data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The endpoints of a line segment $MN$ with a fixed length of $4$ move on the parabola $y^{2}=x$. Let $P$ be the midpoint of the line segment $MN$. The minimum distance from point $P$ to the $y$-axis is ______. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{4}"
} | {
"split": "train",
"index": 23649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of triangle \\( ABC \\) is 1. On the rays \\( AB \\), \\( BC \\), and \\( CA \\), points \\( B' \\), \\( C' \\), and \\( A' \\) are marked respectively, such that:\n\n\\[ BB' = AB, \\quad CC' = 2BC, \\quad AA' = 3CA \\]\n\nCalculate the area of triangle \\( A'B'C' \\).... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 23650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $f(x)$ is a function such that\n\\[f(xy) + x = xf(y) + f(x)\\]for all real numbers $x$ and $y.$ If $f(-1) = 5$ then compute $f(-1001).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2005"
} | {
"split": "train",
"index": 23651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the coefficient of the term containing $x^4$ in the expansion of $(x-1)(x-2)(x-3)(x-4)(x-5)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"split": "train",
"index": 23652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the graph of the parabola $y=ax^2 - 3x + 3 \\ (a \\neq 0)$ is symmetric with the graph of the parabola $y^2 = 2px \\ (p > 0)$ with respect to the line $y = x + m$. Find the product of the real numbers $a$, $p$, and $m$. Let's think step... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 23653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 23654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, ge... | math | {
"style": "rule",
"ground_truth": "7174"
} | {
"split": "train",
"index": 23655,
"solution": "It is not difficult to see that the maximum possible value of $V$ can be achieved when the dimensions are $(0.5+\\epsilon) \\times(0.5+\\epsilon) \\times\\left(8070-\\epsilon^{\\prime}\\right)=2017.5-\\epsilon^{\\prime \\prime}$ for some very small reals $\\epsilon... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overrightarrow{OA}=(-2,4)$, $\\overrightarrow{OB}=(-a,2)$, $\\overrightarrow{OC}=(b,0)$, $a > 0$, $b > 0$, and $O$ is the coordinate origin. If points $A$, $B$, and $C$ are collinear, find the minimum value of $\\frac{1}{a}+\\frac{1}{b}$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{3 + 2\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 23656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a jumbo bag of bows, $\\frac{1}{5}$ are red, $\\frac{1}{2}$ are blue, $\\frac{1}{10}$ are green and the remaining 30 are white. How many of the bows are green? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 23657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\overline{AB}$ be a diameter of circle $\\omega$. Extend $\\overline{AB}$ through $A$ to $C$. Point $T$ lies on $\\omega$ so that line $CT$ is tangent to $\\omega$. Point $P$ is the foot of the perpendicular from $A$ to line $CT$. Suppose $\\overline{AB} = 18$, and let $m... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 23658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ali chooses one of the stones from a group of $2005$ stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do n... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 23659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that points $\\mathbf{A}$ and $\\mathbf{B}$ lie on the curves $C_{1}: x^{2} - y + 1 = 0$ and $C_{2}: y^{2} - x + 1 = 0$ respectively, determine the minimum value of $|AB|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 23660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is $S_n$. If $\\frac{S_6}{S_3} = 3$, then calculate $\\frac{S_{12}}{S_{9}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 23661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $d$ and $e$ are digits. For how many pairs of $(d, e)$ is $2.0d06e > 2.006$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 23662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with radius 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Determine the length of the longer side of the rectangle, expressed in centimeters and in terms of $\\pi$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"split": "train",
"index": 23663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\(a, b, c, d\\) are real numbers satisfying \\(a \\geq b \\geq c \\geq d \\geq 0\\), \\(a^2 + d^2 = 1\\), \\(b^2 + c^2 = 1\\), and \\(ac + bd = \\frac{1}{3}\\). Find the value of \\(ab - cd\\). Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 23664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An elastic ball falls from a height of 128m, where each bounce reaches half of its previous height before falling again. Calculate the total distance traveled by the ball when it hits the ground for the 8th time. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "382"
} | {
"split": "train",
"index": 23665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is a function defined on $[1,+\\infty)$, and\n$$\nf(x)=\\begin{cases}\n1-|2x-3|, & 1\\leqslant x < 2, \\\\\n\\frac{1}{2}f\\left(\\frac{1}{2}x\\right), & x\\geqslant 2,\n\\end{cases}\n$$\nthen the number of zeros of the function $y=2xf(x)-3$ in the interval $(1... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 23666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the largest and smallest prime factors of 990? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 23667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "306"
} | {
"split": "train",
"index": 23668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(a, b, c\\) are distinct positive integers such that \\(\\{a+b, b+c, c+a\\} = \\left\\{n^2, (n+1)^2, (n+2)^2\\right\\}\\), where \\(n\\) is a positive integer. What is the minimum value of \\(a^2 + b^2 + c^2\\)? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "1297"
} | {
"split": "train",
"index": 23669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product $8 \\times .25 \\times 2 \\times .125 =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 23670,
"solution": "1. **Convert Decimals to Fractions**: The given decimals are converted to fractions for easier multiplication.\n - $0.25$ as a fraction is $\\frac{1}{4}$.\n - $0.125$ as a fraction is $\\frac{1}{8}$.\n\n2. **Set Up the Product**: Substitute the decimals with th... |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 500 inclusive is $$(\\sin t + i\\cos t)^n = \\sin(2nt) + i\\cos(2nt)$$ true for all real $t$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 23671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Investigate the formula of \\\\(\\cos nα\\\\) and draw the following conclusions:\n\\\\(2\\cos 2α=(2\\cos α)^{2}-2\\\\),\n\\\\(2\\cos 3α=(2\\cos α)^{3}-3(2\\cos α)\\\\),\n\\\\(2\\cos 4α=(2\\cos α)^{4}-4(2\\cos α)^{2}+2\\\\),\n\\\\(2\\cos 5α=(2\\cos α)^{5}-5(2\\cos α)^{3}+5(2\\c... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 23672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 6:48? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 23673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The larger of two consecutive odd integers is three times the smaller. What is their sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 23674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve \\[\\frac{x^2+2x+3}{x+4}=x+5\\]for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{17}7"
} | {
"split": "train",
"index": 23675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a plane Cartesian coordinate system, a point whose x and y coordinates are both integers is called a \"lattice point.\" How many lattice points are there inside and on the boundaries of the triangle formed by the line $7x + 11y = 77$ and the coordinate axes? Let's think step... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 23676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $C$ with radius 2 has diameter $\\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\\overline{AB}$. The radius of circle $D$ is three times the radius of ... | math | {
"style": "rule",
"ground_truth": "254"
} | {
"split": "train",
"index": 23677,
"solution": "We use the notation of Solution 1 for triangle $\\triangle DEC$ \\[\\sin \\angle EDC = \\frac {EF}{DE} = \\frac {1}{4} \\implies \\cos \\angle EDC = \\frac {\\sqrt{15}}{4}.\\] We use Cosine Law for $\\triangle DEC$ and get: \\[(4r)^2 +(2 – 3r)^2 – 2 \\cdot 4r \\cdo... |
deepscaler | [
{
"role": "user",
"content": "Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\\ldots, P_nP_1$ such that\nthe points $P_1, P_2,\\ldots, P_n$ are coplanar and no three of them are collinear,\neach of the $n$ line segments intersects at least one of the other line segmen... | math | {
"style": "rule",
"ground_truth": "199"
} | {
"split": "train",
"index": 23678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function \\( f \\) is defined on the complex numbers by \\( f(z) = (a + bi)z^2 \\), where \\( a \\) and \\( b \\) are real numbers. The function has the property that for each complex number \\( z \\), \\( f(z) \\) is equidistant from both \\( z \\) and the origin. Given that... | math | {
"style": "rule",
"ground_truth": "\\frac{99}{4}"
} | {
"split": "train",
"index": 23679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two lines $l_{1}$: $x+my+6=0$, and $l_{2}$: $(m-2)x+3y+2m=0$, if the lines $l_{1}\\parallel l_{2}$, then $m=$_______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 23680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\alpha \\in (0, \\frac{\\pi}{2})$, and $\\tan 2\\alpha = \\frac{\\cos \\alpha}{2-\\sin \\alpha}$, calculate the value of $\\tan \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{15}"
} | {
"split": "train",
"index": 23681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the inequality $x^2+ax+1\\geqslant 0$ holds for all $x\\in\\left(0, \\frac{1}{2}\\right]$, then the minimum value of $a$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{2}"
} | {
"split": "train",
"index": 23682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A wheel is rolled without slipping through $15$ laps on a circular race course with radius $7$ . The wheel is perfectly circular and has radius $5$ . After the three laps, how many revolutions around its axis has the wheel been turned through? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate \\( t(0) - t(\\pi / 5) + t\\left((\\pi / 5) - t(3 \\pi / 5) + \\ldots + t\\left(\\frac{8 \\pi}{5}\\right) - t(9 \\pi / 5) \\right) \\), where \\( t(x) = \\cos 5x + * \\cos 4x + * \\cos 3x + * \\cos 2x + * \\cos x + * \\). A math student mentioned that he could compute... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select a number from \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), \\\\(4\\\\), \\\\(5\\\\), \\\\(6\\\\), \\\\(7\\\\), and calculate the probability of the following events:\n\n\\\\((1)\\\\) The selected number is greater than \\\\(3\\\\);\n\n\\\\((2)\\\\) The selected number is divis... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{7}"
} | {
"split": "train",
"index": 23685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Apples are sold at the farmer's market at a rate of $\\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 7 line segments with integer lengths in centimeters: $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, satisfying $a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7$, and any three of these line segments cannot form a triangle. If $a_1 = 1$ cm and $a_7 = 21$ cm, then $a_6 = \\ $... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 23687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Will stands at a point \\(P\\) on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of \\(n^{\\circ}\\) and \\((n+1)^{\\circ}\\) with the tangent at \\(P\\), where \\(n\\) is a positive integer less than 90. ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 23688,
"solution": "Note that we want the path drawn out by the lasers to come back to \\(P\\) in as few steps as possible. Observe that if a laser is fired with an angle of \\(n\\) degrees from the tangent, then the number of points it creates on the circle is \\(\\frac{180}{\\operat... |
deepscaler | [
{
"role": "user",
"content": "In how many distinct ways can I arrange my six keys on a keychain, if my house key must be exactly opposite my car key and my office key should be adjacent to my house key? For arrangement purposes, two placements are identical if one can be obtained from the other through rota... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that sinα + cosα = $\\frac{7}{5}$, find the value of tanα. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 23690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence 1, 1+2, 2+3+4, 3+4+5+6, ..., find the value of the 8th term. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 23691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Rotokas alphabet, consisting of letters A, E, G, I, K, O, P, R, T, U, and V (note S is removed from the original alphabet), is used to design four-letter license plates. How many license plates are possible that start with E, end with O, cannot contain I, and must not have ... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 23692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the numbers 1, 2, 3, 4, 5, there are $5!$ permutations $a_1, a_2, a_3, a_4, a_5$. Find the number of distinct permutations where $a_k \\geq k - 2$ for all $k = 1, 2, 3, 4, 5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 23693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers are in the following list: $$-4, -1, 2, 5,\\ldots, 32$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 23694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? Let's thi... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 23695,
"solution": "1. **Calculate the total amount spent and the individual shares:**\n - Total amount spent by Tom, Dorothy, and Sammy:\n \\[\n 105 + 125 + 175 = 405 \\text{ dollars}\n \\]\n - Since they agreed to split the costs evenly, each person's share is:\n ... |
deepscaler | [
{
"role": "user",
"content": "If physical education is not the first class, and Chinese class is not adjacent to physics class, calculate the total number of different scheduling arrangements for five subjects - mathematics, physics, history, Chinese, and physical education - on Tuesday morning. Let's think... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 23696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jeff's five assignment scores are 89, 92, 88, 95 and 91. What is the arithmetic mean of these five scores? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 23697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has vertices at coordinates (2,2), (5,6) and (6,2). What is the number of units in the length of the longest side of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 23698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\\overline{DE}$, vertex $X$ on $\\overline{DF}$, and vertices $Y$ and $Z$ on $\\overline{EF}$. In terms of the side length $WX = \\epsilon$, the area of $WXYZ$ can be expres... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 23699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A finite set $\\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\\mathcal{S}\\cup\\{1\\}$ is $13$ less than the mean of $\\mathcal{S}$, and the mean of $\\mathcal{S}\\cup\\{2001\\}$ is $27$ more than the mean of $\\mathcal{S}$. Find the mean of $... | math | {
"style": "rule",
"ground_truth": "651"
} | {
"split": "train",
"index": 23700,
"solution": "Since this is a weighted average problem, the mean of $S$ is $\\frac{13}{27}$ as far from $1$ as it is from $2001$. Thus, the mean of $S$ is $1 + \\frac{13}{13 + 27}(2001 - 1) = \\boxed{651}$.\n ~ pi_is_3.14"
} |
deepscaler | [
{
"role": "user",
"content": "For the ellipse $16x^2 - 64x + y^2 + 4y + 4 = 0,$ find the distance between the foci. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{15}"
} | {
"split": "train",
"index": 23701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the sum:\n\\[\n\\sum_{n=1}^\\infty \\frac{n^3 + n^2 - n}{(n+3)!}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 23702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains five identical balls except for the numbers on them, which are labeled from 1 to 5. Three balls are randomly drawn from this bag one at a time without replacement, and the highest number among the drawn balls is denoted as $X$. Calculate the probability $P(X=4)$.... | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"split": "train",
"index": 23703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB=13$, $BC=15$ and $CA=17$. Point $D$ is on $\\overline{AB}$, $E$ is on $\\overline{BC}$, and $F$ is on $\\overline{CA}$. Let $AD=p\\cdot AB$, $BE=q\\cdot BC$, and $CF=r\\cdot CA$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=2/3$ and $p^2+q^2+r^... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 23704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of \\( a \\) such that the roots \\( x_1, x_2, x_3 \\) of the polynomial\n\\[ x^3 - 6x^2 + ax + a \\]\nsatisfy\n\\[ \\left(x_1 - 3\\right)^3 + \\left(x_2 - 3\\right)^3 + \\left(x_3 - 3\\right)^3 = 0. \\] Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 23705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve people arrive at dinner, but the circular table only seats eight. If two seatings, such that one is a rotation of the other, are considered the same, then in how many different ways can we choose eight people, divide them into two groups of four each, and seat each group... | math | {
"style": "rule",
"ground_truth": "1247400"
} | {
"split": "train",
"index": 23706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lawrence runs \\(\\frac{d}{2}\\) km at an average speed of 8 minutes per kilometre.\n\nGeorge runs \\(\\frac{d}{2}\\) km at an average speed of 12 minutes per kilometre.\n\nHow many minutes more did George run than Lawrence? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 23707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\theta$ is an acute angle, and $\\sin 2\\theta=a$, then $\\sin\\theta+\\cos\\theta$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{a+1}"
} | {
"split": "train",
"index": 23708,
"solution": "1. **Expression for \\( x \\)**: Let \\( x = \\sin\\theta + \\cos\\theta \\). We aim to find the value of \\( x \\).\n\n2. **Squaring \\( x \\)**: Squaring both sides, we have:\n \\[\n x^2 = (\\sin\\theta + \\cos\\theta)^2 = \\sin^2\\theta + 2\\sin\\theta\\cos\... |
deepscaler | [
{
"role": "user",
"content": "Mark borrows $10$ dollars from Emily with a simple interest rate of $15\\%$ everyday. What is the least integer number of days after which Mark will have to pay her back at least twice as much as he borrowed? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "7 \\text{ days}"
} | {
"split": "train",
"index": 23709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify: $(\\sqrt{5})^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 23710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a square $A B C D$ on a plane, find the minimum of the ratio $\\frac{O A + O C}{O B + O D}$, where $O$ is an arbitrary point on the plane. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}"
} | {
"split": "train",
"index": 23711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0)=2$, then $f(4)=$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 23712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, $a_1=1$, $a_2=2$, and $a_{n+2} - a_n = 1 + (-1)^n$ ($n \\in \\mathbb{N}^*$), then the sum $a_1 + a_2 + \\ldots + a_{51} =$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "676"
} | {
"split": "train",
"index": 23713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hare and the tortoise had a race over 100 meters, in which both maintained constant speeds. When the hare reached the finish line, it was 75 meters in front of the tortoise. The hare immediately turned around and ran back towards the start line. How far from the finish line... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 23714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of solutions in positive integers of $2x+3y=763$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 23715,
"solution": "1. **Formulate the Equation**: We start with the Diophantine equation given:\n \\[\n 2x + 3y = 763\n \\]\n\n2. **Find a Particular Solution**: We need to find one particular solution $(x_0, y_0)$ in integers. By trial and error or using the extended Euclidean... |
deepscaler | [
{
"role": "user",
"content": "Given that the base edge length of a right prism is $1$ and the side edge length is $2$, and all the vertices of the prism lie on a sphere, find the radius of the sphere. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 23716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "24×12, my approach is to first calculate \\_\\_\\_\\_\\_\\_, then calculate \\_\\_\\_\\_\\_\\_, and finally calculate \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 23717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A jar contains 8 red balls and 2 blue balls. Every minute, a ball is randomly removed. The probability that there exists a time during this process where there are more blue balls than red balls in the jar can be expressed as $\\frac{a}{b}$ for relatively prime integers $a$ and... | math | {
"style": "rule",
"ground_truth": "209"
} | {
"split": "train",
"index": 23718,
"solution": "One can show that the condition in the problem is satisfied if and only the last ball drawn is blue (which happens with probability $\\frac{1}{5}$), or the blue balls are drawn second-to-last and third-to-last (which happens with probability $\\frac{1}{\\binom{10}{... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 23719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a\\in\\{1,3,5\\}$ and $b\\in\\{2,4,8\\}$, find the probability that the function $y=\\log_{\\frac{b}{a}}{\\frac{1}{x}}$ is an increasing function. 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": 23720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 5-digit numbers beginning with $2$ are there that have exactly three identical digits which are not $2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 23721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \\cdots + n?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23722,
"solution": "1. **Sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula:\n \\[\n 1 + 2 + \\cdots + n = \\frac{n(n+1)}{2}\n \\]\n This formula can be derived by pairing terms from the beginning and end of the sequenc... |
deepscaler | [
{
"role": "user",
"content": "Find the product of the divisors of $50$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125,\\!000"
} | {
"split": "train",
"index": 23723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On each side of a right-angled triangle, a semicircle is drawn with that side as a diameter. The areas of the three semicircles are \\( x^{2} \\), \\( 3x \\), and 180, where \\( x^{2} \\) and \\( 3x \\) are both less than 180. What is the area of the smallest semicircle? Let's ... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 23724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=e^{x}(x^{3}-3x+3)-ae^{x}-x$, where $e$ is the base of the natural logarithm, find the minimum value of the real number $a$ such that the inequality $f(x)\\leqslant 0$ has solutions in the interval $x\\in\\[-2,+\\infty)$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "1-\\frac{1}{e}"
} | {
"split": "train",
"index": 23725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**\nLet \\( P \\) be a point on the parabola \\( y = x^2 - 6x + 15 \\), and let \\( Q \\) be a point on the line \\( y = 2x - 7 \\). Find the shortest possible distance \\( PQ \\).\n\n** Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6 \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 23726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a=3 \\sqrt {3}$, $c=2$, $B=150^{\\circ}$, find the length of side $b$ and the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 23727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chloe is baking muffins for a school event. If she divides the muffins equally among 8 of her friends, she'll have 3 muffins left over. If she divides the muffins equally among 5 of her friends, she'll have 2 muffins left over. Assuming Chloe made fewer than 60 muffins, what is... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 23728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos(\\alpha - \\beta) = \\frac{3}{5}$, $\\sin(\\beta) = -\\frac{5}{13}$, where $\\alpha \\in \\left(0, \\frac{\\pi}{2} \\right)$, $\\beta \\in \\left(-\\frac{\\pi}{2}, 0 \\right)$, find the value of $\\sin(\\alpha)$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{33}{65}"
} | {
"split": "train",
"index": 23729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set $S = \\{1, 2, 3, \\ldots , 49, 50\\}$ contains the first $50$ positive integers. After the multiples of 2 and the multiples of 3 are removed, how many integers remain in the set $S$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 23730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When randomly selecting three line segments from the lengths 1, 3, 5, 7, and 9, calculate the probability that the selected line segments cannot form a triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{10}"
} | {
"split": "train",
"index": 23731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Laila had a 10-hour work day and attended three different meetings during this period, the first meeting took 50 minutes, the second meeting twice as long as the first, and the third meeting half as long as the second meeting, calculate the percentage of her work day that... | math | {
"style": "rule",
"ground_truth": "33.33\\%"
} | {
"split": "train",
"index": 23732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( A = \\{0, 1, 2, \\ldots, 9\\} \\). The collection \\( \\{B_1, B_2, \\ldots, B_k\\} \\) is a family of non-empty subsets of \\( A \\). When \\( i \\neq j \\), the intersection \\( B_i \\cap B_j \\) has at most two elements. Find the maximum value of \\( k \\). Le... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 23733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The following is Xiao Liang's problem-solving process. Please read carefully and answer the following questions.\n\nCalculate $({-15})÷({\\frac{1}{3}-3-\\frac{3}{2}})×6$.\n\nSolution: Original expression $=({-15})÷({-\\frac{{25}}{6}})×6\\ldots \\ldots $ First step $=\\left(-15\... | math | {
"style": "rule",
"ground_truth": "\\frac{108}{5}"
} | {
"split": "train",
"index": 23734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that m > 0, p: 0 < x < m, q: x(x - 1) < 0, if p is a sufficient but not necessary condition for q, then the value of m can be _______. (Only one value of m that satisfies the condition is needed) Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 23735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$-2^{3}+|2-3|-2\\times \\left(-1\\right)^{2023}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 23736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the solution set for the inequality $ax^2+ax+2>0$ is $\\mathbb{R}$ (the set of all real numbers), let the set of all numerical values of the real number $a$ be denoted as $M$.\n(1) Find the set $M$.\n(2) If $t>0$, for all $a \\in M$, it holds that $(a^2-2a)t \\leq t^... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 23737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jude repeatedly flips a coin. If he has already flipped $n$ heads, the coin lands heads with probability $\\frac{1}{n+2}$ and tails with probability $\\frac{n+1}{n+2}$. If Jude continues flipping forever, let $p$ be the probability that he flips 3 heads in a row at some point. ... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 23738,
"solution": "Let $p_{n}$ be the probability that the $n$th head is flipped after a tail and Jude has yet to flip 3 heads consecutively to this point. For example, $p_{2}=\\frac{2}{3}$, as it is impossible for 3 heads to be flipped consecutively and the second head comes after a... |
deepscaler | [
{
"role": "user",
"content": "Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32\\sqrt{3}+21\\frac{1}{3}\\pi"
} | {
"split": "train",
"index": 23739,
"solution": "1. **Identify the Geometry and Setup**: We are given a circle with radius $8$ inches and two equal parallel chords that are $8$ inches apart. Let's denote the center of the circle as $O$ and the chords as $BD$ and $CE$. Draw the diameter $A$ perpendicular to the ch... |
deepscaler | [
{
"role": "user",
"content": "The blue parabola shown is the graph of the equation \\( x = ay^2 + by + c \\). The vertex of the parabola is at \\( (5, 3) \\), and it passes through the point \\( (3, 5) \\). Find \\( c \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 23740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fract... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1001}{2003}"
} | {
"split": "train",
"index": 23741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $B(-\\sqrt{5}, 0)$, $C(\\sqrt{5}, 0)$, and the sum of the lengths of the medians on sides $AB$ and $AC$ is $9$.\n(Ⅰ) Find the equation of the trajectory of the centroid $G$ of $\\triangle ABC$.\n(Ⅱ) Let $P$ be any point on the trajectory found in (Ⅰ), find ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{9}"
} | {
"split": "train",
"index": 23742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles with a radius of 15 cm overlap such that each circle passes through the center of the other. Determine the length of the common chord (dotted segment) in centimeters between these two circles. Express your answer in simplest radical form. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "15\\sqrt{3}"
} | {
"split": "train",
"index": 23743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\\{A,B\\}$ of subsets of $S$ such that $A \\cup B = S$, $A \\cap B = \\emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be emp... | math | {
"style": "rule",
"ground_truth": "\\binom{n}{2} + 1"
} | {
"split": "train",
"index": 23744,
"solution": "The maximum is $\\binom{n}{2} + 1$, achieved for instance by a convex $n$-gon: besides the trivial partition (in which all of the points are in one part), each linear partition occurs by drawing a line crossing a unique pair of edges.\n\n\\textbf{First solution:} W... |
deepscaler | [
{
"role": "user",
"content": "The average of the numbers $1, 2, 3, \\dots, 149,$ and $x$ is $150x$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11175}{22499}"
} | {
"split": "train",
"index": 23745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve candidates for mayor are participating in a televised debate. At some point, one of them says, \"So far, we've lied once.\" A second then says, \"Now it's twice.\" A third exclaims, \"Three times now,\" and so on, up to the twelfth who claims that before him, they lied t... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 23746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ $z,$ $v,$ $w$ be positive real numbers such that $x^2 + y^2 + z^2 + v^2 + w^2 = 2016.$ Let $M$ be the maximum value of\n\\[xz + 2yz + 3zv + 7zw,\\]and let $x_M,$ $y_M$, $z_M,$ $v_M,$ $w_M$ be the values of $x,$ $y,$ $z,$ $v,$ $w,$ respectively, that produce the m... | math | {
"style": "rule",
"ground_truth": "52 + 3036 \\sqrt{7}"
} | {
"split": "train",
"index": 23747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "100 people participated in a quick calculation test consisting of 10 questions. The number of people who answered each question correctly is given in the table below:\n\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline \nProblem Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 23748,
"solution": ""
} |
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