data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers. One of the roots of $x^3 + ax + b = 0$ is $1 + i \\sqrt{3}.$ Find $a + b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 6900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the 100th letter in the pattern ABCABCABC...? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "A"
} | {
"split": "train",
"index": 6901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, if $\\frac{a_{11}}{a_{10}} < -1$ and its sum of the first $n$ terms, $S_n$, has a maximum value, find the value of $n$ when $S_n$ takes the minimum positive value. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 6902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle, the base is 12 units; one of the angles at this base is $120^{\\circ}$; the side opposite this angle is 28 units.\n\nFind the third side. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of this figure is $100\\text{ cm}^2$. Its perimeter is \n[asy] draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1)); draw((1,2)--(1,1)--(2,1)--(2,0),dashed); [/asy]\n\n[figure consists of four identical squares] Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "50 cm"
} | {
"split": "train",
"index": 6904,
"solution": "1. **Identify the area of each square**: Given that the total area of the figure is $100\\text{ cm}^2$ and the figure consists of four identical squares, the area of each square is:\n \\[\n \\frac{100\\text{ cm}^2}{4} = 25\\text{ cm}^2\n \\]\n\n2. **Calculate ... |
deepscaler | [
{
"role": "user",
"content": "In a game of rock-paper-scissors with $n$ people, the following rules are used to determine a champion: (a) In a round, each person who has not been eliminated randomly chooses one of rock, paper, or scissors to play. (b) If at least one person plays rock, at least one person p... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{14}"
} | {
"split": "train",
"index": 6905,
"solution": "For each positive integer $n$, let $E_{n}$ denote the expected number of rounds required to determine a winner among $n$ people. Clearly, $E_{1}=0$. When $n=2$, on the first move, there is a $\\frac{1}{3}$ probability that there is a tie, and a $\\frac{2}{3}$ probab... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$, respectively, and $\\frac{\\cos C}{\\cos B} = \\frac{3a-c}{b}$.\n(1) Find the value of $\\sin B$;\n(2) If $b = 4\\sqrt{2}$ and $a = c$, find the area of $\\triangle ABC$. Let's think s... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}"
} | {
"split": "train",
"index": 6906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers from 1 to 1000 inclusive that give the same remainder when divided by 11 and by 12. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "87"
} | {
"split": "train",
"index": 6907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 6908,
"solution": "[asy]defaultpen(fontsize(9)); pair A=expi(-pi/2-acos(475/486)), B=expi(-pi/2+acos(475/486)), C=expi(-pi/2+acos(475/486)+acos(7/18)), D=expi(-pi/2+acos(475/486)+2*acos(7/18)), E=expi(-pi/2+acos(475/486)+3*acos(7/18)), F=expi(-pi/2-acos(475/486)-acos(7/18)); draw(unit... |
deepscaler | [
{
"role": "user",
"content": "Given that $P$ is any point on the hyperbola $\\frac{x^2}{3} - y^2 = 1$, a line perpendicular to each asymptote of the hyperbola is drawn through point $P$, with the feet of these perpendiculars being $A$ and $B$. Determine the value of $\\overrightarrow{PA} \\cdot \\overrighta... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"split": "train",
"index": 6909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On Ming's way to the swimming pool, there are 200 trees. On his round trip, Ming marked some trees with red ribbons. On his way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, and so on, marking every 4th tree. On his way back, he marked the 1st tree ... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 6910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amy writes down four integers \\(a > b > c > d\\) whose sum is 50. The pairwise positive differences of these numbers are \\(2, 3, 4, 5, 7,\\) and \\(10\\). What is the sum of the possible values for \\(a\\)? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 6911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$ such that it is tangent to circle $C$ at point $R$. Meanwhile, a secant from $Q$ intersects $C$ at points $D$ and $E$, such that $QD < QE$. If $QD = 4$ and $QR = ED - QD$, then what is $QE$? Let's think step ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a memorable $9$-digit telephone number defined as $d_1d_2d_3d_4-d_5d_6d_7d_8d_9$. A number is memorable if the prefix sequence $d_1d_2d_3d_4$ is exactly the same as either of the sequences $d_5d_6d_7d_8$ or $d_6d_7d_8d_9$. Each digit $d_i$ can be any of the ten decimal... | math | {
"style": "rule",
"ground_truth": "199990"
} | {
"split": "train",
"index": 6913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a mathematics competition consisting of three problems, A, B, and C, among the 39 participants, each person solved at least one problem. Among those who solved problem A, there are 5 more people who only solved A than those who solved A and any other problems. Among those wh... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 6914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ewan writes out a sequence where he counts by 11s starting at 3. The resulting sequence is $3, 14, 25, 36, \\ldots$. What is a number that will appear in Ewan's sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 6915,
"solution": "Ewan's sequence starts with 3 and each following number is 11 larger than the previous number. Since every number in the sequence is some number of 11s more than 3, this means that each number in the sequence is 3 more than a multiple of 11. Furthermore, every such ... |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of \\( n \\) between 1 and 999 inclusive does the decimal representation of \\( \\frac{n}{1000} \\) terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"split": "train",
"index": 6916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an equilateral triangle and a point M inside the triangle such that $MA^2 = MB^2 +MC^2$ . Draw an equilateral triangle $ACD$ where $D \\ne B$ . Let the point $N$ inside $\\vartriangle ACD$ such that $AMN$ is an equilateral triangle. Determine $\\angle ... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 6917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The endpoints of a diameter of circle $M$ are $(-1,-4)$ and $(-7,6)$. What are the coordinates of the center of circle $M$? Express your answer as an ordered pair. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-4,1)"
} | {
"split": "train",
"index": 6918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $1^{234} + 4^6 \\div 4^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 6919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eric builds a small pyramid for a school project. His pyramid has a height of twelve inches and a square base that measures ten inches on each side. Eric wants to find the smallest cube-shaped box to put his pyramid in so that he can safely bring it to school right side up. Wha... | math | {
"style": "rule",
"ground_truth": "1728"
} | {
"split": "train",
"index": 6920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?\n\nNote: The 1 at the top is often labelled the \"zeroth\" row of Pascal's Triangle, by convention. So to count a total of 20 ... | math | {
"style": "rule",
"ground_truth": "\\frac{39}{210}"
} | {
"split": "train",
"index": 6921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ where $a>0$ and $b>0$. If the point $F_2$ is symmetric with respect to the asymptote line and lies on the hyperbola, calculate the eccentricity of the hyperbola. Let's ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 6922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ such that $(n + i), (n + i)^3,$ and $(n + i)^4$ are the vertices of a triangle in the complex plane whose area is greater than 3000. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$, $y$, $z$ satisfying $\\begin{cases} xy+2z=1 \\\\ x^{2}+y^{2}+z^{2}=5 \\end{cases}$, the minimum value of $xyz$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9 \\sqrt {11}-32"
} | {
"split": "train",
"index": 6924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all $n \\geq 1$, let \\[ a_n = \\sum_{k=1}^{n-1} \\frac{\\sin \\left( \\frac{(2k-1)\\pi}{2n} \\right)}{\\cos^2 \\left( \\frac{(k-1)\\pi}{2n} \\right) \\cos^2 \\left( \\frac{k\\pi}{2n} \\right)}. \\] Determine \\[ \\lim_{n \\to \\infty} \\frac{a_n}{n^3}. \\] Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{\\pi^3}"
} | {
"split": "train",
"index": 6925,
"solution": "The answer is $\\frac{8}{\\pi^3}$.\n\nBy the double angle and sum-product identities for cosine, we have \\begin{align*} 2\\cos^2\\left(\\frac{(k-1)\\pi}{2n}\\right) - 2\\cos^2 \\left(\\frac{k\\pi}{2n}\\right) &= \\cos\\left(\\frac{(k-1)\\pi}{n}\\right) - \\cos\\lef... |
deepscaler | [
{
"role": "user",
"content": "Petya inscribed two squares in a right triangle with sides 3, 4, and 5. One vertex of the first square coincides with the right-angle vertex of the triangle, while one side of the second square lies on the hypotenuse. Petya found the side lengths of each square, expressed their... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 6926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy: $|\\overrightarrow{a}| = |\\overrightarrow{b}| = 1$, and $|k\\overrightarrow{a} + \\overrightarrow{b}| = \\sqrt{3}|\\overrightarrow{a} - k\\overrightarrow{b}| (k > 0)$. Find the maximum value of the ang... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 6927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the area of the crescent moon enclosed by the portion of the circle of radius 4 centered at (0,0) that lies in the first quadrant, the portion of the circle with radius 2 centered at (0,1) that lies in the first quadrant, and the line segment from (0,0) to (4,0). Let'... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 6928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the figure below, a circular park consists of an outer-ring path for walkers (white) and a ring-shaped flower garden (gray) surrounding a central circular fountain (black). The walking path is six feet wide in all places, the garden ring is eight feet wide in all pl... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 6929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest and smallest fractions $F = \\frac{y-x}{x+4y}$ \nif the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0 \\leq \\frac{y-x}{x+4y} \\leq 4"
} | {
"split": "train",
"index": 6930,
"solution": "\nGiven the equation \\( x^2y^2 + xy + 1 = 3y^2 \\), we need to determine the largest and smallest values of the fraction \\( F = \\frac{y-x}{x+4y} \\).\n\n### Step 1: Analyze the Given Equation\nTo simplify the problem, we first explore the given equation:\n\\[\nx^... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\Gamma$: $\\dfrac {x^{2}}{a^{2}}+y^{2}=1(a > 1)$, its left focus is $F_{1}$, the right vertex is $A_{1}$, and the top vertex is $B_{1}$. The circle $P$ that passes through points $F_{1}$, $A_{1}$, and $B_{1}$ has its center coordinates at $\\left( \\dfrac { ... | math | {
"style": "rule",
"ground_truth": "\\dfrac { \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 6931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many integer palindromes are between 200 and 700. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 6932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can I arrange 3 different math books and 5 different history books on my bookshelf, if I require there to be a math book on both ends? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4,\\!320"
} | {
"split": "train",
"index": 6933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integers \\( a, b, c, \\) and \\( d \\) such that \\( a > b > c > d \\) and \\( a + b + c + d = 2004 \\), as well as \\( a^2 - b^2 + c^2 - d^2 = 2004 \\), what is the minimum value of \\( a \\)? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "503"
} | {
"split": "train",
"index": 6934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of pairs of union/intersection operations $\\left(\\square_{1}, \\square_{2}\\right) \\in\\{\\cup, \\cap\\}^{2}$ satisfying the condition: for any sets $S, T$, function $f: S \\rightarrow T$, and subsets $X, Y, Z$ of $S$, we have equality of sets $f(X) \\square_... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 6935,
"solution": "If and only if $\\square_{1}=\\square_{2}=\\cup$. See http://math.stackexchange.com/questions/359693/overview-of-"
} |
deepscaler | [
{
"role": "user",
"content": "Alex thought of a two-digit number (from 10 to 99). Grisha tries to guess it by naming two-digit numbers. It is considered that he guessed the number if he correctly guessed one digit, and the other digit is off by no more than one (for example, if the number thought of is 65, ... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 6936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integer values of the parameter \\(a\\) for which the system\n\\[\n\\begin{cases}\nx - 2y = y^2 + 2, \\\\\nax - 2y = y^2 + x^2 + 0.25a^2\n\\end{cases}\n\\]\nhas at least one solution. In the answer, indicate the sum of the found values of the parameter \\(a\\). Let's t... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $b=1$, $c= \\sqrt {3}$, and $\\angle C= \\frac {2}{3}\\pi$, find the area $S_{\\triangle ABC}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}}{4}"
} | {
"split": "train",
"index": 6938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Read the following material: The overall idea is a common thinking method in mathematical problem solving: Here is a process of a student factorizing the polynomial $(x^{2}+2x)(x^{2}+2x+2)+1$. Regard \"$x^{2}+2x$\" as a whole, let $x^{2}+2x=y$, then the original expression $=y^... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 6939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer \\(n\\), such that \\(-180 < n < 180\\), for which \\(\\tan n^\\circ = \\tan 276^\\circ.\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 6940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $m$ such that the equation $10x^2 - mx + 660 = 0$ has integral solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "170"
} | {
"split": "train",
"index": 6941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest possible number of digits in the product of a 4-digit whole number and a 3-digit whole number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 6942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[f(a,b) = \\left\\{\n\\renewcommand{\\arraystretch}{3}\n\\begin{array}{cl}\n\\dfrac{ab - a + 2}{2a} & \\text{if $a + b \\le 3$}, \\\\\n\\dfrac{ab - b - 2}{-2b} & \\text{if $a + b > 3$}.\n\\end{array}\n\\renewcommand{\\arraystretch}{1}\n\\right.\\]Find $f(2,1) + f(2,4).$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 6943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each of the possible outcomes of rolling three coins (TTT, THT, TTH, HHT, HTH, HHH), a fair die is rolled for each two heads that appear together. What is the probability that the sum of the die rolls is odd? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 6944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $a$ such that $ax^2+15x+4$ is the square of a binomial. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{225}{16}"
} | {
"split": "train",
"index": 6945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n$$\\sum_{k=1}^{2000} k(\\lceil \\log_{\\sqrt{3}}{k}\\rceil - \\lfloor\\log_{\\sqrt{3}}{k} \\rfloor).$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1999907"
} | {
"split": "train",
"index": 6946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the four digits 2, 4, 6, and 7, how many different positive two-digit integers can be formed using these digits if a digit may not be repeated in an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 6947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If there are $1, $2, and $3 bills in the board game \"Silly Bills\" and let x be the number of $1 bills, then x+11, x-18, and x+11+(x-18) = 2x-7 are the respective number of $2 and $3 bills, determine the value of x when the total amount of money is $100. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 6948,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{n = 1}^\\infty \\frac{2n - 1}{n(n + 1)(n + 2)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 6949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random a... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 6950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Attach a single digit to the left and right of the eight-digit number 20222023 so that the resulting 10-digit number is divisible by 72. (Specify all possible solutions.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3202220232"
} | {
"split": "train",
"index": 6951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hexagon with the R is colored red. Each hexagon is colored either red, yellow or green, such that no two hexagons with a common side are colored the same color. In how many different ways can the figure be colored?\n\n[asy]\npath a=(0,0)--(10,0)--(15,8.7)--(10,17.3)--(0... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 6952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of an isosceles obtuse triangle is 8, and the median drawn to one of its equal sides is $\\sqrt{37}$. Find the cosine of the angle at the vertex. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{5}"
} | {
"split": "train",
"index": 6953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of all coefficients in the expansion of $(x+ \\frac {1}{x})^{2n}$ is greater than the sum of all coefficients in the expansion of $(3 3x -x)^{n}$ by $240$.\n$(1)$ Find the constant term in the expansion of $(x+ \\frac {1}{x})^{2n}$ (answer with a number);\n$(... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 128 players in a single elimination tennis tournament, where exactly 32 players receive a bye in the first round. Calculate the total number of matches played in this tournament. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 6955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{p}$ be the projection of vector $\\mathbf{v}$ onto vector $\\mathbf{u},$ and let $\\mathbf{q}$ be the projection of $\\mathbf{p}$ onto $\\mathbf{u}.$ If $\\frac{\\|\\mathbf{p}\\|}{\\|\\mathbf{v}\\|} = \\frac{3}{4},$ then find $\\frac{\\|\\mathbf{q}\\|}{\\|\\mathbf... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"split": "train",
"index": 6956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of three consecutive one-digit, positive, odd integers is one-seventh of the product of the same three integers. What is the middle integer when the three integers are listed in ascending order? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 6957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the start of this month, Mathilde and Salah each had 100 coins. For Mathilde, this was $25 \\%$ more coins than she had at the start of last month. For Salah, this was $20 \\%$ fewer coins than he had at the start of last month. What was the total number of coins that they h... | math | {
"style": "rule",
"ground_truth": "205"
} | {
"split": "train",
"index": 6958,
"solution": "Suppose that Mathilde had $m$ coins at the start of last month and Salah had $s$ coins at the start of last month. From the given information, 100 is $25 \\%$ more than $m$, so $100=1.25 m$ which means that $m=\\frac{100}{1.25}=80$. From the given information, 100 i... |
deepscaler | [
{
"role": "user",
"content": "I have a drawer with 4 shirts, 5 pairs of shorts, and 6 pairs of socks in it. If I reach in and randomly remove three articles of clothing, what is the probability that I get one shirt, one pair of shorts, and one pair of socks? (Treat pairs of socks as one article of clothing.... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{91}"
} | {
"split": "train",
"index": 6959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that the $73$ fractions $\\frac{19}{n+21}, \\frac{20}{n+22},\\frac{21}{n+23},...,\\frac{91}{n+93}$ are all irreducible. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 6960,
"solution": "\nTo solve this problem, we need to find the smallest positive integer \\( n \\) such that all 73 fractions of the form \\(\\frac{k}{n+k+20}\\) for \\( k = 19, 20, \\ldots, 91 \\) are irreducible. A fraction \\(\\frac{a}{b}\\) is irreducible if and only if \\(\\gcd(... |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral has vertices at $(0,1)$, $(3,4)$, $(4,3)$ and $(3,0)$. Its perimeter can be expressed in the form $a\\sqrt2+b\\sqrt{10}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of\n\\[\\frac{x + y}{xyz}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers are grouped as follows:\n\n\\( A_1 = \\{1\\}, A_2 = \\{2, 3, 4\\}, A_3 = \\{5, 6, 7, 8, 9\\} \\), and so on.\n\nIn which group does 2009 belong? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 6963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The variables \\(a, b, c, d, e\\), and \\(f\\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that\n\\[\n\\begin{aligned}\n& a + b = c, \\\\\n& b + c = d, \\\\\n& c + e = f,\n\\end{aligned}\n\\]\nDetermine the value of \\(a + c + f\\). Let's think step b... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 6964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A journalist wants to take a photo of 5 volunteers and 2 elderly individuals they are assisting, requiring them to line up in a row. The 2 elderly individuals must stand next to each other but not at the ends. The total number of different arrangements is ____. Let's think step... | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 6965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\\\(a < 0\\\\), \\\\((3x^{2}+a)(2x+b) \\geqslant 0\\\\) holds true over the interval \\\\((a,b)\\\\), then the maximum value of \\\\(b-a\\\\) is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"split": "train",
"index": 6966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "29 boys and 15 girls came to the ball. Some of the boys danced with some of the girls (at most once with each person in the pair). After the ball, each individual told their parents how many times they danced. What is the maximum number of different numbers that the children co... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 6967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Let's think st... | math | {
"style": "rule",
"ground_truth": "647"
} | {
"split": "train",
"index": 6968,
"solution": "Let $P_n$ be the probability of getting consecutive $1,2,3$ rolls in $n$ rolls and not rolling $1,2,3$ prior to the nth roll.\n Let $x = P_3+P_5+...=1-(P_4+P_6+..)$. Following Solution 2, one can see that \\[P_{n+1}=P_{n}-\\frac{P_{n-2}}{6^3}\\] for all positive int... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all ordered triple of integers $(a_1,a_2,a_3)$ with $1 \\le a_1,a_2,a_3 \\le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\\cdot | a_{n-2}-a_{n-3} |$ for all $n\\ge 4$. Find the number of such sequences for which $... | math | {
"style": "rule",
"ground_truth": "494"
} | {
"split": "train",
"index": 6969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside the circle \\(\\omega\\), there is a circle \\(\\omega_{1}\\) that touches it at point \\(K\\). Circle \\(\\omega_{2}\\) touches circle \\(\\omega_{1}\\) at point \\(L\\) and intersects circle \\(\\omega\\) at points \\(M\\) and \\(N\\). It turns out that points \\(K\\),... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 6970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $t$ that satisfies $\\frac{1}{t+2} + \\frac{2t}{t+2} - \\frac{3}{t+2} = 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 6971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are several balls of the same shape and size in a bag, including $a+1$ red balls, $a$ yellow balls, and $1$ blue ball. Now, randomly draw a ball from the bag, with the rule that drawing a red ball earns $1$ point, a yellow ball earns $2$ points, and a blue ball earns $3$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 6972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $27^{-\\frac{1}{3}} + 32^{-\\frac{2}{5}}$. Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"split": "train",
"index": 6973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Integers a, b, c, d, and e satisfy the following three properties: \n(i) $2 \\le a < b <c <d <e <100$ \n(ii) $ \\gcd (a,e) = 1 $ (iii) a, b, c, d, e form a geometric sequence. \nWhat is the value of c? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 6974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of \\[\\cot(\\cot^{-1}5 + \\cot^{-1}11 + \\cot^{-1}17 + \\cot^{-1}23).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{97}{40}"
} | {
"split": "train",
"index": 6975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are 10 streetlights numbered from 1 to 10, two of which will be turned off under the conditions that two adjacent lights cannot be turned off at the same time and the lights at both ends cannot be turned off either, calculate the number of ways to turn off the ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 6976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "[asy]size(8cm);\n\nreal w = 2.718; // width of block\nreal W = 13.37; // width of the floor\nreal h = 1.414; // height of block\nreal H = 7; // height of block + string\nreal t = 60; // measure of theta\n\npair apex = (w/2, H); // point where the strings meet\n\npath block = (0... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 6977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each unit square of a $4 \\times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 6978,
"solution": "Notice that in each $2 \\times 2$ square contained in the grid, we can form 4 L-trominoes. By the pigeonhole principle, some color appears twice among the four squares, and there are two trominoes which contain both. Therefore each $2 \\times 2$ square contains at m... |
deepscaler | [
{
"role": "user",
"content": "A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and ... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"split": "train",
"index": 6979,
"solution": "\nWe are given a square with vertices at $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$, and we need to find the radius $d$ such that the probability a randomly chosen point within the square is within $d$ units of a lattice point is $\\frac{1}{2}$. \n\n#### Ste... |
deepscaler | [
{
"role": "user",
"content": "Solve the equations:\n1. $2x^{2}+4x+1=0$ (using the method of completing the square)\n2. $x^{2}+6x=5$ (using the formula method) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3-\\sqrt{14}"
} | {
"split": "train",
"index": 6980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have a triangle $\\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{39}"
} | {
"split": "train",
"index": 6981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the rightmost non-zero digit of the expansion of (20)(13!). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6982,
"solution": "We can rewrite this as $(10 \\times 2)(13 \\times 12 \\times 11 \\times 10 \\times 9 \\times 8 \\times 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1)=\\left(10^{3}\\right)(2 \\times 13 \\times 12 \\times 11 \\times 9 \\times 8 \\times 7 \\times 6 \\t... |
deepscaler | [
{
"role": "user",
"content": "Nine hundred forty-three minus eighty-seven equals what number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "856"
} | {
"split": "train",
"index": 6983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $452_8$ and $164_8$ in base $8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "636_8"
} | {
"split": "train",
"index": 6984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what real value of $b$ is the expression $\\frac{1}{2}b^2 + 5b - 3$ minimized? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 6985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a five-digit palindromic number is equal to the product of 45 and a four-digit palindromic number (i.e., $\\overline{\\mathrm{abcba}} = 45 \\times \\overline{\\text{deed}}$), find the largest possible value of the five-digit palindromic number. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "59895"
} | {
"split": "train",
"index": 6986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At exactly noon, Anna Kuzminichna looked out the window and saw Klava, the village shop clerk, going on a break. Two minutes past twelve, Anna Kuzminichna looked out the window again, and no one was at the closed store. Klava was absent for exactly 10 minutes, and when she retu... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 6987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a\\_n\\}$ that satisfies $a\\_n-(-1)^{n}a\\_{n-1}=n$ $(n\\geqslant 2)$, and $S\\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\\_{40}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 6988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of four-element subsets of $\\{1,2,3,4,\\dots, 20\\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\\{3,5,13,19\\}$ and $\\{6,10,20,18\\}$ are two such subset... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 6989,
"solution": "Let's say our four elements in our subset are $a,b,c,d$. We have two cases. Note that the order of the elements / the element letters themselves don't matter since they are all on equal grounds at the start.\n$\\textrm{Case } 1 \\textrm{:}$ $a+b = 16$ and $c+d = 24$... |
deepscaler | [
{
"role": "user",
"content": "Among A, B, C, and D comparing their heights, the sum of the heights of two of them is equal to the sum of the heights of the other two. The average height of A and B is 4 cm more than the average height of A and C. D is 10 cm taller than A. The sum of the heights of B and C is... | math | {
"style": "rule",
"ground_truth": "139"
} | {
"split": "train",
"index": 6990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function on $\\mathbb{R}$, when $x\\geqslant 0$, $f(x)= \\begin{cases} \\log _{\\frac {1}{2}}(x+1),0\\leqslant x < 1 \\\\ 1-|x-3|,x\\geqslant 1\\end{cases}$. Find the sum of all the zeros of the function $y=f(x)+\\frac {1}{2}$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}-1"
} | {
"split": "train",
"index": 6991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the curves $C_{1}: \\begin{cases} x=2+\\cos t \\\\ y=\\sin t-1 \\end{cases}$ (with $t$ as the parameter), and $C_{2}: \\begin{cases} x=4\\cos \\alpha \\\\ y=\\sin \\alpha \\end{cases}$ (with $\\alpha$ as the parameter), in the polar coordinate system with the origin $O$ a... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {2}-1}{ \\sqrt {2}}"
} | {
"split": "train",
"index": 6992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum number of connections required to organize a wired communication network of 10 nodes, so that if any two nodes fail, it still remains possible to transmit information between any two remaining nodes (at least through a chain via other nodes)? Let's think ste... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest positive integer that is not the sum of a positive integral multiple of $37$ and a positive composite integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 6994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-1,2)"
} | {
"split": "train",
"index": 6995,
"solution": "1. **Identify the Arithmetic Progression (AP) Relationship**: Given that $a$, $b$, and $c$ form an arithmetic progression, we can express $b$ and $c$ in terms of $a$ and a common difference $d$. Thus, $b = a + d$ and $c = a + 2d$.\n\n2. **Substitute in Line Equation... |
deepscaler | [
{
"role": "user",
"content": "The acronym XYZ is displayed on a rectangular grid similarly spaced 1 unit apart. The acronym starts X having length 2 units at the top and the bottom and slanted sides making triangles on both ends. Y is made with a vertical line of 3 units and two slanted lines extending from... | math | {
"style": "rule",
"ground_truth": "13 + 4\\sqrt{2} + \\sqrt{10}"
} | {
"split": "train",
"index": 6996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean (average), the median, and the mode of the five numbers \\( 12, 9, 11, 16, x \\) are all equal. 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": "12"
} | {
"split": "train",
"index": 6997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The TV station continuously plays 5 advertisements, consisting of 3 different commercial advertisements and 2 different Olympic promotional advertisements. The requirements are that the last advertisement must be an Olympic promotional advertisement, and the 2 Olympic promotion... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 6998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{5}}{5}$, and its right latus rectum equation is $x=5$.\n(1) Find the equation of the ellipse;\n(2) A line $l$ with a slope of $1$ passes through the right foc... | math | {
"style": "rule",
"ground_truth": "\\frac{16\\sqrt{10}}{9}"
} | {
"split": "train",
"index": 6999,
"solution": ""
} |
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