data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Six distinct integers are picked at random from $\\{1,2,3,\\ldots,10\\}$ . What is the probability that, among those selected, the second smallest is $3$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 5400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), the angle at vertex \\(A\\) is \\(60^{\\circ}\\). A circle is drawn through points \\(B\\), \\(C\\), and point \\(D\\), which lies on side \\(AB\\), intersecting side \\(AC\\) at point \\(E\\). Find \\(AE\\) if \\(AD = 3\\), \\(BD = 1\\), and \\(EC = 4\\)... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 5401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "<u>Set 4</u> **p10.** Eve has nine letter tiles: three $C$ ’s, three $M$ ’s, and three $W$ ’s. If she arranges them in a random order, what is the probability that the string “ $CMWMC$ ” appears somewhere in the arrangement?**p11.** Bethany’s Batteries sells two kinds of bat... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{28}"
} | {
"split": "train",
"index": 5402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the definite integral:\n\n$$\n\\int_{1}^{e^{2}} \\frac{\\ln ^{2} x}{\\sqrt{x}} \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24e - 32"
} | {
"split": "train",
"index": 5403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the square \\(ABCD\\), point \\(E\\) is on side \\(AD\\) such that \\(AE = 3ED\\), and point \\(F\\) is on side \\(DC\\). When the area of triangle \\(\\triangle BEF\\) is minimized, what is the ratio of the area of \\(\\triangle BEF\\) to the area of square \\(ABCD\\)? Let'... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 5404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Construct a cylindrical iron barrel with a volume of $V$. The lid of the barrel is made of aluminum alloy, and the price of aluminum alloy per unit area is three times that of iron. To minimize the cost of this container, the ratio of the bottom radius $r$ of the iron barrel to... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 5405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest number that can be written as a sum of $2$ squares in $3$ ways? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "325"
} | {
"split": "train",
"index": 5406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{987654321}{2^{30}\\cdot 5^3}$ as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 5407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using arithmetic operation signs, write the largest natural number using two twos. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 5408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are elements of the set ${ 1,2,3,4,5,6 }$ and $|a-b| \\leqslant 1$, calculate the probability that any two people playing this game form a \"friendly pair\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{9}"
} | {
"split": "train",
"index": 5409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest integer \\( k \\) such that when \\( 5 \\) is multiplied by a number consisting of \\( k \\) digits of \\( 7 \\) (i.e., \\( 777\\ldots7 \\) with \\( k \\) sevens), the resulting product has digits summing to \\( 800 \\).\nA) 86\nB) 87\nC) 88\nD) 89\nE) 90 Let'... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 5410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many values of $x$ is the expression $\\frac{x^2-9}{(x^2+2x-3)(x-3)}$ undefined? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 5411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the circumcircle of $\\triangle ABC$, denoted as circle $O$ with a radius of $2$, and $\\vec{AB} + \\vec{AC} = 2\\vec{AO}$, $|\\vec{AB}| = |\\vec{AO}|$, find the value of $\\vec{CA} \\cdot \\vec{CB}$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 5412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Any five points are taken inside or on a rectangle with dimensions 2 by 1. Let b be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than b. What is b? ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 5413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $8-\frac{6}{4-2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 5414,
"solution": "Evaluating, $8-\frac{6}{4-2}=8-\frac{6}{2}=8-3=5$."
} |
deepscaler | [
{
"role": "user",
"content": "When 542 is multiplied by 3, what is the ones (units) digit of the result? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 5415,
"solution": "Since \\( 542 \\times 3 = 1626 \\), the ones digit of the result is 6."
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of\n$\\displaystyle \\left(\\left((2+1)^{-1}+1\\right)^{-1}+1\\right)^{-1}+1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{7}"
} | {
"split": "train",
"index": 5416,
"solution": "1. **Understanding the expression and using the property of reciprocals**: We start by recognizing that for any nonzero number $a$, the reciprocal is given by $a^{-1} = \\frac{1}{a}$. We apply this to the given expression:\n \\[\n \\left(\\left((2+1)^{-1}+1\\rig... |
deepscaler | [
{
"role": "user",
"content": "An geometric sequence $\\{a_n\\}$ has 20 terms, where the product of the first four terms is $\\frac{1}{128}$, and the product of the last four terms is 512. The product of all terms in this geometric sequence is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 5417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A very large number $x$ is equal to $2^23^34^45^56^67^78^89^9$. What is the smallest positive integer that, when multiplied with $x$, produces a product that is a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 5418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain business district parking lot, the temporary parking fee is charged by time period. The fee is 6 yuan for each car if the parking duration does not exceed 1 hour, and each additional hour (or a fraction thereof is rounded up to a full hour) costs 8 yuan. It is know... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 5419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \\leq 343$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 5420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square of side length 4 , a point on the interior of the square is randomly chosen and a circle of radius 1 is drawn centered at the point. What is the probability that the circle intersects the square exactly twice? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi+8}{16}"
} | {
"split": "train",
"index": 5421,
"solution": "Consider the two intersection points of the circle and the square, which are either on the same side of the square or adjacent sides of the square. In order for the circle to intersect a side of the square twice, it must be at distance at most 1 from that side and a... |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If Bob has the ball, there is a 2/5 chance that he will toss it to Alice, and if he doesn't toss it t... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{20}"
} | {
"split": "train",
"index": 5422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$ . Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$ . If $\\angle MPN = 40^\\circ$ , ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 5423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonal $KM$ of trapezoid $KLMN$ is 3 times the length of segment $KP$ on this diagonal. The base $KN$ of the trapezoid is 3 times the length of the base $LM$. Find the ratio of the area of trapezoid $KLMN$ to the area of triangle $KPR$, where $R$ is the point of intersect... | math | {
"style": "rule",
"ground_truth": "32/3"
} | {
"split": "train",
"index": 5424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$, $B$, $C$ and $D$ are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square? [asy]\npair a=(0,1),b=(1,0),c=(0,-1),d=(-1,0);\ndraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle);\ndraw(a--b--c--d--cycle);\nlabel... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 5425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Billy Bones has two coins — one gold and one silver. One of these coins is fair, while the other is biased. It is unknown which coin is biased but it is known that the biased coin lands heads with a probability of $p=0.6$.\n\nBilly Bones tossed the gold coin and it landed heads... | math | {
"style": "rule",
"ground_truth": "5/9"
} | {
"split": "train",
"index": 5426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ordered pairs $(m, n)$ that satisfy $m$ and $n \\in \\{-1,0,1,2,3\\}$, and the equation $mx^2 + 2x + n = 0$ has real solutions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 5427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A geometric sequence of positive integers is formed for which the first term is 3 and the fifth term is 375. What is the sixth term of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9375"
} | {
"split": "train",
"index": 5428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the first $1000$ positive integers can be expressed in the form\n\\[\\lfloor 2x \\rfloor + \\lfloor 4x \\rfloor + \\lfloor 6x \\rfloor + \\lfloor 8x \\rfloor\\]where $x$ is a real number, and $\\lfloor z \\rfloor$ denotes the greatest integer less than or equal to $... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 5429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the width and length of the pan be $w$ and $l$ respectively. If the number of interior pieces is twice the number of perimeter pieces, then find the greatest possible value of $w \\cdot l$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "294"
} | {
"split": "train",
"index": 5430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\n2011 warehouses are connected by roads in such a way that from any warehouse you can reach any other, possibly passing through several roads. There are $x_{1}, \\ldots, x_{2011}$ kg of cement in the warehouses, respectively. In one trip, an arbitrary amount of cement can be t... | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"split": "train",
"index": 5431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the ent... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 5432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C:\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{3}=1$, let ${F}_{1}$ and ${F}_{2}$ be its left and right foci, respectively. A line $l$ passing through point ${F}_{2}$ with a slope of $1$ intersects ellipse $C$ at two distinct points $M$ and $N$. Calculate the area of t... | math | {
"style": "rule",
"ground_truth": "\\frac{12\\sqrt{2}}{7}"
} | {
"split": "train",
"index": 5433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solutions to $z^4 = -16i$ can be expressed in the form\n\\begin{align*}\nz_1 &= r_1 (\\cos \\theta_1 + i \\sin \\theta_1), \\\\\nz_2 &= r_2 (\\cos \\theta_2 + i \\sin \\theta_2), \\\\\nz_3 &= r_3 (\\cos \\theta_3 + i \\sin \\theta_3), \\\\\nz_4 &= r_4 (\\cos \\theta_4 + i \... | math | {
"style": "rule",
"ground_truth": "810^\\circ"
} | {
"split": "train",
"index": 5434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An infinite sequence $ \\,x_{0},x_{1},x_{2},\\ldots \\,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \\,C\\,$ such that $ \\, \\vert x_{i} \\vert \\leq C\\,$ for every $ \\,i\\geq 0$. Given any real number $ \\,a > 1,\\,$ construct a bounded infinite s... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 5435,
"solution": "\nTo solve this problem, we need to construct a bounded sequence of real numbers \\( x_0, x_1, x_2, \\ldots \\) such that for any two distinct nonnegative integers \\( i \\) and \\( j \\), the condition\n\n\\[\n|x_i - x_j| \\cdot |i - j|^a \\geq 1\n\\]\n\nis satisfi... |
deepscaler | [
{
"role": "user",
"content": "\nSergey, while being a student, worked part-time at a student cafe throughout the year. Sergey's salary was 9000 rubles per month. In the same year, Sergey paid 100000 rubles for his treatment at a healthcare institution and purchased medication prescribed by a doctor for 2000... | math | {
"style": "rule",
"ground_truth": "14040"
} | {
"split": "train",
"index": 5436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five soccer teams play a match where each team plays every other team exactly once. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After all matches have been played, the total points of the five teams are found... | math | {
"style": "rule",
"ground_truth": "13213"
} | {
"split": "train",
"index": 5437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p(x)$ be a monic quartic polynomial such that $p(1) = 2,$ $p(2) = 5,$ $p(3) = 10,$ and $p(4) = 17.$ Find $p(5).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 5438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. Additionally, for this four-digit number, one prime factor minus five times another prime factor is equal to two times the third prime factor. What is this number? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "1221"
} | {
"split": "train",
"index": 5439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the volume in cubic inches of a right, rectangular prism with side, front and bottom faces having an area 15 square inches, 10 square inches and 6 square inches, respectively? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 5440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the... | math | {
"style": "rule",
"ground_truth": "2 - \\frac{6}{\\pi}"
} | {
"split": "train",
"index": 5441,
"solution": "The probability is $2 - \\frac{6}{\\pi}$.\n\nSet coordinates so that the original tiling includes the (filled) square \n$S = \\{(x,y): 0 \\leq x,y \\leq 1 \\}$. It is then equivalent to choose the second square by first choosing a point uniformly at random in $S$ to... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos (\\sqrt{3}x+\\varphi)$, where $\\varphi \\in (-\\pi, 0)$. If the function $g(x)=f(x)+f'(x)$ (where $f'(x)$ is the derivative of $f(x)$) is an even function, determine the value of $\\varphi$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi }{3}"
} | {
"split": "train",
"index": 5442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $-15 \\leq n \\leq 10$ satisfy $(n-1)(n+3)(n + 7) < 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 5443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point P is any point on the surface of the circumscribed sphere of a cube ABCD-A1B1C1D1 with edge length 2. What is the maximum volume of the tetrahedron P-ABCD? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4(1+\\sqrt{3})}{3}"
} | {
"split": "train",
"index": 5444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the following expressions:\n\n(1) $2 \\sqrt{12} -6 \\sqrt{ \\frac{1}{3}} + \\sqrt{48}$\n\n(2) $(\\sqrt{3}-\\pi)^{0}-\\frac{\\sqrt{20}-\\sqrt{15}}{\\sqrt{5}}+(-1)^{2017}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 2"
} | {
"split": "train",
"index": 5445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{2a^{-1}+\\frac{a^{-1}}{2}}{a}$ when $a= \\frac{1}{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5446,
"solution": "1. **Identify the expression and substitute $a$:**\n Given the expression $\\frac{2a^{-1}+\\frac{a^{-1}}{2}}{a}$, we need to evaluate it at $a = \\frac{1}{2}$.\n\n2. **Calculate $a^{-1}$:**\n Since $a^{-1}$ is the reciprocal of $a$, when $a = \\frac{1}{2}$, we h... |
deepscaler | [
{
"role": "user",
"content": "One of the eight faces of a hexagonal prism will be transformed into the base of a new pyramid. Calculate the maximum sum of the number of exterior faces, vertices, and edges of the resultant composite shape (combining the prism and pyramid) when the pyramid is added to each ty... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 5447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer not exceeding the number $\\left( 1 + \\frac{\\sqrt 2 + \\sqrt 3 + \\sqrt 4}{\\sqrt 2 + \\sqrt 3 + \\sqrt 6 + \\sqrt 8 + 4}\\right)^{10}$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 5448,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the product of the base nine numbers $35_9$ and $47_9$, express it in base nine, and find the base nine sum of the digits of this product. Additionally, subtract $2_9$ from the sum of the digits. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "22_9"
} | {
"split": "train",
"index": 5449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Abbot writes the letter $A$ on the board. Every minute, he replaces every occurrence of $A$ with $A B$ and every occurrence of $B$ with $B A$, hence creating a string that is twice as long. After 10 minutes, there are $2^{10}=1024$ letters on the board. How many adjacent pairs ... | math | {
"style": "rule",
"ground_truth": "341"
} | {
"split": "train",
"index": 5450,
"solution": "Let $a_{n}$ denote the number of adjacent pairs of letters that are the same after $n$ minutes, and $b_{n}$ the number of adjacent pairs that are different. Lemma 1. $a_{n}=b_{n-1}$ for all $n \\geq 0$. Proof. Any adjacent pair of identical letters $X X$ at stage $n... |
deepscaler | [
{
"role": "user",
"content": "Add $123.4567$ to $98.764$ and round your answer to the nearest hundredth. Then, subtract $0.02$ from the rounded result. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "222.20"
} | {
"split": "train",
"index": 5451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of three numbers, all equally likely to be $1$, $2$, $3$, or $4$, drawn from an urn with replacement, is $9$, calculate the probability that the number $3$ was drawn each time. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{13}"
} | {
"split": "train",
"index": 5452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial equation \\[x^4 + dx^2 + ex + f = 0,\\] where \\(d\\), \\(e\\), and \\(f\\) are rational numbers, has \\(3 - \\sqrt{5}\\) as a root. It also has two integer roots. Find the fourth root. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 5453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 1500$ and $AD = 3000$. Let $\\angle A = 30^\\circ$, $\\angle D = 60^\\circ$, and $P$ and $Q$ be the midpoints of $\\overline{BC}$ and $\\overline{AD}$, respectively. Determine the length $PQ$. Let's th... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 5454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $n$ such that $\\frac{n}{n+150}$ is equal to a terminating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the following pair of equations:\n\\[120x^4 + ax^3 + bx^2 + cx + 18 = 0\\] and \n\\[18x^5 + dx^4 + ex^3 + fx^2 + gx + 120 = 0\\]\nThese equations have a common rational root $k$ which is not an integer and is positive. Determine $k$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 5456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle was cut into three rectangles, two of which have dimensions 9 m x 12 m and 10 m x 15 m. What is the maximum possible area of the original rectangle? Express your answer in square meters. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "330"
} | {
"split": "train",
"index": 5457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \\{1,2,\\dots,2p\\}$ are there, the sum of whose elements is divisible by $ p$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{1}{p} \\left(\\binom{2p}{p} - 2 \\right)"
} | {
"split": "train",
"index": 5458,
"solution": "\nLet \\( p \\) be an odd prime number. We are tasked with finding the number of \\( p \\)-element subsets \\( A \\) of the set \\(\\{1, 2, \\dots, 2p\\}\\) such that the sum of the elements in \\( A \\) is divisible by \\( p \\).\n\n### Step 1: Representation of Su... |
deepscaler | [
{
"role": "user",
"content": "A positive number is increased by $60\\%$. By what percentage should the result be decreased to return to the original value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37.5\\%"
} | {
"split": "train",
"index": 5459,
"solution": "Solution 1: Suppose that the original number is 100. When 100 is increased by $60\\%$, the result is 160. To return to the original value of 100, 160 must be decreased by 60. This percentage is $\\frac{60}{160} \\times 100\\%=\\frac{3}{8} \\times 100\\%=37.5\\%$. So... |
deepscaler | [
{
"role": "user",
"content": "James has a total of 66 dollars in his piggy bank. He only has one dollar bills and two dollar bills in his piggy bank. If there are a total of 49 bills in James's piggy bank, how many one dollar bills does he have? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 5460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular prism measures 10-inches by 20-inches by 10-inches. What is the length, in inches, of the diagonal connecting point A and point B? Express your answer in simplest radical form. [asy]\nunitsize(0.75cm);\ndefaultpen(linewidth(0.7pt)+fontsize(10pt));\ndotfactor=4;\n\... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{6}"
} | {
"split": "train",
"index": 5461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number such that $2^{x}=3$. Determine the value of $4^{3 x+2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11664"
} | {
"split": "train",
"index": 5462,
"solution": "We have $$4^{3 x+2}=4^{3 x} \\cdot 4^{2}=\\left(2^{2}\\right)^{3 x} \\cdot 16=2^{6 x} \\cdot 16=\\left(2^{x}\\right)^{6} \\cdot 16=3^{6} \\cdot 16=11664$$"
} |
deepscaler | [
{
"role": "user",
"content": "(1) If 7 students stand in a row, and students A and B must stand next to each other, how many different arrangements are there? \n(2) If 7 students stand in a row, and students A, B, and C must not stand next to each other, how many different arrangements are there? \n(3) If... | math | {
"style": "rule",
"ground_truth": "3720"
} | {
"split": "train",
"index": 5463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $A(1,1)$ is a point on the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and $F\\_1$, $F\\_2$ are the two foci of the ellipse such that $|AF\\_1|+|AF\\_2|=4$.\n(I) Find the standard equation of the ellipse;\n(II) Find the equation of the t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 5464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$?\n Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "799"
} | {
"split": "train",
"index": 5465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be an odd function defined on $(-\\infty, +\\infty)$, and $f(x+2) = -f(x)$. Given that $f(x) = x$ for $0 \\leq x \\leq 1$, find $f(3\\pi)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 - 3\\pi"
} | {
"split": "train",
"index": 5466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define\n\\[\nd_k = k + \\cfrac{1}{3k + \\cfrac{1}{3k + \\cfrac{1}{3k + \\dotsb}}}.\n\\]\nCalculate $\\sum_{k = 1}^{10} (d_k^2 + 2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 5467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$.\n Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 5468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( P \\) is a point on the hyperbola \\( C: \\frac{x^{2}}{4}-\\frac{y^{2}}{12}=1 \\), \\( F_{1} \\) and \\( F_{2} \\) are the left and right foci of \\( C \\), and \\( M \\) and \\( I \\) are the centroid and incenter of \\(\\triangle P F_{1} F_{2}\\) respectively, ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 5469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Group the set of positive odd numbers {1, 3, 5, ...} from smallest to largest, where the $n$-th group contains $2n-1$ odd numbers. That is, the first group, the second group, the third group... consist of the sets {1}, {3, 5, 7}, {9, 11, 13, 15, 17}, ..., respectively. In which... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 5470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In △ABC, B = $$\\frac{\\pi}{3}$$, AB = 8, BC = 5, find the area of the circumcircle of △ABC. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49\\pi}{3}"
} | {
"split": "train",
"index": 5471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(A, B, C\\), and \\(D\\) be four points that are not coplanar. A plane passes through the centroid of triangle \\(ABC\\) that is parallel to the lines \\(AB\\) and \\(CD\\). In what ratio does this plane divide the median drawn to the side \\(CD\\) of triangle \\(ACD\\)? ... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 5472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graphs of the equations\n$y=k, \\qquad y=\\sqrt{3}x+2k, \\qquad y=-\\sqrt{3}x+2k,$\nare drawn in the coordinate plane for $k=-10,-9,-8,\\ldots,9,10.\\,$ These 63 lines cut part of the plane into equilateral triangles of side $2/\\sqrt{3}.\\,$ How many such triangles are for... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"split": "train",
"index": 5473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six congruent circles form a ring with each circle externally tangent to the two circles adjacent to it. All six circles are internally tangent to a circle $\\cal C$ with radius 30. Let $K$ be the area of the region inside $\\cal C$ and outside all of the six circles in the r... | math | {
"style": "rule",
"ground_truth": "942"
} | {
"split": "train",
"index": 5474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real number \\( x \\), let \\( [x] \\) denote the greatest integer less than or equal to \\( x \\). Find the positive integer \\( n \\) such that \\(\\left[\\log _{2} 1\\right] + \\left[\\log _{2} 2\\right] + \\left[\\log _{2} 3\\right] + \\cdots + \\left[\\log _{2} n\\righ... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"split": "train",
"index": 5475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be a positive integer whose decimal representation contains 11235 as a contiguous substring, and let $k$ be a positive integer such that $10^{k}>N$. Find the minimum possible value of $$ \\frac{10^{k}-1}{\\operatorname{gcd}\\left(N, 10^{k}-1\\right)} $$ Let's think step... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 5476,
"solution": "Set $m=\\frac{10^{k}-1}{\\operatorname{gcd}\\left(N, 10^{k}-1\\right)}$. Then, in lowest terms, $\\frac{N}{10^{k}-1}=\\frac{a}{m}$ for some integer $a$. On the other hand, the decimal expansion of $\\frac{N}{10^{k}-1}$ simply consists of the decimal expansion of $N$... |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = 4\\cos x\\sin \\left(x+ \\frac{\\pi}{6}\\right)-1$.\n(Ⅰ) Determine the smallest positive period of $f(x)$;\n(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $\\left[- \\frac{\\pi}{6}, \\frac{\\pi}{4}\\right]$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $2175_{9} + 1714_{9} + 406_9$. Express your answer in base $9$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4406_{9}"
} | {
"split": "train",
"index": 5478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Greater Fourteen Basketball League has two divisions, each containing seven teams. Each team plays each of the other teams in its own division twice and every team in the other division twice. How many league games are scheduled? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 5479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $7 \\cdot 9\\frac{2}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65\\frac{4}{5}"
} | {
"split": "train",
"index": 5480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a standard dice, the sum of the numbers of pips on opposite faces is always 7. Four standard dice are glued together as shown. What is the minimum number of pips that could lie on the whole surface?\nA) 52\nB) 54\nC) 56\nD) 58\nE) 60 Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 5481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)2(\\sqrt{3}-\\sqrt{5})+3(\\sqrt{3}+\\sqrt{5})$;<br/>$(2)-{1}^{2}-|1-\\sqrt{3}|+\\sqrt[3]{8}-(-3)×\\sqrt{9}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11 - \\sqrt{3}"
} | {
"split": "train",
"index": 5482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Given a point P(-4, 3) on the terminal side of angle $\\alpha$, calculate the value of $$\\frac {\\cos( \\frac {\\pi}{2}+\\alpha)\\sin(-\\pi-\\alpha)}{\\cos( \\frac {11\\pi}{2}-\\alpha )\\sin( \\frac {9\\pi}{2}+\\alpha )}$$.\n(2) If $\\sin x= \\frac {m-3}{m+5}$ and $\\cos x... | math | {
"style": "rule",
"ground_truth": "-\\frac {5}{12}"
} | {
"split": "train",
"index": 5483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( w \\) be a complex number such that \\( |w - 3 + 2i| = 4 \\). Find the minimum value of\n\\[\n|w + 1 + 2i|^2 + |w - 7 - 2i|^2.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 5484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any real number \\(x\\), \\(\\lfloor x \\rfloor\\) denotes the largest integer less than or equal to \\(x\\). For example, \\(\\lfloor 4.2 \\rfloor = 4\\) and \\(\\lfloor 0.9 \\rfloor = 0\\).\n\nIf \\(S\\) is the sum of all integers \\(k\\) with \\(1 \\leq k \\leq 999999\\)... | math | {
"style": "rule",
"ground_truth": "999999000"
} | {
"split": "train",
"index": 5485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 5486,
"solution": "1. **Identify the condition for the area to be less than the circumference**:\n Given a circle with diameter $d$, the circumference $C$ and area $A$ are given by:\n \\[\n C = \\pi d \\quad \\text{and} \\quad A = \\pi \\left(\\frac{d}{2}\\right)^2 = \\frac{\\pi... |
deepscaler | [
{
"role": "user",
"content": "In a park, there is a row of flags arranged in the sequence of 3 yellow flags, 2 red flags, and 4 pink flags. Xiaohong sees that the row ends with a pink flag. Given that the total number of flags does not exceed 200, what is the maximum number of flags in this row? Let's think... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 5487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose Lucy picks a letter at random from the extended set of characters 'ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'. What is the probability that the letter she picks is in the word 'MATHEMATICS123'? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{36}"
} | {
"split": "train",
"index": 5488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive integers such that $(2a+b)(2b+a)=4752$ . Find the value of $ab$ .\n\n*Proposed by James Lin* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "520"
} | {
"split": "train",
"index": 5489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Divide the sequence $\\{2n+1\\}$ cyclically into one-term, two-term, three-term, four-term groups as follows: $(3), (5,7), (9,11,13), (15,17,19,21), (23), (25,27), (29,31,33), (35,37,39,41), (43), \\cdots$. What is the sum of the numbers in the 100th group? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "1992"
} | {
"split": "train",
"index": 5490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the root $x$ of the equation $\\log x = 4 - x$ where $x \\in (k, k+1)$, and $k \\in \\mathbb{Z}$. What is the value of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "k = 3"
} | {
"split": "train",
"index": 5491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be a triangle with a right angle $\\angle ABC$ . Let $D$ be the midpoint of $\\overline{BC}$ , let $E$ be the midpoint of $\\overline{AC}$ , and let $F$ be the midpoint of $\\overline{AB}$ . Let $G$ be the midpoint of $\\overline{EC}$ . On... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 5492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin(2x+\\frac{\\pi}{6})+\\sin(2x-\\frac{\\pi}{6})+\\cos{2x}+a$ (where $a \\in \\mathbb{R}$ and $a$ is a constant):\n\n(1) Find the smallest positive period of the function and the intervals of monotonic increase.\n\n(2) If the minimum value of $f(x)$ ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a polar coordinate system, the midpoint of the line segment whose endpoints are $\\left( 8, \\frac{5 \\pi}{12} \\right)$ and $\\left( 8, -\\frac{3 \\pi}{12} \\right)$ is the point $(r, \\theta).$ Enter $(r, \\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$ Let's think... | math | {
"style": "rule",
"ground_truth": "\\left( 4, \\frac{\\pi}{12} \\right)"
} | {
"split": "train",
"index": 5494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A total area of \\( 2500 \\, \\mathrm{m}^2 \\) will be used to build identical houses. The construction cost for a house with an area \\( a \\, \\mathrm{m}^2 \\) is the sum of the material cost \\( 100 p_{1} a^{\\frac{3}{2}} \\) yuan, labor cost \\( 100 p_{2} a \\) yuan, and ot... | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 5495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$ . Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "672"
} | {
"split": "train",
"index": 5496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of scalene triangles where all sides are integers and have a perimeter less than 20. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 5497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of $x+2$ to $2x+2$ is equal to the ratio of $4x+3$ to $7x+3$. What is the product of all real x which satisfy this statement? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 5498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the units digit of the product of the first three positive composite numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 5499,
"solution": ""
} |
Subsets and Splits
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