, two pairs of identical isosceles triangles are cut off of square $ABCD$, leaving rectangle $PQRS$. The total area cut off is $200 \\text{ m}^2$. What is the length of $PR$, in meters? [asy]\nsize(5cm);\n\npair a = (0, 1); pair b = (1, 1); pair c = (1, 0); pair d... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the two solutions to the equation $54-15x-x^2=0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"split": "train",
"index": 36360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 10 red marbles and 6 blue marbles. Three marbles are selected at random and without replacement. What is the probability that one marble is red and two are blue? Express your answer as a common fraction. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{56}"
} | {
"split": "train",
"index": 36361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest possible value of $k$ for which $3^{13}$ is expressible as the sum of $k$ consecutive positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1458"
} | {
"split": "train",
"index": 36362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Welcome to the USAYNO, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them all correct, you will receive $\\max (0,(n-1)(n-2))$ points. If any of them are wrong, you will receive 0 points. Y... | math | {
"style": "rule",
"ground_truth": "NNNYYY"
} | {
"split": "train",
"index": 36363,
"solution": "Answer: NNNYYY"
} |
deepscaler | [
{
"role": "user",
"content": "Let $a > 1$ and $x > 1$ satisfy $\\log_a(\\log_a(\\log_a 2) + \\log_a 24 - 128) = 128$ and $\\log_a(\\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "896"
} | {
"split": "train",
"index": 36364,
"solution": "The first condition implies\n\\[a^{128} = \\log_a\\log_a 2 + \\log_a 24 - 128\\]\n\\[128+a^{128} = \\log_a\\log_a 2^{24}\\]\n\\[a^{a^{128}a^{a^{128}}} = 2^{24}\\]\n\\[\\left(a^{a^{128}}\\right)^{\\left(a^{a^{128}}\\right)} = 2^{24} = 8^8\\]\n So $a^{a^{128}} = 8$.\... |
deepscaler | [
{
"role": "user",
"content": "Given that the complex number $z_1$ corresponds to the point $(-1,1)$ on the complex plane, and the complex number $z_2$ satisfies $z_1z_2=-2$, find the value of $|z_2+2i|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 36365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the distinct prime factors of $7^7 - 7^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 36366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Madam Mim has a deck of $52$ cards, stacked in a pile with their backs facing up. Mim separates the small pile consisting of the seven cards on the top of the deck, turns it upside down, and places it at the bottom of the deck. All cards are again in one pile, but not all of ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 36367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of... | math | {
"style": "rule",
"ground_truth": "351"
} | {
"split": "train",
"index": 36368,
"solution": "Let $E_n$ denotes the number of sequences with length $n$ that ends at $E$. Define similarly for the other vertices. We seek for a recursive formula for $E_n$. \\begin{align*} E_n&=P_{3_{n-1}}+P_{4_{n-1}} \\\\ &=P_{2_{n-2}}+P_{5_{n-2}} \\\\ &=P_{1_{n-3}}+P_{3_{n-3}... |
deepscaler | [
{
"role": "user",
"content": "A convex polyhedron $P$ has $26$ vertices, $60$ edges, and $36$ faces, $24$ of which are triangular and $12$ of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $... | math | {
"style": "rule",
"ground_truth": "241"
} | {
"split": "train",
"index": 36369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that square PQRS has dimensions 3 × 3, points T and U are located on side QR such that QT = TU = UR = 1, and points V and W are positioned on side RS such that RV = VW = WS = 1, find the ratio of the shaded area to the unshaded area. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"split": "train",
"index": 36370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the definite integral:\n\n$$\n\\int_{0}^{2} \\frac{(4 \\sqrt{2-x}-\\sqrt{3 x+2}) d x}{(\\sqrt{3 x+2}+4 \\sqrt{2-x})(3 x+2)^{2}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{32} \\ln 5"
} | {
"split": "train",
"index": 36371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 8 balls are randomly and independently painted either red or blue with equal probability, find the probability that exactly 4 balls are red and exactly 4 balls are blue, and all red balls come before any blue balls in the order they were painted. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{256}"
} | {
"split": "train",
"index": 36372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the positive real numbers \\(a\\) and \\(b\\) satisfy \\(\\frac{1}{a} + \\frac{1}{b} \\leq 2 \\sqrt{2}\\) and \\((a - b)^2 = 4 (ab)^3\\), then \\(\\log_a b =\\) ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the sides of a square $S_1$ with area $16$ is bisected, and a smaller square $S_2$ is constructed using the bisection points as vertices. The same process is carried out on $S_2$ to construct an even smaller square $S_3$. What is the area of $S_3$? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36374,
"solution": "1. **Understanding the problem**: We are given a square $S_1$ with area 16. Each side of $S_1$ is bisected, and a smaller square $S_2$ is constructed using these bisection points. This process is repeated to construct $S_3$ from $S_2$.\n\n2. **Calculating the side ... |
deepscaler | [
{
"role": "user",
"content": "Find a positive integer that is divisible by 18 and has a square root between 26 and 26.2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "684"
} | {
"split": "train",
"index": 36375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the equation of curve $C$ is $x^{2}+y^{2}=16$, and the parameter equation of the line is $\\left\\{\\begin{array}{l}x=1+t\\cos \\theta \\\\y=2+t\\sin \\theta \\end{array}\\right.$ (where $t$ is the parameter). If the line $l$ intersects... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 36376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct solutions are there to the equation $|x-7| = |x+1|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 36377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of adventurers is showing their loot. It is known that exactly 5 adventurers have rubies; exactly 11 have emeralds; exactly 10 have sapphires; exactly 6 have diamonds. Additionally, it is known that:\n\n- If an adventurer has diamonds, then they have either emeralds or ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ .\n Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 36379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 15$, $y = 5$ and the line $x = 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 36380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $u$ and $v$ are complex numbers such that $|u+v|=2$ and $|u^2+v^2|=8,$ find the smallest possible value of $|u^3+v^3|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer $n,$ $-90 \\le n \\le 90,$ such that $\\sin n^\\circ = \\sin 604^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-64"
} | {
"split": "train",
"index": 36382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadrilateral \\(P Q R S\\) is inscribed in a circle. Diagonals \\(P R\\) and \\(Q S\\) are perpendicular and intersect at point \\(M\\). It is known that \\(P S = 13\\), \\(Q M = 10\\), and \\(Q R = 26\\). Find the area of the quadrilateral \\(P Q R S\\). Let's think step ... | math | {
"style": "rule",
"ground_truth": "319"
} | {
"split": "train",
"index": 36383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a positive integer \\( n \\) less than 2006 such that \\( 2006n \\) is a multiple of \\( 2006 + n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1475"
} | {
"split": "train",
"index": 36384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three merchants - Sosipatra Titovna, Olympiada Karpovna, and Poliksena Uvarovna - sat down to drink tea. Olympiada Karpovna and Sosipatra Titovna together drank 11 cups, Poliksena Uvarovna and Olympiada Karpovna drank 15 cups, and Sosipatra Titovna and Poliksena Uvarovna drank ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five glass bottles can be recycled to make a new bottle. Additionally, for every 20 new bottles created, a bonus bottle can be made from residual materials. Starting with 625 glass bottles, how many total new bottles can eventually be made from recycling and bonuses? (Keep coun... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"split": "train",
"index": 36386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a $3 \\times 3$ grid (each cell is a $1 \\times 1$ square), place two identical chess pieces, with at most one piece per cell. There are ___ different ways to arrange the pieces (if two arrangements can overlap by rotation, they are considered the same arrangement). Let's th... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all five-digit numbers \\(\\overline{abcde}\\) that are divisible by 9, and \\(\\overline{ace} - \\overline{bda} = 760\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81828"
} | {
"split": "train",
"index": 36388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of the parallelepiped generated by the vectors $\\begin{pmatrix} 3 \\\\ 4 \\\\ 5 \\end{pmatrix}$, $\\begin{pmatrix} 2 \\\\ m \\\\ 3 \\end{pmatrix}$, and $\\begin{pmatrix} 2 \\\\ 3 \\\\ m \\end{pmatrix}$ is 20. Find $m$, where $m > 0$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "3 + \\frac{2\\sqrt{15}}{3}"
} | {
"split": "train",
"index": 36389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:\n 1) The marksman first chooses a column from which a target is to be broken... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 36390,
"solution": "Clearly, the marksman must shoot the left column three times, the middle column two times, and the right column three times.\n From left to right, suppose that the columns are labeled $L,M,$ and $R,$ respectively. We consider the string $LLLMMRRR:$\n Since the lett... |
deepscaler | [
{
"role": "user",
"content": "Given Karl's rectangular garden measures \\(30\\) feet by \\(50\\) feet with a \\(2\\)-feet wide uniformly distributed pathway and Makenna's garden measures \\(35\\) feet by \\(55\\) feet with a \\(3\\)-feet wide pathway, compare the areas of their gardens, assuming the pathway... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 36391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the surface of a sphere with a radius of $2$, there is a triangular prism with an equilateral triangle base and lateral edges perpendicular to the base. All vertices of the prism are on the sphere's surface. Determine the maximum lateral area of this triangular prism. Let's ... | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"split": "train",
"index": 36392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos 2x+2 \\sqrt {3}\\sin x\\cos x$,\n(1) Find the range of the function $f(x)$ and write out the interval where the function $f(x)$ is strictly increasing;\n(2) If $0 < θ < \\dfrac {π}{6}$ and $f(θ)= \\dfrac {4}{3}$, compute the value of $\\cos 2θ$. L... | math | {
"style": "rule",
"ground_truth": "\\dfrac { \\sqrt {15}+2}{6}"
} | {
"split": "train",
"index": 36393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $b$ does there exist a polynomial function with integer coefficients such that $f(2)=2010$ and $f(b)=8$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 36394,
"solution": "We can take $f(x)=-\\frac{2002}{d}(x-b)+2010$ for all divisors $d$ of -2002. To see that we can't get any others, note that $b-2$ must divide $f(b)-f(2)$, so $b-2$ divides -2002 (this is because $b-2$ divides $b^{n}-2^{n}$ and hence any sum of numbers of the form $... |
deepscaler | [
{
"role": "user",
"content": "The product of all the positive integer divisors of an integer is $2^{120} \\cdot 3^{60} \\cdot 5^{90}$. What could this integer be? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18000"
} | {
"split": "train",
"index": 36395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many six-digit numbers of the form ababab are there, which are the product of six different prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $ABC$ with internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $A=2C$.\n\n(Ⅰ) If $\\triangle ABC$ is an acute triangle, find the range of $\\frac{a}{c}$.\n\n(Ⅱ) If $b=1, c=3$, find the area of $\\triangle ABC$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 36397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, given that $\\sin A = 10 \\sin B \\sin C$ and $\\cos A = 10 \\cos B \\cos C$, what is the value of $\\tan A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 36398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the town of Gearville, each bike license plate consists of three letters. The first letter is chosen from the set $\\{B, F, G, T, Y\\}$, the second from $\\{E, U\\}$, and the third from $\\{K, S, W\\}$. Gearville decided to increase the number of possible license plates by a... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 36399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "215"
} | {
"split": "train",
"index": 36400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of integers $2 \\leq n \\leq 2016$ such that $n^{n}-1$ is divisible by $2,3,5,7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 36401,
"solution": "Only $n \\equiv 1(\\bmod 210)$ work. Proof: we require $\\operatorname{gcd}(n, 210)=1$. Note that $\\forall p \\leq 7$ the order of $n$ $(\\bmod p)$ divides $p-1$, hence is relatively prime to any $p \\leq 7$. So $n^{n} \\equiv 1(\\bmod p) \\Longleftrightarrow n \\... |
deepscaler | [
{
"role": "user",
"content": "The base 5 number $34x1_5$ is divisible by 31. What is the digit $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a table tennis team of 5 players, which includes 2 veteran players and 3 new players, we need to select 3 players to be ranked as No. 1, No. 2, and No. 3 for a team competition. The selection must ensure that among the 3 chosen players, there is at least 1 veteran player and... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 36403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p(x) = x^2 + bx + c,$ where $b$ and $c$ are integers. If $p(x)$ is factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^ 2+ 28x + 5,$ what is $p(1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x - y = 6$ and $x + y = 12$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 36405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r$ be the roots of $x^3 - x^2 + x - 2 = 0.$ Find $p^3 + q^3 + r^3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36406,
"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 $A=1, B=2, C=3, D=4, E=5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "BEE"
} | {
"split": "train",
"index": 36407,
"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 wi... |
deepscaler | [
{
"role": "user",
"content": "For non-negative integers $n$, the function $f(n)$ is defined by $f(0) = 0$, $f(1) = 1$, and $f(n) = f\\left(\\left\\lfloor \\frac{1}{2} n \\right\\rfloor \\right) + n - 2\\left\\lfloor \\frac{1}{2} n \\right\\rfloor$. Find the maximum value of $f(n)$ for $0 \\leq n \\leq 1997$... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point $A(1,2)$ and circle $C: x^{2}+y^{2}+2mx+2y+2=0$. \n$(1)$ If there are two tangents passing through point $A$, find the range of $m$. \n$(2)$ When $m=-2$, a point $P$ on the line $2x-y+3=0$ is chosen to form two tangents $PM$ and $PN$ to the circle. Find the minimu... | math | {
"style": "rule",
"ground_truth": "\\frac{7\\sqrt{15}}{5}"
} | {
"split": "train",
"index": 36409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Sanchez's students were asked to add two positive integers. Juan subtracted by mistake and got 2. Maria mistakenly multiplied and got 120. What was the correct answer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 36410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered integer pairs \\((a, b)\\) such that the equation \\(x^{2} + a x + b = 167 y\\) has integer solutions \\((x, y)\\), where \\(1 \\leq a, b \\leq 2004\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2020032"
} | {
"split": "train",
"index": 36411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a bag contains only green, yellow, and red marbles in the ratio $3: 4: 2$ and 63 of the marbles are not red, how many red marbles are in the bag? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 36412,
"solution": "Since the ratio of green marbles to yellow marbles to red marbles is $3: 4: 2$, then we can let the numbers of green, yellow and red marbles be $3n, 4n$ and $2n$ for some positive integer $n$. Since 63 of the marbles in the bag are not red, then $3n+4n=63$ and so $... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( a \\) is a real number, and for any \\( k \\in [-1,1] \\), when \\( x \\in (0,6] \\), the following inequality is always satisfied:\n\\[ 6 \\ln x + x^2 - 8 x + a \\leq k x. \\]\n\nFind the maximum value of \\( a \\). Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "6 - 6 \\ln 6"
} | {
"split": "train",
"index": 36413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $A=120^{\\circ}$, $AB=5$, $BC=7$, find the value of $\\sin B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}}{14}"
} | {
"split": "train",
"index": 36414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six test scores have a mean of 85, a median of 88, and a mode of 90. The highest score exceeds the second highest by 5 points. Find the sum of the three highest scores. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "275"
} | {
"split": "train",
"index": 36415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a sequence $x_1,x_2,\\cdots x_{12}$ of real numbers such that $x_1=1$ and for $n=1,2,\\dots,10$ let \\[ x_{n+2}=\\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \\] Suppose $x_n>0$ for $n=1,2,\\dots,11$ and $x_{12}=0$ . Then the value of $x_2$ can be written as $\\fr... | math | {
"style": "rule",
"ground_truth": "622"
} | {
"split": "train",
"index": 36416,
"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? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "1971"
} | {
"split": "train",
"index": 36417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $(2b-c)\\cos A=a\\cos C$.\n\n(1) Find the measure of angle $A$;\n\n(2) If $a=3$ and $b=2c$, find the area of $\\triangle ABC$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 36418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point O is the circumcenter of triangle ABC, and |BA|=2, |BC|=6, calculate the value of the dot product of the vectors BO and AC. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle configuration, each row consists of increasing multiples of 3 unit rods. The number of connectors in a triangle always forms an additional row than the rods, with connectors enclosing each by doubling the requirements of connecting joints from the previous triangl... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 36420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest three-digit number in Pascal's triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 36421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn a football tournament, each team is supposed to play one match against each of the other teams. However, during the tournament, half of the teams were disqualified and did not participate further. As a result, a total of 77 matches were played, and the disqualified teams m... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 36422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, three circles of radius 2 with centers $P$, $Q$, and $R$ are tangent to one another and to two sides of $\\triangle ABC$, as shown. Assume the centers $P$, $Q$, and $R$ form a right triangle, with $PQ$ as the hypotenuse.\n\nFind the perimeter of triangle $ABC$. ... | math | {
"style": "rule",
"ground_truth": "8 + 4\\sqrt{2}"
} | {
"split": "train",
"index": 36423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(2^8 + 4^5)(2^3 - (-2)^2)^{11}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5368709120"
} | {
"split": "train",
"index": 36424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the first site, high-class equipment was used, while at the second site, first-class equipment was used, with the amount of high-class equipment being less than that of the first-class. Initially, 30% of the equipment from the first site was transferred to the second site. T... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 36425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD_{}^{}$ has sides $\\overline {AB}$ of length 4 and $\\overline {CB}$ of length 3. Divide $\\overline {AB}$ into 168 congruent segments with points $A_{}^{}=P_0, P_1, \\ldots, P_{168}=B$, and divide $\\overline {CB}$ into 168 congruent segments with points $C_{}^... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 36426,
"solution": "First, count the diagonal which has length $5$. For the rest of the segments, think about pairing them up so that each pair makes $5$. For example, the parallel lines closest to the diagonal would have length $\\frac{167}{168}\\cdot{5}$ while the parallel line clos... |
deepscaler | [
{
"role": "user",
"content": "At the beginning of the first day, a box contains 1 black ball, 1 gold ball, and no other balls. At the end of each day, for each gold ball in the box, 2 black balls and 1 gold ball are added to the box. If no balls are removed from the box, how many balls are in the box at the... | math | {
"style": "rule",
"ground_truth": "383"
} | {
"split": "train",
"index": 36427,
"solution": "At the beginning of the first day, the box contains 1 black ball and 1 gold ball. At the end of the first day, 2 black balls and 1 gold ball are added, so the box contains 3 black balls and 2 gold balls. At the end of the second day, $2 \\times 2=4$ black balls and... |
deepscaler | [
{
"role": "user",
"content": "An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"split": "train",
"index": 36428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous o... | math | {
"style": "rule",
"ground_truth": "2^{n-1}"
} | {
"split": "train",
"index": 36429,
"solution": "\nTo solve this problem, we need to determine how many distinct ways the child can build a wall with \\( n \\) identical cubes. Each cube can be placed in such a way that it shares a face with the previous cube. This can be done either by placing the new cube on th... |
deepscaler | [
{
"role": "user",
"content": "(The full score for this question is 12 points) Given $f(x+1)=x^2-1$, \n(1) Find $f(x)$ \n(2) Find the maximum and minimum values of $f(x)$, and specify the corresponding values of $x$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each real number \\( x \\), \\( f(x) \\) is defined to be the minimum of the values of \\( 2x + 3 \\), \\( 3x - 2 \\), and \\( 25 - x \\). What is the maximum value of \\( f(x) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{53}{3}"
} | {
"split": "train",
"index": 36431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a school cafeteria line, there are 16 students alternating between boys and girls (starting with a boy, followed by a girl, then a boy, and so on). Any boy, followed immediately by a girl, can swap places with her. After some time, all the girls end up at the beginning of th... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of the fifth and sixth elements in Row 20 of Pascal's triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20349"
} | {
"split": "train",
"index": 36433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that in a randomly chosen arrangement of the numbers and letters in \"HMMT2005,\" one can read either \"HMMT\" or \"2005\" from left to right? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23/144"
} | {
"split": "train",
"index": 36434,
"solution": "To read \"HMMT,\" there are $\\binom{8}{4}$ ways to place the letters, and $\\frac{4!}{2}$ ways to place the numbers. Similarly, there are $\\binom{8}{4} \\frac{4!}{2}$ arrangements where one can read \"2005.\" The number of arrangements in which one can read both ... |
deepscaler | [
{
"role": "user",
"content": "To monitor the skier's movements, the coach divided the track into three sections of equal length. It was found that the skier's average speeds on these three separate sections were different. The time required for the skier to cover the first and second sections together was 4... | math | {
"style": "rule",
"ground_truth": "58.5"
} | {
"split": "train",
"index": 36435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( 9210 - 9124 = 210 - \\square \\), the value represented by the \\( \\square \\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 36436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=x^{2}+x-2$, determine the characteristics of the function $f(x)$ in the interval $[-1,1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{4}"
} | {
"split": "train",
"index": 36437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A room is shaped like an 'L'. One part is a rectangle that is 23 feet long and 15 feet wide, and a square with a side of 8 feet is attached at one end of the rectangle, extending its width. Calculate the ratio of the total length of the room to its perimeter. Let's think step b... | math | {
"style": "rule",
"ground_truth": "1:4"
} | {
"split": "train",
"index": 36438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ and $y$ satisfy the equation $(x-1)^{2}+(y+2)^{2}=4$, find the maximum and minimum values of $S=3x-y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 - 2\\sqrt{10}"
} | {
"split": "train",
"index": 36439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the roots of the equation \\((x-4)(x-2)+(x-2)(x-6)=0\\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36440,
"solution": "Since the two terms have a common factor, then we factor and obtain \\((x-2)((x-4)+(x-6))=0\\). This gives \\((x-2)(2x-10)=0\\). Therefore, \\(x-2=0\\) (which gives \\(x=2\\)) or \\(2x-10=0\\) (which gives \\(x=5\\)). Therefore, the two roots of the equation are \\... |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = \\frac{3}{9^x + 3}.$ Find\n\\[f \\left( \\frac{1}{1001} \\right) + f \\left( \\frac{2}{1001} \\right) + f \\left( \\frac{3}{1001} \\right) + \\dots + f \\left( \\frac{1000}{1001} \\right).\\] Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 36441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all ordered pairs of integers $(x, y)$ such that $3^{x} 4^{y}=2^{x+y}+2^{2(x+y)-1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0,1), (1,1), (2,2)"
} | {
"split": "train",
"index": 36442,
"solution": "The right side is $2^{x+y}\\left(1+2^{x+y-1}\\right)$. If the second factor is odd, it needs to be a power of 3 , so the only options are $x+y=2$ and $x+y=4$. This leads to two solutions, namely $(1,1)$ and $(2,2)$. The second factor can also be even, if $x+y-1=0$.... |
deepscaler | [
{
"role": "user",
"content": "There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two gre... | math | {
"style": "rule",
"ground_truth": "310"
} | {
"split": "train",
"index": 36443,
"solution": "Split the problem into two cases:\n Case 1 - Both poles have blue flags: There are 9 ways to place the 10 blue flags on the poles. In each of these configurations, there are 12 spots that a green flag could go. (either in between two blues or at the tops or bottoms... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of positive terms $\\{a\\_n\\}$, with $a\\_1=2$, $(a\\_n+1)a_{n+2}=1$, and $a\\_2=a\\_6$, find the value of $a_{11}+a_{12}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}+\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 36444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 8a = 0$$ have integer solutions for $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 36445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The symbol $\\diamond$ is defined so that $a \\diamond b=\\frac{a+b}{a \\times b}$. What is the value of $3 \\diamond 6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 36446,
"solution": "Using the definition of the symbol, $3 \\diamond 6=\\frac{3+6}{3 \\times 6}=\\frac{9}{18}=\\frac{1}{2}$."
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), point \\(K\\) on side \\(AB\\) and point \\(M\\) on side \\(AC\\) are positioned such that \\(AK:KB = 3:2\\) and \\(AM:MC = 4:5\\). Determine the ratio in which the line through point \\(K\\) parallel to side \\(BC\\) divides segment \\(BM\\). Let's think... | math | {
"style": "rule",
"ground_truth": "18/7"
} | {
"split": "train",
"index": 36447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade, the second card is a 10, and the third card is a queen? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{11050}"
} | {
"split": "train",
"index": 36448,
"solution": ""
} |
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