data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $\\pi$ be a permutation of $\\{1,2, \\ldots, 2015\\}$. With proof, determine the maximum possible number of ordered pairs $(i, j) \\in\\{1,2, \\ldots, 2015\\}^{2}$ with $i<j$ such that $\\pi(i) \\cdot \\pi(j)>i \\cdot j$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\binom{2014}{2}"
} | {
"split": "train",
"index": 26849,
"solution": "Let $n=2015$. The only information we will need about $n$ is that $n>5 \\sqrt[4]{4}$. For the construction, take $\\pi$ to be the $n$-cycle defined by $\\pi(k)= \\begin{cases}k+1 & \\text { if } 1 \\leq k \\leq n-1 \\\\ 1 & \\text { if } k=n\\end{cases}$. Then $\\p... |
deepscaler | [
{
"role": "user",
"content": "Yura has unusual clocks with several minute hands, moving in different directions. Yura counted that in one hour, the minute hands coincided exactly 54 times in pairs. What is the maximum number of minute hands that Yura's clock can have? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 26850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\left\\lceil\\sqrt{3}\\,\\right\\rceil+\\left\\lceil\\sqrt{33}\\,\\right\\rceil+\\left\\lceil\\sqrt{333}\\,\\right\\rceil$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 26851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of th... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26852,
"solution": "To solve this problem, we need to find the common sum of the numbers in each row, column, and diagonal of a $5 \\times 5$ square matrix using the integers from $-10$ to $14$ inclusive.\n\n1. **Calculate the Total Sum of All Numbers:**\n The integers from $-10$ to... |
deepscaler | [
{
"role": "user",
"content": "A set of $n$ numbers has the sum $s$. Each number of the set is increased by $20$, then multiplied by $5$, and then decreased by $20$. The sum of the numbers in the new set thus obtained is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5s + 80n"
} | {
"split": "train",
"index": 26853,
"solution": "Let the original set of numbers be $\\{x_1, x_2, \\ldots, x_n\\}$, and the sum of these numbers is given by $s = x_1 + x_2 + \\ldots + x_n$.\n\nEach number in the set undergoes the following transformations:\n1. Increased by $20$: $x_i + 20$\n2. Multiplied by $5$: ... |
deepscaler | [
{
"role": "user",
"content": "Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\\sqrt{113}$ units and diagonals that differ by 10 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 26854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cafeteria is tiled with the same floor tiles, and it takes 630 tiles to cover an area with 18 square decimeters of tiles. How many tiles will it take if we switch to square tiles with a side length of 6 decimeters? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 26855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the complex number \\(1 + i \\sqrt{3}\\) into its exponential form \\(re^{i \\theta}\\) and find \\(\\theta\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 26856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean of a set of $60$ numbers is $42$. If three numbers from the set, $48$, $58$, and $52$, are removed, find the arithmetic mean of the remaining set of numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41.4"
} | {
"split": "train",
"index": 26857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Yao Ming has a free throw shooting percentage of 90% during games. What is the probability that he misses one free throw out of three attempts? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.243"
} | {
"split": "train",
"index": 26858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One of the factors of $x^4+4$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2-2x+2"
} | {
"split": "train",
"index": 26859,
"solution": "To solve the problem, we need to verify if any of the given options is a factor of the polynomial $x^4 + 4$. We will test each option by either direct substitution or polynomial division.\n\n#### Option (A) $x^2 + 2$\nWe can quickly check if $x^2 + 2$ is a factor o... |
deepscaler | [
{
"role": "user",
"content": "Determine the volume of the right rectangular parallelepiped whose edges are formed by the distances from the orthocenter to the vertices of a triangle, where the radius of the circumcircle $r = 2.35$ and the angles are: $\\alpha = 63^{\\circ} 18^{\\prime} 13^{\\prime \\prime},... | math | {
"style": "rule",
"ground_truth": "12.2"
} | {
"split": "train",
"index": 26860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{1}{8}$ of $2^{32}$ equals $8^y$, what is the value of $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9.67"
} | {
"split": "train",
"index": 26861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 26862,
"solution": "1. **Calculate the total surface area of the larger cube**: \n The formula for the surface area of a cube is $6s^2$, where $s$ is the length of a side of the cube. For a cube with side length 3 inches, the surface area is:\n \\[\n 6 \\times 3^2 = 6 \\times 9... |
deepscaler | [
{
"role": "user",
"content": "In the xy-plane, consider a right triangle $ABC$ with the right angle at $C$. The hypotenuse $AB$ is of length $50$. The medians through vertices $A$ and $B$ are described by the lines $y = x + 5$ and $y = 2x + 2$, respectively. Determine the area of triangle $ABC$. Let's think... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 26863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g_{1}(x)=\\frac{1}{3}\\left(1+x+x^{2}+\\cdots\\right)$ for all values of $x$ for which the right hand side converges. Let $g_{n}(x)=g_{1}\\left(g_{n-1}(x)\\right)$ for all integers $n \\geq 2$. What is the largest integer $r$ such that $g_{r}(x)$ is defined for some real n... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 26864,
"solution": "Notice that the series is geometric with ratio $x$, so it converges if $-1<x<1$. Also notice that where $g_{1}(x)$ is defined, it is equal to $\\frac{1}{3(1-x)}$. The image of $g_{1}(x)$ is then the interval $\\left(\\frac{1}{6}, \\infty\\right)$. The image of $g_{... |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with sides opposite to the angles \\(A\\), \\(B\\), and \\(C\\) being \\(a\\), \\(b\\), and \\(c\\) respectively, and it is known that \\( \\sqrt {3}\\sin A-\\cos (B+C)=1\\) and \\( \\sin B+\\sin C= \\dfrac {8}{7}\\sin A\\) with \\(a=7\\):\n(Ⅰ) Find t... | math | {
"style": "rule",
"ground_truth": "\\dfrac {15 \\sqrt {3}}{4}"
} | {
"split": "train",
"index": 26865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $k > 0$, let $I_k = 10\\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?\n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 7\\qquad \\textbf{(C)}\... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 26866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amy and Bob choose numbers from $0,1,2,\\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 26867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$''. Moreover, both these two-digit primes are greater than 50 but less than 100. Find the total number of possible house numbers for Dr. Math. Le... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 26868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In \\(\\triangle ABC\\), \\(AB : AC = 4 : 3\\) and \\(M\\) is the midpoint of \\(BC\\). \\(E\\) is a point on \\(AB\\) and \\(F\\) is a point on \\(AC\\) such that \\(AE : AF = 2 : 1\\). It is also given that \\(EF\\) and \\(AM\\) intersect at \\(G\\) with \\(GF = 72 \\mathrm{~... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 26869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\\overline{PQ}$, points $C$ and $D$ lie on $\\overline{QR}$, and points $E$ and $F$ lie on $\\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$.\nDiagram\n[asy] dot((0,0));... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 26870,
"solution": "Note that $\\triangle{PQR}$ has area $150$ and is a $3$-$4$-$5$ right triangle. Then, by similar triangles, the altitude from $B$ to $QC$ has length $3$ and the altitude from $A$ to $FP$ has length $4$, so $[QBC]+[DRE]+[AFP]=\\frac{15}{2}+\\frac{25}{2}+\\frac{20}{2... |
deepscaler | [
{
"role": "user",
"content": "Calculate\n\\[\\prod_{n = 1}^{13} \\frac{n(n + 2)}{(n + 4)^2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{161840}"
} | {
"split": "train",
"index": 26871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four of the following test scores are Henry's and the other four are Julia's: 88, 90, 92, 94, 95, 97, 98, 99. Henry's mean score is 94. What is Julia's mean score? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "94.25"
} | {
"split": "train",
"index": 26872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\\left\\{\\begin{array}{l}x=2t+1,\\\\ y=2t\\end{array}\\right.$ (where $t$ is a parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{14}}{2}"
} | {
"split": "train",
"index": 26873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive solution to\n$\\frac 1{x^2-10x-29}+\\frac1{x^2-10x-45}-\\frac 2{x^2-10x-69}=0$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 26874,
"solution": "We could clear out the denominators by multiplying, though that would be unnecessarily tedious.\n To simplify the equation, substitute $a = x^2 - 10x - 29$ (the denominator of the first fraction). We can rewrite the equation as $\\frac{1}{a} + \\frac{1}{a - 16} - \... |
deepscaler | [
{
"role": "user",
"content": "Given that $x > 0$, $y > 0$, and ${\\!\\!}^{2x+2y}=2$, find the minimum value of $\\frac{1}{x}+\\frac{1}{y}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 26875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic function $y=-x^{2}+bx+c$ where $b$ and $c$ are constants. \n$(1)$ If $y=0$ and the corresponding values of $x$ are $-1$ and $3$, find the maximum value of the quadratic function. \n$(2)$ If $c=-5$, and the quadratic function $y=-x^{2}+bx+c$ intersects the li... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 26876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the base of a hemisphere is $100\\pi$. What is the total surface area of the hemisphere? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300\\pi"
} | {
"split": "train",
"index": 26877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carl only eats food in the shape of equilateral pentagons. Unfortunately, for dinner he receives a piece of steak in the shape of an equilateral triangle. So that he can eat it, he cuts off two corners with straight cuts to form an equilateral pentagon. The set of possible peri... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{3}-6"
} | {
"split": "train",
"index": 26878,
"solution": "Assume that the triangle has side length 1. We will show the pentagon side length $x$ is in $\\left[2 \\sqrt{3}-3, \\frac{1}{2}\\right)$. Call the triangle $A B C$ and let corners $B, C$ be cut. Choose $P$ on $A B, Q, R$ on $B C$, and $S$ on $A C$ such that $A P Q ... |
deepscaler | [
{
"role": "user",
"content": "There exist constants $a_1,$ $a_2,$ $a_3,$ $a_4,$ $a_5$ such that\n\\[\\cos^5 \\theta = a_1 \\cos \\theta + a_2 \\cos 2 \\theta + a_3 \\cos 3 \\theta + a_4 \\cos 4 \\theta + a_5 \\cos 5 \\theta\\]for all angles $\\theta.$ Find $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2.$ Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{63}{128}"
} | {
"split": "train",
"index": 26879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jillian drives along a straight road that goes directly from her house $(J)$ to her Grandfather's house $(G)$. Some of this road is on flat ground and some is downhill or uphill. Her car travels downhill at $99 \\mathrm{~km} / \\mathrm{h}$, on flat ground at $77 \\mathrm{~km} /... | math | {
"style": "rule",
"ground_truth": "308"
} | {
"split": "train",
"index": 26880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 42 stepping stones in a pond, arranged along a circle. You are standing on one of the stones. You would like to jump among the stones so that you move counterclockwise by either 1 stone or 7 stones at each jump. Moreover, you would like to do this in such a way that y... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 26881,
"solution": "Number the stones $0,1, \\ldots, 41$, treating the numbers as values modulo 42, and let $r_{n}$ be the length of your jump from stone $n$. If you jump from stone $n$ to $n+7$, then you cannot jump from stone $n+6$ to $n+7$ and so must jump from $n+6$ to $n+13$. Tha... |
deepscaler | [
{
"role": "user",
"content": "For real numbers $t,$ the point\n\\[(x,y) = \\left( e^t + e^{-t}, 3 (e^t - e^{-t}) \\right)\\]is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option.\n\nNote: $e$... | math | {
"style": "rule",
"ground_truth": "\\text{(E)}"
} | {
"split": "train",
"index": 26882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive difference between the two solutions to the equation $\\displaystyle\\sqrt[3]{4 - \\frac{x^2}{3}} = -2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $n$, let \\langle n\\rangle denote the closest integer to \\sqrt{n}. Evaluate\n\\[\\sum_{n=1}^\\infty \\frac{2^{\\langle n\\rangle}+2^{-\\langle n\\rangle}}{2^n}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 26884,
"solution": "Since $(k-1/2)^2 = k^2-k+1/4$ and $(k+1/2)^2 = k^2+k+1/4$, we have that $\\langle n \\rangle = k$ if and only if $k^2-k+1 \\leq n \\leq k^2+k$. Hence\n\\begin{align*}\n\\sum_{n=1}^\\infty \\frac{2^{\\langle n \\rangle} + 2^{-\\langle n \\rangle}}{2^n}\n&= \\sum_{k... |
deepscaler | [
{
"role": "user",
"content": "Vitya has five math lessons per week, one on each day from Monday to Friday. Vitya knows that with a probability of \\(1 / 2\\), the teacher will not check his homework at all during the week, and with a probability of \\(1 / 2\\), the teacher will check it, but only once on an... | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"split": "train",
"index": 26885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each integer $n \\ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\\le x \\le n$ and $0\\le y \\le x \\left\\lfloor \\sqrt x \\right\\rfloor$, where $\\left\\lfloor \\sqrt x \\right\\rfloor$ is the greatest integer not excee... | math | {
"style": "rule",
"ground_truth": "483"
} | {
"split": "train",
"index": 26886,
"solution": "By considering the graph of this function, it is shown that the graph is composed of trapezoids ranging from $a^2$ to $(a+1)^2$ with the top made of diagonal line $y=ax$. The width of each trapezoid is $3, 5, 7$, etc. Whenever $a$ is odd, the value of $A(n)$ increa... |
deepscaler | [
{
"role": "user",
"content": "Marcus has two numbers, $a$ and $b$. When he divides $a$ by 45 he gets a remainder of 37. When he divides $b$ by 30 he gets a remainder of $9$. What remainder does he get when he divides $a+b$ by 15? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 26887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \\%,$ and $m = a... | math | {
"style": "rule",
"ground_truth": "87"
} | {
"split": "train",
"index": 26888,
"solution": "Case 1:\n Case 2:\n We draw a number line representing the time interval. If mathematician $M_1$ comes in at the center of the time period, then the two mathematicions will meet if $M_2$ comes in somewhere between $m$ minutes before and after $M_1$ comes (a total r... |
deepscaler | [
{
"role": "user",
"content": "The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if a is less than or equal to zero; the symbol $<$ means \"less than\"; \nthe symbol $>$ means \"greater than.\" \nThe set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such th... | math | {
"style": "rule",
"ground_truth": "-1<x<7"
} | {
"split": "train",
"index": 26889,
"solution": "The problem involves solving the inequality $|3-x| < 4$. The absolute value inequality $|a| < b$ where $a$ is an expression and $b$ is a positive number, can be rewritten as $-b < a < b$. Applying this to our inequality:\n\n1. Rewrite the inequality:\n \\[\n |3... |
deepscaler | [
{
"role": "user",
"content": "Find all pairs of real numbers $(x,y)$ satisfying the system of equations\n\\begin{align*}\n \\frac{1}{x} + \\frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\\\\n \\frac{1}{x} - \\frac{1}{2y} &= 2(y^4-x^4).\n\\end{align*} Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "x = (3^{1/5}+1)/2, y = (3^{1/5}-1)/2"
} | {
"split": "train",
"index": 26890,
"solution": "By adding and subtracting the two given equations, we obtain the equivalent pair of equations\n\\begin{align*}\n2/x &= x^4 + 10x^2y^2 + 5y^4 \\\\\n1/y &= 5x^4 + 10x^2y^2 + y^4.\n\\end{align*}\nMultiplying the former by $x$ and the latter by $y$, then adding and sub... |
deepscaler | [
{
"role": "user",
"content": "A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is no less than half the number of beetles in the tenth jar. How many beetles are in the sixth jar? Let's t... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $y = -16t^2 + 34t + 25$ describes the height (in feet) of a ball thrown upwards at $34$ feet per second from $25$ feet above the ground. Determine the time (in seconds) when the ball will hit the ground. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{8}"
} | {
"split": "train",
"index": 26892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) that is not less than 9, such that for any \\( n \\) integers (which can be the same) \\( a_{1}, a_{2}, \\cdots, a_{n} \\), there always exist 9 numbers \\( a_{i_{1}}, a_{i_{2}}, \\cdots, a_{i_{9}} \\) (where \\(1 \\leq i_{1} < i_{2}... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 26893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person's commute times (in minutes) for 5 trips were 12, 8, 10, 11, and 9, respectively. The standard deviation of this set of data is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 26894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the regression equation $y = 0.849x - 85.712$, where $x$ represents the height in cm and $y$ represents the weight in kg, determine the predicted weight of a female student who is 172 cm tall. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60.316"
} | {
"split": "train",
"index": 26895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16.67\\%"
} | {
"split": "train",
"index": 26896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two vertices of a square with an area of \\( 256 \\, \\text{cm}^2 \\) lie on a circle, while the other two vertices lie on a tangent to this circle. Find the radius of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integer multiples of $77$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \\leq i < j \\leq 49$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 26898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\mathbf{a} = \\begin{pmatrix} 2 \\\\ 3 \\\\ 1 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 1 \\\\ -2 \\\\ 0 \\end{pmatrix},$ determine the scalar $s$ such that\n\\[\\begin{pmatrix} 5 \\\\ -4 \\\\ 1 \\end{pmatrix} = s(\\mathbf{a} \\times \\mathbf{b}) + p\\... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{45}"
} | {
"split": "train",
"index": 26899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the center of the circle with equation $9x^2-18x+9y^2+36y+44=0.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1,-2)"
} | {
"split": "train",
"index": 26900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When evaluated, the sum of the digits of the integer equal to \\(10^{2021} - 2021\\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18185"
} | {
"split": "train",
"index": 26901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(CD\\) be a chord of a circle \\(\\Gamma_{1}\\) and \\(AB\\) a diameter of \\(\\Gamma_{1}\\) perpendicular to \\(CD\\) at \\(N\\) with \\(AN > NB\\). A circle \\(\\Gamma_{2}\\) centered at \\(C\\) with radius \\(CN\\) intersects \\(\\Gamma_{1}\\) at points \\(P\\) and \\(... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 26902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lines tangent to a circle with center $O$ at points $A$ and $B$ intersect at point $M$. Find the chord $AB$ if the segment $MO$ is divided by it into segments equal to 2 and 18. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the number $2^{1000}$ is divided by $13$, the remainder in the division is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 26904,
"solution": "1. **Application of Fermat's Little Theorem**: Fermat's Little Theorem states that if $p$ is a prime number and $a$ is an integer not divisible by $p$, then $a^{p-1} \\equiv 1 \\pmod{p}$. Here, $p = 13$ and $a = 2$, so we have:\n \\[\n 2^{12} \\equiv 1 \\pmod{1... |
deepscaler | [
{
"role": "user",
"content": "From 6 sprinters, 4 are to be selected to participate in a 4×100 m relay. If among them, Athlete A cannot run the first leg, and Athlete B cannot run the fourth leg, how many different ways are there to form the team? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 26905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_n\\}$ is an arithmetic sequence, and if $\\frac{a_{12}}{a_{11}} < -1$, find the maximum value of $n$ for which the sum of its first $n$ terms, $s_n$, is greater than $0$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 26906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At an exchange point, there are two types of transactions:\n1) Give 2 euros - receive 3 dollars and a candy as a gift.\n2) Give 5 dollars - receive 3 euros and a candy as a gift.\n\nWhen the wealthy Buratino came to the exchange point, he only had dollars. When he left, he had ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function \\( y = \\sin(w x) \\) with \\( w > 0 \\) attains its maximum value at least 50 times in the interval \\([0,1]\\), what is the minimum value of \\( w \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100 \\pi"
} | {
"split": "train",
"index": 26908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many vertical asymptotes does the equation $y=\\frac{x-1}{x^2+6x-7}$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 26909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the tenth positive integer that is both odd and a multiple of 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 26910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $VWX$ shown, $VX = \\sqrt{13}$ and $VW = 3$. What is $\\tan V$?\n\n[asy]\npair V,W,X;\nW = (0,0);\nV = (0,3);\nX = (2,0);\ndraw(V--W--X--V);\ndraw(rightanglemark(V,W,X,7));\nlabel(\"$W$\",W,SW);\nlabel(\"$V$\",V,NW);\nlabel(\"$X$\",X,SE);\nlabel(\"$3$\",(V+W)/2,NW);... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 26911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sine and cosine values of angle $α$ are both negative, and $\\cos(75^{\\circ}+α)=\\frac{1}{3}$, find the value of $\\cos(105^{\\circ}-α)+\\sin(α-105^{\\circ})$ = \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}-1}{3}"
} | {
"split": "train",
"index": 26912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a rectangle $ABCD$, and inside it are four squares with non-overlapping interiors. Two squares have the same size and an area of 4 square inches each, located at corners $A$ and $C$ respectively. There is another small square with an area of 1 square inch, and a larger... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To obtain the graph of the function $$y=2\\sin(x+ \\frac {\\pi}{6})\\cos(x+ \\frac {\\pi}{6})$$, determine the horizontal shift required to transform the graph of the function $y=\\sin 2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{6}"
} | {
"split": "train",
"index": 26914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the number of different rectangles with sides parallel to the grid that can be formed by connecting four of the dots in a $5\\times 5$ square array of dots. (Two rectangles are different if they do not share all four vertices.) Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 26915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right-angled geometric setup, $\\angle ABC$ and $\\angle ADB$ are both right angles. The lengths of segments are given as $AC = 25$ units and $AD = 7$ units. Determine the length of segment $DB$. 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": 26916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five cards labeled A, B, C, D, and E are placed consecutively in a row. How many ways can they be re-arranged so that no card is moved more than one position away from where it started? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 26917,
"solution": "The only things we can do is leave cards where they are or switch them with adjacent cards. There is 1 way to leave them all where they are, 4 ways to switch just one adjacent pair, and 3 ways to switch two different adjacent pairs, for 8 possibilities total."
} |
deepscaler | [
{
"role": "user",
"content": "Given circle $O$, points $E$ and $F$ are located such that $E$ and $F$ are on opposite sides of the diameter $\\overline{AB}$. If $\\angle AOE = 60^\\circ$ and $\\angle BOF = 30^\\circ$, find the ratio of the area of sector $EOF$ to the area of the circle. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 26918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line $l$ is drawn through the fixed point $M(4,0)$, intersecting the parabola $y^{2}=4x$ at points $A$ and $B$. $F$ is the focus of the parabola. Calculate the minimum area of the triangle $\\triangle AFB$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 26919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the even positive divisors of 1000. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2184"
} | {
"split": "train",
"index": 26920,
"solution": "Notice that $2 k$ is a divisor of 1000 iff $k$ is a divisor of 500, so we need only find the sum of the divisors of 500 and multiply by 2. This can be done by enumerating the divisors individually, or simply by using the formula: $\\sigma\\left(2^{2} \\cdot 5^{3}\\... |
deepscaler | [
{
"role": "user",
"content": "Let \\( OP \\) be the diameter of the circle \\( \\Omega \\), and \\( \\omega \\) be a circle with center at point \\( P \\) and a radius smaller than that of \\( \\Omega \\). The circles \\( \\Omega \\) and \\( \\omega \\) intersect at points \\( C \\) and \\( D \\). The chord... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 26921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)\n\nWhat is the sum of the $x$-coordinates of all points where $f(x) = 1.8$?\n\n[asy]\n\nsize(150);\n\nreal ticklen=3;\n\nreal ti... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 26922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin (2x+\\varphi)$, if the graph is shifted to the left by $\\dfrac {\\pi}{6}$ units and the resulting graph is symmetric about the $y$-axis, determine the possible value of $\\varphi$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\dfrac {\\pi}{6}"
} | {
"split": "train",
"index": 26923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no tw... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{256}"
} | {
"split": "train",
"index": 26924,
"solution": "\nWe approach this problem by counting the number of ways ants can do their desired migration, and then multiply this number by the probability that each case occurs.\n\nLet the octahedron be labeled as $ABCDEF$, with points $B, C, D, E$ being coplanar. Then the an... |
deepscaler | [
{
"role": "user",
"content": "How many points $(x, y)$, with $x$ and $y$ both integers, are on the line with equation $y=4x+3$ and inside the region bounded by $x=25, x=75, y=120$, and $y=250$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 26925,
"solution": "We determine the number of integers $x$ with $25 \\leq x \\leq 75$ for which $120 \\leq 4x+3 \\leq 250$. When $x=30$, $4x+3=123$ and when $x=61$, $4x+3=247$. Therefore, $4x+3$ is between 120 and 250 exactly when $30 \\leq x \\leq 61$. There are $61-30+1=32$ such va... |
deepscaler | [
{
"role": "user",
"content": "Six bags contain 18, 19, 21, 23, 25, and 34 marbles, respectively. One of the bags contains marbles with cracks, while the remaining five bags contain marbles without cracks. Jenny took three of the bags, and George took two of the other bags, leaving the bag with the cracked m... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 26926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What value should the real number $m$ take for the complex number $z = m + 1 + (m - 1)i$ to be: \n(1) a real number; (2) a complex number; (3) a pure imaginary number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the value of $20$ quarters and $10$ dimes equals the value of $10$ quarters and $n$ dimes, then $n=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 26928,
"solution": "1. **Set up the equation based on the problem statement:** \n The problem states that the value of $20$ quarters and $10$ dimes is equal to the value of $10$ quarters and $n$ dimes. We know that the value of a quarter is $25$ cents and the value of a dime is $10... |
deepscaler | [
{
"role": "user",
"content": "The roots of\n\\[z^7 = -\\frac{1}{\\sqrt{2}} - \\frac{i}{\\sqrt{2}}\\]are $\\text{cis } \\theta_1$, $\\text{cis } \\theta_2$, $\\dots$, $\\text{cis } \\theta_7$, where $0^\\circ \\le \\theta_k < 360^\\circ$ for all $1 \\le k \\le 7$. Find $\\theta_1 + \\theta_2 + \\dots + \\th... | math | {
"style": "rule",
"ground_truth": "1305^\\circ"
} | {
"split": "train",
"index": 26929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a quadrilateral with $A=(3,4), B=(9,-40), C=(-5,-12), D=(-7,24)$. Let $P$ be a point in the plane (not necessarily inside the quadrilateral). Find the minimum possible value of $A P+B P+C P+D P$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "16 \\sqrt{17}+8 \\sqrt{5}"
} | {
"split": "train",
"index": 26930,
"solution": "By the triangle inequality, $A P+C P \\geq A C$ and $B P+D P \\geq B D$. So $P$ should be on $A C$ and $B D$; i.e. it should be the intersection of the two diagonals. Then $A P+B P+C P+D P=A C+B D$, which is easily computed to be $16 \\sqrt{17}+8 \\sqrt{5}$ by the ... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2}, ... , ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 26931,
"solution": "Let $T_1$ stand for $AP_1Q_1$, and $T_k = AP_kQ_k$. All triangles $T$ are similar by AA. Let the area of $T_1$ be $x$. The next trapezoid will also have an area of $x$, as given. Therefore, $T_k$ has an area of $kx$. The ratio of the areas is equal to the square of... |
deepscaler | [
{
"role": "user",
"content": "The \"Middle School Eight\" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the \"M... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 26932,
"solution": "1. **Calculate the number of games within the conference:**\n - There are 8 teams in the conference.\n - Each team plays every other team twice (once at home and once away).\n - The number of ways to choose 2 teams from 8 is given by the combination formula $... |
deepscaler | [
{
"role": "user",
"content": "How many integers $n$ satisfy $(n+2)(n-8) \\le 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 26933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( x, y, \\) and \\( k \\) are positive real numbers such that \n\\[\n5 = k^2\\left(\\dfrac{x^2}{y^2}+\\dfrac{y^2}{x^2}\\right) + k\\left(\\dfrac{x}{y}+\\dfrac{y}{x}\\right),\n\\]\nfind the maximum possible value of \\( k \\). Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{-1+\\sqrt{22}}{2}"
} | {
"split": "train",
"index": 26934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest whole number that is larger than the sum\n\\[2\\dfrac{1}{2}+3\\dfrac{1}{3}+4\\dfrac{1}{4}+5\\dfrac{1}{5}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26935,
"solution": "\n1. **Break down the mixed numbers**: \n Each term in the sum \\(2\\dfrac{1}{2}+3\\dfrac{1}{3}+4\\dfrac{1}{4}+5\\dfrac{1}{5}\\) can be separated into its integer and fractional parts:\n \\[\n 2\\dfrac{1}{2} = 2 + \\dfrac{1}{2}, \\quad 3\\dfrac{1}{3} = 3 + \\... |
deepscaler | [
{
"role": "user",
"content": "Given right $\\triangle ABC$ with legs $BC=3, AC=4$. Find the length of the shorter angle trisector from $C$ to the hypotenuse: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12\\sqrt{3}-9}{13}"
} | {
"split": "train",
"index": 26936,
"solution": "1. **Identify the triangle and its properties**: We are given a right triangle $\\triangle ABC$ with legs $BC = 3$ and $AC = 4$. Since it is a right triangle, we can use the Pythagorean theorem to find the hypotenuse $AB$:\n \\[\n AB = \\sqrt{BC^2 + AC^2} = \\s... |
deepscaler | [
{
"role": "user",
"content": "The graphs of $y=\\log_3 x$, $y=\\log_x 3$, $y=\\log_{\\frac{1}{3}} x$, and $y=\\log_x \\dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 26937,
"solution": "To solve this problem, we need to analyze the intersections of the given logarithmic functions. We start by rewriting each function in terms of a common logarithmic base or expression for easier comparison.\n\n1. **Rewriting the functions:**\n - $y = \\log_3 x$\n... |
deepscaler | [
{
"role": "user",
"content": "On the refrigerator, MATHEMATICS is spelled out with $11$ magnets, one letter per magnet. Two vowels and four consonants fall off and are put away in a bag. If the T's, M's, and A's are indistinguishable, how many distinct possible collections of letters could be put in the bag... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 26938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\begin{vmatrix} x & \\frac {1}{x} \\\\ -2 & 1\\end{vmatrix}$ ($x>0$), its range is set $A$, \n(1) If the universal set $U=\\mathbb{R}$, find $C_UA$; \n(2) For any $x \\in (0, \\frac {1}{2}]$, the inequality $f(x)+a \\geq 0$ always holds, find the ra... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 26940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin (2x+ \\frac {π}{3})- \\sqrt {3}\\sin (2x- \\frac {π}{6})$\n(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;\n(2) When $x\\in\\[- \\frac {π}{6}, \\frac {π}{3}\\]$, find the maximum and minimum ... | math | {
"style": "rule",
"ground_truth": "-\\sqrt {3}"
} | {
"split": "train",
"index": 26941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cat and Claire are having a conversation about Cat's favorite number.\n \nCat says, \"My favorite number is a two-digit positive integer with distinct nonzero digits, $\\overline{AB}$ , such that $A$ and $B$ are both factors of $\\overline{AB}$ .\"\n \nClaire says, ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 26942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct trees with exactly 7 vertices exist? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 26943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1,$ $a_2,$ $\\dots$ be a sequence of positive real numbers such that\n\\[a_n = 11a_{n - 1} - n\\]for all $n > 1.$ Find the smallest possible value of $a_1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21}{100}"
} | {
"split": "train",
"index": 26944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $S=1!+2!+3!+\\cdots +99!$, then the units' digit in the value of S is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 26945,
"solution": "1. **Identify the relevant factorials**: We need to find the units' digit of the sum $S = 1! + 2! + 3! + \\cdots + 99!$. We start by observing the units' digits of the first few factorials:\n - $1! = 1$\n - $2! = 2$\n - $3! = 6$\n - $4! = 24$\n - $5! = 12... |
deepscaler | [
{
"role": "user",
"content": "Compute the sum $i^{-103} + i^{-102} + \\cdots + i^{-1} + i^0 + i^1 + \\cdots + i^{102} + i^{103} + \\sum_{n=1}^{103} n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5355"
} | {
"split": "train",
"index": 26946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_5+a_6 > 0$, and $a_5a_6 < 0$, calculate the maximum natural number $n$ for which the sum of the first $n$ terms $S_n > 0$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 26947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 26948,
"solution": "Suppose that the two identical digits are both $1$. Since the thousands digit must be $1$, only one of the other three digits can be $1$. This means the possible forms for the number are\n$11xy,\\qquad 1x1y,\\qquad1xy1$\n Because the number must have exactly two id... |
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