data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "For any 2016 complex numbers \\( z_1, z_2, \\ldots, z_{2016} \\), it holds that\n\\[ \n\\sum_{k=1}^{2016} |z_k|^2 \\geq \\lambda \\min_{1 \\leq k \\leq 2016} \\{ |z_{k+1} - z_k|^2 \\}, \n\\]\nwhere \\( z_{2017} = z_1 \\). Find the maximum value of \\( \\lambda \\). Let's think ... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 39249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The school club sells 200 tickets for a total of $2500. Some tickets are sold at full price, and the rest are sold for one-third the price of the full-price tickets. Determine the amount of money raised by the full-price tickets. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "1250"
} | {
"split": "train",
"index": 39250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = a\\cos x - \\left(x - \\frac{\\pi}{2}\\right)\\sin x$, where $x \\in \\left[0, \\frac{\\pi}{2}\\right]$. \n$(1)$ When $a = -1$, find the range of the derivative ${f'}(x)$ of the function $f(x)$. \n$(2)$ If $f(x) \\leq 0$ always holds, find the maximum value of the... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 39251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle is drawn with a side of length $a$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit... | math | {
"style": "rule",
"ground_truth": "6a"
} | {
"split": "train",
"index": 39252,
"solution": "1. **Identify the sequence of perimeters**: \n - The perimeter of the first equilateral triangle is $3a$ since each side is $a$ and there are three sides.\n - Each subsequent triangle is formed by joining the midpoints of the sides of the previous triangle, thu... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB=10$, $AC=8$, and $BC=6$. Circle $P$ passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection of circle $P$ with sides $AC$ and $BC$ (excluding $C$). The length of segment $QR$ is Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "4.8"
} | {
"split": "train",
"index": 39253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b$, and $c$ be real numbers such that $a+b+c=100$, $ab+bc+ca=20$, and $(a+b)(a+c)=24$. Compute all possible values of $bc$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "224, -176"
} | {
"split": "train",
"index": 39254,
"solution": "We first expand the left-hand-side of the third equation to get $(a+b)(a+c)=a^{2}+ac+ab+bc=24$. From this, we subtract the second equation to obtain $a^{2}=4$, so $a=\\pm 2$. If $a=2$, plugging into the first equation gives us $b+c=98$ and plugging into the second ... |
deepscaler | [
{
"role": "user",
"content": "Four-digit \"progressive numbers\" are arranged in ascending order, determine the 30th number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1359"
} | {
"split": "train",
"index": 39255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), angle \\(A\\) is the largest angle. Points \\(M\\) and \\(N\\) are symmetric to vertex \\(A\\) with respect to the angle bisectors of angles \\(B\\) and \\(C\\) respectively. Find \\(\\angle A\\) if \\(\\angle MAN = 50^\\circ\\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 39256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer \\( n \\) is the smallest positive number that satisfies the following conditions:\n1. \\( n \\) is a multiple of 75.\n2. \\( n \\) has exactly 75 positive divisors (including 1 and itself).\n\nFind the value of \\( \\frac{n}{75} \\). Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 39257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. Determine the largest possible area of pentagon $ABCDE$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9 + 3\\sqrt{2}"
} | {
"split": "train",
"index": 39258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A, B, C, D) \\in\\{1,2, \\ldots, N\\}^{4}$ (not necessarily distinct) such that for every integer $n, A n^{3}+B n^{2}+2 C n+D$ is divisible by $N$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 39259,
"solution": "Note that $n^{0}=\\binom{n}{0}, n^{1}=\\binom{n}{1}, n^{2}=2\\binom{n}{2}+\\binom{n}{1}, n^{3}=6\\binom{n}{3}+6\\binom{n}{2}+\\binom{n}{1}$. Thus the polynomial rewrites as $6 A\\binom{n}{3}+(6 A+2 B)\\binom{n}{2}+(A+B+2 C)\\binom{n}{1}+D\\binom{n}{0}$ which by the... |
deepscaler | [
{
"role": "user",
"content": "Solve the equations:<br/>$(1)x^{2}-10x-10=0$;<br/>$(2)3\\left(x-5\\right)^{2}=2\\left(5-x\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{3}"
} | {
"split": "train",
"index": 39260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ball was floating in a lake when the lake froze. The ball was removed, leaving a hole $32$ cm across at the top and $16$ cm deep. What was the radius of the ball (in centimeters)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bottoms of two vertical poles are 20 feet apart on a flat ground. One pole is 8 feet tall and the other is 18 feet tall. Simultaneously, the ground between the poles is sloped, with the base of the taller pole being 2 feet higher than the base of the shorter pole due to the... | math | {
"style": "rule",
"ground_truth": "\\sqrt{544}"
} | {
"split": "train",
"index": 39262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{2}{\\sqrt{5} - 1}}"
} | {
"split": "train",
"index": 39263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 10. Points $V$, $W$, and $X$ are the midpoints of edges $PR$, $RQ$, and $QT$, respectively. Determine the perimeter of triangle $VWX$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "5 + 10\\sqrt{5}"
} | {
"split": "train",
"index": 39264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle \\(A B C\\) has side lengths \\(A B = 65\\), \\(B C = 33\\), and \\(A C = 56\\). Find the radius of the circle tangent to sides \\(A C\\) and \\(B C\\) and to the circumcircle of triangle \\(A B C\\). Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 39265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$, points $P$, $Q$, $R$, and $S$ are the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively, and point $T$ is the midpoint of segment $SR$. Given that the area of parallelogram $ABCD$ is 120 square centimeters, what is the area of $\\triangle PQT$... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let (b_1, b_2, ... b_7) be a list of the first 7 odd positive integers such that for each 2 ≤ i ≤ 7, either b_i + 2 or b_i - 2 (or both) must appear before b_i in the list. How many such lists are there? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 39267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cone and a cylinder with equal base radii and heights, if the axis section of the cone is an equilateral triangle, calculate the ratio of the lateral surface areas of this cone and cylinder. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 39268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Roll a die twice. Let $X$ be the maximum of the two numbers rolled. Which of the following numbers is closest to the expected value $E(X)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 39269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the parabola with equation $y=x^2$ and let $Q = (20, 14)$. There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s.$ What is $r + s$? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 39270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Hooligan Vasya likes to run on the escalator in the metro. He runs downward twice as fast as upward. If the escalator is not working, it will take Vasya 6 minutes to run up and down. If the escalator is moving downward, it will take Vasya 13.5 minutes to run up and down. How ma... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 39271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{\\sqrt{507}}{\\sqrt{48}}-\\frac{\\sqrt{175}}{\\sqrt{112}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 39273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR$, $PQ = 8$, $QR = 15$, $PR = 17$, and $QS$ is the angle bisector. Find the length of $QS$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{87.04}"
} | {
"split": "train",
"index": 39274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya and Vasya came up with ten fifth-degree polynomials. Then, Vasya sequentially called out consecutive natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice, writing the resulting values on the board ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 39275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The largest ocean in the world is the Pacific Ocean, with an area of 17,996,800 square kilometers. Rewrite this number in terms of \"ten thousand\" as the unit, and round it to the nearest \"ten thousand\" square kilometers. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"split": "train",
"index": 39276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right cone with a base radius of \\(15\\) cm and a height of \\(30\\) cm, a sphere is inscribed. The radius of the sphere can be expressed as \\(b\\sqrt{d} - b\\) cm. Find the value of \\(b + d\\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"split": "train",
"index": 39277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shopkeeper purchases 2000 pens at a cost of $0.15 each. If the shopkeeper wants to sell them for $0.30 each, calculate the number of pens that need to be sold to make a profit of exactly $120.00. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1400"
} | {
"split": "train",
"index": 39278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos(2x+\\frac{\\pi}{3})+1$, in triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively.\n\n$(I)$ If angles $A$, $B$, and $C$ form an arithmetic sequence, find the value of $f(B)$;\n\n$(II)$ If $f\\left(\\frac{B}{... | math | {
"style": "rule",
"ground_truth": "3+\\sqrt{2}"
} | {
"split": "train",
"index": 39279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the quotient when $7x^3 + 3x^2 - 5x - 8$ is divided by $x + 2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7x^2 - 11x + 17"
} | {
"split": "train",
"index": 39280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers $X_1, X_2, \\dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$ . If the expected value of $\\min(X_1,X_2,\\dots, X_{10})^4$ can be expressed as a rational number $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , what... | math | {
"style": "rule",
"ground_truth": "1002"
} | {
"split": "train",
"index": 39281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\ln x-xe^{x}+ax$ where $a\\in \\mathbb{R}$.\n(Ⅰ) If the function $f(x)$ is monotonically decreasing on $\\left[1,+\\infty \\right)$, find the range of real number $a$.\n(Ⅱ) If $a=1$, find the maximum value of $f(x)$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 39282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ , $M$ , and $C$ be nonnegative integers such that $A+M+C=10$ . Find the maximum value of $A\\cdot M\\cdot C+A\\cdot M+M\\cdot C+C\\cdot A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 39283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, it is known that $BC=1$, $B= \\frac{\\pi}{3}$, and the area of $\\triangle ABC$ is $\\sqrt{3}$. Determine the length of $AC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 39284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $6$'s and $9$'s, with at least one of each. Find the last four digits of $m$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "6996"
} | {
"split": "train",
"index": 39285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ADEH$, points $B$ and $C$ trisect $\\overline{AD}$, and points $G$ and $F$ trisect $\\overline{HE}$. In addition, $AH=AC=2$. What is the area of quadrilateral $WXYZ$ shown in the figure?\n\n[asy]\nunitsize(1cm);\npair A,B,C,D,I,F,G,H,U,Z,Y,X;\nA=(0,0);\nB=(1,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 39286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An oreo shop now sells $5$ different flavors of oreos, $3$ different flavors of milk, and $2$ different flavors of cookies. Alpha and Gamma decide to purchase some items. Since Alpha is picky, he will order no more than two different items in total, avoiding replicas. To be equ... | math | {
"style": "rule",
"ground_truth": "2100"
} | {
"split": "train",
"index": 39287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The increasing geometric sequence $x_{0},x_{1},x_{2},\\ldots$ consists entirely of integral powers of $3.$ Given that\n$\\sum_{n=0}^{7}\\log_{3}(x_{n}) = 308$ and $56 \\leq \\log_{3}\\left ( \\sum_{n=0}^{7}x_{n}\\right ) \\leq 57,$\n find $\\log_{3}(x_{14}).$ Let's think step b... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 39288,
"solution": "All these integral powers of $3$ are all different, thus in base $3$ the sum of these powers would consist of $1$s and $0$s. Thus the largest value $x_7$ must be $3^{56}$ in order to preserve the givens. Then we find by the given that $x_7x_6x_5\\dots x_0 = 3^{308}... |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 6 distinct beads be placed on a bracelet? (Note that two arrangements are the same if one can be rotated or reflected to produce the other.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 39289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of adventurers displays their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Additionally, it is known that:\n\n- If an adventurer has diamonds, they either have rubies or sapphires ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 39290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has a perimeter of 30 units and its dimensions are whole numbers. What is the maximum possible area of the rectangle in square units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 39291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of even digits in the base-7 representation of $403_{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\[(3a^3 - 7a^2 + a - 5)(4a - 6)\\] for $a = 2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"split": "train",
"index": 39293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the nonnegative difference between the roots for the equation $x^2+30x+180=-36$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 39294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of line $l$ is $\\rho\\cos(\\theta+ \\frac{\\pi}{4})= \\frac{\\sqrt{2}}{2}$, and the parametr... | math | {
"style": "rule",
"ground_truth": "\\sqrt{34}+2"
} | {
"split": "train",
"index": 39295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers $a$ , $b$ , $c$ which are differ from $1$ satisfies the following conditions;\n(1) $abc =1$ (2) $a^2+b^2+c^2 - \\left( \\dfrac{1}{a^2} + \\dfrac{1}{b^2} + \\dfrac{1}{c^2} \\right) = 8(a+b+c) - 8 (ab+bc+ca)$ Find all possible values of expression $\\dfrac{1... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 39296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lil writes one of the letters \\( \\text{P}, \\text{Q}, \\text{R}, \\text{S} \\) in each cell of a \\( 2 \\times 4 \\) table. She does this in such a way that, in each row and in each \\( 2 \\times 2 \\) square, all four letters appear. In how many ways can she do this?\n\nA) 1... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 39297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We want to design a new chess piece, the American, with the property that (i) the American can never attack itself, and (ii) if an American $A_{1}$ attacks another American $A_{2}$, then $A_{2}$ also attacks $A_{1}$. Let $m$ be the number of squares that an American attacks whe... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 39298,
"solution": "Since one of the Americans must be in the top left corner, that eliminates $m$ squares from consideration for placing additional Americans. So $m+n$ is at most 64, which implies $m n$ can be at most 1024. To achieve 1024, we can color a chessboard the normal way, a... |
deepscaler | [
{
"role": "user",
"content": "The base of isosceles $\\triangle XYZ$ is 30 units and its area is 60 square units. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{241}"
} | {
"split": "train",
"index": 39299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $a_n$ defined by the piecewise function: \n$\\begin{eqnarray*} a_n =\\left\\{ \\begin{array}{lr} 11, & \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }13\\ \\text{and\\ }14;\\\\ 13, & \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }14\\ \\text{and\\ }11;\\\\ 14... | math | {
"style": "rule",
"ground_truth": "448"
} | {
"split": "train",
"index": 39300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 3 math clubs in the school district, with 5, 7, and 8 students respectively. Each club has two co-presidents. If I randomly select a club, and then randomly select three members of that club to give a copy of $\\emph{Introduction to} \\allowbreak\\ \\emph{Counting and... | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{60}"
} | {
"split": "train",
"index": 39301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\(2^{2021}\\) and \\(5^{2021}\\) are written out one after the other. How many digits were written in total? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"split": "train",
"index": 39302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $(2-x)^{5}=a\\_{0}+a\\_{1}x+a\\_{2}x^{2}+…+a\\_{5}x^{5}$, find the value of $\\frac{a\\_0+a\\_2+a\\_4}{a\\_1+a\\_3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{61}{60}"
} | {
"split": "train",
"index": 39303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have a standard deck of 52 cards, with 4 cards in each of 13 ranks. We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such as 33355 or AAAKK). What is the probability that five cards chosen at random form a full house... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{4165}"
} | {
"split": "train",
"index": 39304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( r, s, \\) and \\( t \\) be the roots of the equation \\( x^3 - 15x^2 + 13x - 6 = 0 \\). Find the value of \\( \\frac{r}{\\frac{1}{r}+st} + \\frac{s}{\\frac{1}{s}+tr} + \\frac{t}{\\frac{1}{t}+rs} \\). Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{199}{7}"
} | {
"split": "train",
"index": 39305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the numerator and the denominator when $0.\\overline{0012}$ is expressed as a fraction in lowest terms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13332"
} | {
"split": "train",
"index": 39306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n$, let $p(n)$ denote the product of the positive integer factors of $n$. Determine the number of factors $n$ of 2310 for which $p(n)$ is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 39307,
"solution": "Note that $2310=2 \\times 3 \\times 5 \\times 7 \\times 11$. In general, we see that if $n$ has $d(n)$ positive integer factors, then $p(n)=n^{\\frac{d}{2}}$ since we can pair factors $\\left(d, \\frac{n}{d}\\right)$ which multiply to $n$. As a result, $p(n)$ is a ... |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\\circ}, 180^{\\circ},$ and $270^{\\circ}$ counterclockwise around the origin, reflection ac... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39308,
"solution": "We are given a triangle $T$ with vertices at $(0,0), (4,0),$ and $(0,3)$ and asked to determine how many sequences of three transformations from the set of rotations by $90^\\circ, 180^\\circ, 270^\\circ$ counterclockwise around the origin, and reflections across t... |
deepscaler | [
{
"role": "user",
"content": "When $x=3$ and $y=4$, what is the value of the expression $xy-x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 39309,
"solution": "When $x=3$ and $y=4$, we get $xy-x=3 \\times 4-3=12-3=9$. Alternatively, $xy-x=x(y-1)=3 \\times 3=9$."
} |
deepscaler | [
{
"role": "user",
"content": "A basketball player made the following number of free throws in 8 successive games: 6, 18, 15, 14, 19, 12, 19, and 15. What is the median number of successful free throws? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Xiao Ming ran a lap on a 360-meter circular track at a speed of 5 meters per second in the first half of the time and 4 meters per second in the second half of the time, determine the time taken to run in the second half of the distance. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 39311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the derivative of the function $f(x)$ is $f'(x) = 2 + \\sin x$, and $f(0) = -1$. The sequence $\\{a_n\\}$ is an arithmetic sequence with a common difference of $\\frac{\\pi}{4}$. If $f(a_2) + f(a_3) + f(a_4) = 3\\pi$, calculate $\\frac{a_{2016}}{a_{2}}$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 39312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin (2x+φ)$, where $|φ| < \\dfrac{π}{2}$, the graph is shifted to the left by $\\dfrac{π}{6}$ units and is symmetric about the origin. Determine the minimum value of the function $f(x)$ on the interval $[0, \\dfrac{π}{2}]$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{ \\sqrt{3}}{2}"
} | {
"split": "train",
"index": 39313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\begin{cases} 2x,& x > 0 \\\\ f(x+1),& x\\leqslant 0 \\\\ \\end{cases}$, find $f(- \\frac {4}{3})=$\\_\\_\\_\\_\\_\\_ and the maximum value of the real number $x_{0}$ that satisfies $f(f(x_{0}))=2$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 39314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that\n$\\frac 1{2!17!}+\\frac 1{3!16!}+\\frac 1{4!15!}+\\frac 1{5!14!}+\\frac 1{6!13!}+\\frac 1{7!12!}+\\frac 1{8!11!}+\\frac 1{9!10!}=\\frac N{1!18!}$\n find the greatest integer that is less than $\\frac N{100}$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "137"
} | {
"split": "train",
"index": 39315,
"solution": "Let $f(x) = (1+x)^{19}.$ Applying the binomial theorem gives us $f(x) = \\binom{19}{19} x^{19} + \\binom{19}{18} x^{18} + \\binom{19}{17} x^{17}+ \\cdots + \\binom{19}{0}.$ Since $\\frac 1{2!17!}+\\frac 1{3!16!}+\\dots+\\frac 1{8!11!}+\\frac 1{9!10!} = \\frac{\\fra... |
deepscaler | [
{
"role": "user",
"content": "We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 39316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 39317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given tha... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 39318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a value such that $8x^2 + 7x - 1 = 0$ and $24x^2+53x-7 = 0.$ What is the value of $x$? Express your answer as a simplified common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{8}"
} | {
"split": "train",
"index": 39319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange 1, 2, 3, a, b, c in a row such that letter 'a' is not at either end and among the three numbers, exactly two are adjacent. The probability is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 39320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-4.5, 4.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 39321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 39322,
"solution": "1. **Establishing the Coordinate System and Points**:\n Let's assume the side length of the hexagon is $6$ meters for simplicity. We place the hexagon in a 3D coordinate system with $A$ at the origin, i.e., $A = (0, 0, 0)$. The coordinates of $B$ and $C$ can be c... |
deepscaler | [
{
"role": "user",
"content": "If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 39323,
"solution": "1. **Substitute $x = 2$ into the equation**: Given that $2$ is a root of the polynomial $x^3 + hx + 10 = 0$, we substitute $2$ for $x$:\n \\[\n 2^3 + h \\cdot 2 + 10 = 0\n \\]\n\n2. **Simplify the equation**: Calculate $2^3$ and simplify the left-hand side:\n... |
deepscaler | [
{
"role": "user",
"content": "Let $\\{a_n\\}$ be an arithmetic sequence, and $S_n$ be the sum of its first $n$ terms, given that $S_5 < S_6$, $S_6=S_7 > S_8$, then the correct conclusion(s) is/are \\_\\_\\_\\_\\_\\_ \n$(1) d < 0$ \n$(2) a_7=0$ \n$(3) S_9 > S_5$ \n$(4) S_6$ and $S_7$ are both the maximum... | math | {
"style": "rule",
"ground_truth": "(1)(2)(4)"
} | {
"split": "train",
"index": 39324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x) = \\frac {1}{2}\\sin x - \\frac {\\sqrt {3}}{2}\\cos x$ defined on the interval $[a, b]$, the range of $f(x)$ is $[-\\frac {1}{2}, 1]$. Find the maximum value of $b-a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{3}"
} | {
"split": "train",
"index": 39325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which one of the following points is not on the graph of $y=\\dfrac{x}{x+1}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-1,1)"
} | {
"split": "train",
"index": 39326,
"solution": "To determine which point is not on the graph of the function $y = \\frac{x}{x+1}$, we need to substitute each point's $x$-coordinate into the function and check if the corresponding $y$-coordinate matches.\n\n1. **Point (A) $(0,0)$:**\n \\[\n y = \\frac{0}{0+1}... |
deepscaler | [
{
"role": "user",
"content": "A relay race preparatory team is composed of members from 8 school basketball teams and 2 school football teams, totaling 10 people. If 2 people are randomly selected, find the probability that, given one of them is a football team member, the other one is also a football team ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{17}"
} | {
"split": "train",
"index": 39327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers \\( x, y, z, w \\) satisfy \\( x + y + z + w = 1 \\). Determine the maximum value of \\( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 39328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Right $\\triangle ABC$ has $AB=3$, $BC=4$, and $AC=5$. Square $XYZW$ is inscribed in $\\triangle ABC$ with $X$ and $Y$ on $\\overline{AC}$, $W$ on $\\overline{AB}$, and $Z$ on $\\overline{BC}$. What is the side length of the square?\n\n[asy]\npair A,B,C,W,X,Y,Z;\nA=(-9,0); B=... | math | {
"style": "rule",
"ground_truth": "\\frac{60}{37}"
} | {
"split": "train",
"index": 39329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x+1)$ is an odd function, $f(x-1)$ is an even function, and $f(0)=2$, find $f(4)=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 39330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $P$, $Q$, and $R$ are externally tangent to each other and internally tangent to circle $S$. Circles $Q$ and $R$ are congruent. Circle $P$ has radius 2 and passes through the center of $S$. What is the radius of circle $Q$? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"split": "train",
"index": 39331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a square with side length $1$. How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39332,
"solution": "\nLet \\(ABCD\\) be a square with side length \\(1\\). We are tasked to determine the number of points \\(P\\) inside the square such that the square can be partitioned into \\(10\\) triangles of equal area, all having \\(P\\) as a common vertex.\n\nTo solve this p... |
deepscaler | [
{
"role": "user",
"content": "For how many values of $n$ in the set $\\{101, 102, 103, ..., 200\\}$ is the tens digit of $n^2$ even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 39333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\triangle ABF$, $\\triangle BCF$, and $\\triangle CDF$ are right-angled, with $\\angle ABF=\\angle BCF = 90^\\circ$ and $\\angle CDF = 45^\\circ$, and $AF=36$. Find the length of $CF$.\n[asy]\npair A, B, C, D, F;\nA=(0,25);\nB=(0,0);\nC=(0,-12);\nD=(12, -12);\... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 39334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle with a perimeter of 100 cm was divided into 70 identical smaller rectangles by six vertical cuts and nine horizontal cuts. What is the perimeter of each smaller rectangle if the total length of all cuts equals 405 cm? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 39335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fly is on the edge of a ceiling of a circular room with a radius of 58 feet. The fly walks straight across the ceiling to the opposite edge, passing through the center of the circle. It then walks straight to another point on the edge of the circle but not back through the ce... | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 39336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A zoo houses five different pairs of animals, each pair consisting of one male and one female. To maintain a feeding order by gender alternation, if the initial animal fed is a male lion, how many distinct sequences can the zookeeper follow to feed all the animals? Let's think ... | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"split": "train",
"index": 39337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use $ \\log_{10} 2 \\equal{} 0.301,\\ \\log_{10} 3 \\equal{} 0.477,\\ \\log_{10} 7 \\equal{} 0.845$ to find the value of $ \\log_{10} (10!)$ .\r\nNote that you must answer according to the rules:fractional part of $ 0.5$ and higher is rounded up, and everything strictly le... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 39338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a set consisting of three blue cards labeled $X$, $Y$, $Z$ and three orange cards labeled $X$, $Y$, $Z$, two cards are randomly drawn. A winning pair is defined as either two cards with the same label or two cards of the same color. What is the probability of drawing a win... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 39339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n \\ge 2$ be an integer. Alex writes the numbers $1, 2, ..., n$ in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment $s$ we denote by $d_s$ ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 39340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy? Let's think step by... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39341,
"solution": "To find the minimum number of small bottles needed to completely fill a large bottle, we need to divide the capacity of the large bottle by the capacity of the small bottle and then round up to the nearest whole number if necessary.\n\n1. **Calculate the number of ... |
deepscaler | [
{
"role": "user",
"content": "Given a finite arithmetic sequence \\(\\left\\{a_{n}\\right\\}\\) with the first term equal to 1 and the last term \\(a_{n} = 1997\\) (where \\(n > 3\\)), and the common difference being a natural number, find the sum of all possible values of \\(n\\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "3501"
} | {
"split": "train",
"index": 39342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the trapezoid \\(ABCD\\), if \\(AB = 8\\), \\(DC = 10\\), the area of \\(\\triangle AMD\\) is 10, and the area of \\(\\triangle BCM\\) is 15, then the area of trapezoid \\(ABCD\\) is \\(\\quad\\). Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 39343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10a}{r}"
} | {
"split": "train",
"index": 39344,
"solution": "1. **Identify the rate of travel**: The automobile travels $\\frac{a}{6}$ feet in $r$ seconds. Thus, the rate of travel is:\n \\[\n \\text{Rate} = \\frac{\\frac{a}{6} \\text{ feet}}{r \\text{ seconds}}\n \\]\n\n2. **Convert the rate to yards per second**: Sin... |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, $f(x) = \\begin{cases} \\log_{2}(x+1) & \\text{for } x \\geq 0 \\\\ g(x) & \\text{for } x < 0 \\\\ \\end{cases}$. Find the value of $g\\left(f(-7)\\right)$. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 39345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair 6-sided die and a fair 8-sided die are rolled once each. Calculate the probability that the sum of the numbers rolled is greater than 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"split": "train",
"index": 39346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many natural numbers between 200 and 400 are divisible by 8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 39347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x=11, y=8$, and $2x+3z=5y$, what is the value of $z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 39348,
"solution": "Since $x=11, y=8$ and $2x+3z=5y$, then $2 \\times 11+3z=5 \\times 8$ or $3z=40-22$. Therefore, $3z=18$ and so $z=6$."
} |
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