data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the \"base\" $-n+i$ using the integers $1,2,\\ldots,n^2$ as \"digits.\" That is, the equation\\[r+si=a_m(-n+i)^m+a_{m-1}(-n+i... | math | {
"style": "rule",
"ground_truth": "490"
} | {
"split": "train",
"index": 5500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$ with side lengths $AB = 13,$ $BC = 14,$ and $CA = 15,$ let $M$ be the midpoint of $\\overline{BC}.$ Let $P$ be the point on the circumcircle of $\\triangle ABC$ such that $M$ is on $\\overline{AP}.$ There exists a unique point $Q$ on segment $\\overline{AM}$... | math | {
"style": "rule",
"ground_truth": "247"
} | {
"split": "train",
"index": 5501,
"solution": "We use the law of Cosine and get \\[AB^2 = AM^2 + BM^2 - 2 AM \\cdot BM \\cos \\angle AMB,\\] \\[AC^2 = AM^2 + CM^2 + 2 AM \\cdot CM \\cos \\angle AMB \\implies\\] \\[AM^2 = \\frac {AB^2 + AC^2}{2}- BM^2 = \\sqrt{148} \\approx 12.\\] We use the power of point $M$ wi... |
deepscaler | [
{
"role": "user",
"content": "\nA circle \\( K \\) goes through the vertices \\( A \\) and \\( C \\) of triangle \\( ABC \\). The center of circle \\( K \\) lies on the circumcircle of triangle \\( ABC \\). Circle \\( K \\) intersects side \\( AB \\) at point \\( M \\). Find the angle \\( BAC \\) if \\( AM ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 5502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers are between $(10.2)^3$ and $(10.3)^3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 5503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB=\\sqrt{5}$, $BC=1$, and $AC=2$. $I$ is the incenter of $\\triangle ABC$ and the circumcircle of $\\triangle IBC$ intersects $AB$ at $P$. Find $BP$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5} - 2"
} | {
"split": "train",
"index": 5504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos^4x-2\\sin x\\cos x-\\sin^4x.$ \n$(1)$ Find the smallest positive period of $f(x)$. \n$(2)$ When $x\\in\\left[0, \\frac{\\pi}{2}\\right]$, find the minimum value of $f(x)$ and the set of $x$ values for which this minimum is achieved. Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{8}"
} | {
"split": "train",
"index": 5505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, there is a sequence of curves \\(P_{0}, P_{1}, P_{2}, \\cdots\\). It is given that \\(P_{0}\\) is an equilateral triangle with an area of 1. Each \\(P_{k+1}\\) is obtained from \\(P_{k}\\) by performing the following operations: each side of \\(P_{k}\\)... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"split": "train",
"index": 5506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the blackboard, Amy writes 2017 in base-$a$ to get $133201_{a}$. Betsy notices she can erase a digit from Amy's number and change the base to base-$b$ such that the value of the number remains the same. Catherine then notices she can erase a digit from Betsy's number and cha... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 5507,
"solution": "$2017=133201_{4}=13201_{6}=1201_{12}$"
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g,$ and the $x$-coordinate of the vertex of $g$ is a root of $f,$ and both $f$ and $g$ have the same minimum va... | math | {
"style": "rule",
"ground_truth": "-400"
} | {
"split": "train",
"index": 5508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the graph in 3-space of $0=xyz(x+y)(y+z)(z+x)(x-y)(y-z)(z-x)$. This graph divides 3-space into $N$ connected regions. What is $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 5509,
"solution": "Note that reflecting for each choice of sign for $x, y, z$, we get new regions. Therefore, we can restrict to the case where $x, y, z>0$. In this case, the sign of the expression only depends on $(x-y)(y-z)(z-x)$. It is easy to see that for this expression, every on... |
deepscaler | [
{
"role": "user",
"content": "Calculate the monotonic intervals of $F(x)=\\int_{0}^{x}{(t^{2}+2t-8)dt}$ for $x > 0$.\n\n(1) Determine the monotonic intervals of $F(x)$.\n\n(2) Find the maximum and minimum values of the function $F(x)$ on the interval $[1,2]$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "-\\frac{28}{3}"
} | {
"split": "train",
"index": 5510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circular piece of metal with a radius of 10 units has a maximum square cut out from it. Then, the largest possible circle is cut from this square. Calculate the total amount of metal wasted.\nA) $50\\pi - 200$\nB) $200 - 50\\pi$\nC) $100\\pi$\nD) $50\\pi$\nE) None of these Le... | math | {
"style": "rule",
"ground_truth": "50\\pi - 200"
} | {
"split": "train",
"index": 5511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles have centers that are \\( d \\) units apart, and each has a diameter \\( \\sqrt{d} \\). For any \\( d \\), let \\( A(d) \\) be the area of the smallest circle that contains both of these circles. Find \\( \\lim _{d \\rightarrow \\infty} \\frac{A(d)}{d^{2}} \\). Let'... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 5512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point \\( D \\) is marked on the altitude \\( BH \\) of triangle \\( ABC \\). Line \\( AD \\) intersects side \\( BC \\) at point \\( E \\), and line \\( CD \\) intersects side \\( AB \\) at point \\( F \\). It is known that \\( BH \\) divides segment \\( FE \\) in the ratio ... | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"split": "train",
"index": 5513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a\\in[0,\\pi]$, $\\beta\\in\\left[-\\frac{\\pi}{4},\\frac{\\pi}{4}\\right]$, $\\lambda\\in\\mathbb{R}$, and $\\left(\\alpha -\\frac{\\pi}{2}\\right)^{3}-\\cos \\alpha -2\\lambda =0$, $4\\beta^{3}+\\sin \\beta \\cos \\beta +\\lambda =0$, then the value of $\\cos \\left(\\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{ \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 5514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number is divisible by $8$ if the number formed by its last $3$ digits is divisible by $8.$ For example, the number $47\\,389\\,248$ is divisible by $8$ because $248$ is divisible by $8.$ However, $47\\,389\\,284$ is not divisible by $8$ because $284$ is not divisible by $8.$... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\\\(\\theta\\\\) is an angle in the fourth quadrant, and \\\\(\\sin (\\theta+ \\frac {\\pi}{4})= \\frac {3}{5}\\\\), then \\\\(\\tan (\\theta- \\frac {\\pi}{4})=\\\\) \\_\\_\\_\\_\\_\\_ . Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "- \\frac {4}{3}"
} | {
"split": "train",
"index": 5516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A three-digit number has digits a, b, and c in the hundreds, tens, and units place respectively. If a < b and b > c, then the number is called a \"convex number\". If you randomly select three digits from 1, 2, 3, and 4 to form a three-digit number, what is the probability that... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 5517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a four-digit number $\\overline{ABCD}$ such that $\\overline{ABCD} + \\overline{AB} \\times \\overline{CD}$ is a multiple of 1111, what is the minimum value of $\\overline{ABCD}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1729"
} | {
"split": "train",
"index": 5518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any pri... | math | {
"style": "rule",
"ground_truth": "682"
} | {
"split": "train",
"index": 5519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the operation $\\spadesuit$ as $a\\,\\spadesuit\\,b = |a- b|$ . What is the value of $2\\, \\spadesuit\\,(4\\,\\spadesuit\\,7)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 5520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the set of all complex numbers $z$ where $z = w - \\frac{1}{w}$ for some complex number $w$ of absolute value $2$. Determine the area inside the curve formed by $T$ in the complex plane. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4} \\pi"
} | {
"split": "train",
"index": 5521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first term of a sequence is 1. Each subsequent term is 4 times the square root of the sum of all preceding terms plus 4. What is the sum of the first 1971 terms in the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15531481"
} | {
"split": "train",
"index": 5522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A survey conducted at a conference found that 70% of the 150 male attendees and 75% of the 850 female attendees support a proposal for new environmental legislation. What percentage of all attendees support the proposal? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "74.2\\%"
} | {
"split": "train",
"index": 5523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function \\( f(x) = 3 \\sin x + 2 \\cos x + 1 \\). If real numbers \\( a, b, c \\) satisfy \\( a f(x) + b f(x-c) = 1 \\) for all real numbers \\( x \\), find the value of \\( \\frac{b \\cos c}{a} \\). Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the Horner's method to write out the process of calculating the value of $f(x) = 1 + x + 0.5x^2 + 0.16667x^3 + 0.04167x^4 + 0.00833x^5$ at $x = -0.2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.81873"
} | {
"split": "train",
"index": 5525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of \\( \\cos (\\angle OBC + \\angle OCB) \\) in triangle \\( \\triangle ABC \\), where angle \\( \\angle A \\) is an obtuse angle, \\( O \\) is the orthocenter, and \\( AO = BC \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 5526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the legs of a right triangle are in the ratio $3:4$, find the ratio of the areas of the two triangles created by dropping an altitude from the right-angle vertex to the hypotenuse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"split": "train",
"index": 5527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square $ABCD$, points $E$ and $H$ lie on $\\overline{AB}$ and $\\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\\overline{BC}$ and $\\overline{CD}$, respectively, and points $I$ and $J$ lie on $\\overline{EH}$ so that $\\overline{FI} \\perp \\overl... | math | {
"style": "rule",
"ground_truth": "8-4\\sqrt{2}"
} | {
"split": "train",
"index": 5528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of 0s and 1s of length 23 that begins with a 0, ends with a 0, contains no two consecutive 0s, and contains no four consecutive 1s, determine the number of such sequences. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 5529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a\\sin A + c\\sin C - \\sqrt{2}a\\sin C = b\\sin B$.\n1. Find $B$.\n2. If $\\cos A = \\frac{1}{3}$, find $\\sin C$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{4 + \\sqrt{2}}{6}"
} | {
"split": "train",
"index": 5530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCompute the definite integral:\n\n$$\n\\int_{0}^{1} \\frac{x^{4}}{\\left(2-x^{2}\\right)^{3 / 2}} \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2} - \\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 5531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} x-2 & \\quad \\text{ if } x < 4 \\\\ \\sqrt{x} & \\quad \\text{ if } x \\ge 4 \\end{aligned} \\right.\\]has an inverse $f^{-1}.$ Find the value of $f^{-1}(-5) + f^{-1}(-4) + \\dots + f^{-1}(4) + f^{-1}(5).$ Let's think step by s... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 5532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Stan drove 300 miles in 5 hours, 20 minutes. Next, he drove 360 miles in 6 hours, 40 minutes. What was Stan's average speed in miles per hour for the total trip? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 5533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quartic equation \\( x^{4} + a x^{3} + b x^{2} + a x + 1 = 0 \\) has a real root. Find the minimum value of \\( a^{2} + b^{2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"split": "train",
"index": 5534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive number $\\lambda$ such that for any triangle with side lengths $a, b, c$, given $a \\geqslant \\frac{b+c}{3}$, it holds that\n$$\na c + b c - c^{2} \\leqslant \\lambda\\left(a^{2} + b^{2} + 3 c^{2} + 2 a b - 4 b c\\right).\n$$ Let's think step by step... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2} + 1}{7}"
} | {
"split": "train",
"index": 5535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are digits for which\n$\\begin{array}{ccc}& 2 & a\\ \\times & b & 3\\ \\hline & 6 & 9\\ 9 & 2 & \\ \\hline 9 & 8 & 9\\end{array}$\nthen $a+b =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 5536,
"solution": "1. **Identify the multiplication setup**: The multiplication setup is given as:\n \\[\n \\begin{array}{ccc}\n & 2 & a\\\\ \n \\times & b & 3\\\\ \n \\hline \n & 6 & 9\\\\ \n 9 & 2\\\\ \n \\hline \n 9 & 8 & 9\n \\end{array}\n \\]\n This repres... |
deepscaler | [
{
"role": "user",
"content": "Given a hyperbola with eccentricity $2$ and equation $\\dfrac{x^2}{a^2} - \\dfrac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, the right focus $F_2$ of the hyperbola is the focus of the parabola $y^2 = 8x$. A line $l$ passing through point $F_2$ intersects the right branch of the hyperbola... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\sqrt{7}}{7}"
} | {
"split": "train",
"index": 5537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a bag containing 7 apples and 1 orange, the probability of randomly picking an apple is ______, and the probability of picking an orange is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 5538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A uniform solid semi-circular disk of radius $R$ and negligible thickness rests on its diameter as shown. It is then tipped over by some angle $\\gamma$ with respect to the table. At what minimum angle $\\gamma$ will the disk lose balance and tumble over? Express your ans... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 5539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The surface area ratio of an equilateral cone to a sphere is 3:1. How many times larger is the volume of the equilateral cone compared to the volume of the sphere? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 5540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The total length of the highway from Lishan Town to the provincial capital is 189 kilometers, passing through the county town. The county town is 54 kilometers away from Lishan Town. In the morning at 8:30, a bus departs from Lishan Town to the county town and arrives at 9:15. ... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 5541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p, q, r, s$ be distinct primes such that $p q-r s$ is divisible by 30. Find the minimum possible value of $p+q+r+s$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 5542,
"solution": "The key is to realize none of the primes can be 2,3, or 5, or else we would have to use one of them twice. Hence $p, q, r, s$ must lie among $7,11,13,17,19,23,29, \\ldots$. These options give remainders of $1(\\bmod 2)$ (obviously), $1,-1,1,-1,1,-1,-1, \\ldots$ modu... |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $n$, with $n \\leq 100$, is $n^{3}+5n^{2}$ the square of an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 5543,
"solution": "For $n^{3}+5n^{2}$ to be the square of an integer, $\\sqrt{n^{3}+5n^{2}}$ must be an integer. We know that $\\sqrt{n^{3}+5n^{2}}=\\sqrt{n^{2}(n+5)}=\\sqrt{n^{2}} \\sqrt{n+5}=n \\sqrt{n+5}$. For $n \\sqrt{n+5}$ to be an integer, $\\sqrt{n+5}$ must be an integer. In o... |
deepscaler | [
{
"role": "user",
"content": "A bivalent metal element is used in a chemical reaction. When 3.5g of the metal is added into 50g of a dilute hydrochloric acid solution with a mass percentage of 18.25%, there is some metal leftover after the reaction finishes. When 2.5g of the metal is added into the same mas... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 5544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many times does the digit 9 appear in the list of all integers from 1 to 1000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 5545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThere are 256 players in a tennis tournament who are ranked from 1 to 256, with 1 corresponding to the highest rank and 256 corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability \\(\\... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 5546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$ .\n\n\n*2021 CCA Math Bonanza Lightning Round #2.4* Let's think step by s... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The school now introduces a new color, silver, for the flag design. Crestview's school colors are now purple, gold, and silver. The students are designing a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different ... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 5548,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair coin is flipped 8 times. What is the probability that at least 6 consecutive flips come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{128}"
} | {
"split": "train",
"index": 5549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\( ABCD \\), point \\( X \\) is taken on the base \\( BC \\) such that segments \\( XA \\) and \\( XD \\) divide the trapezoid into three similar but pairwise unequal, non-isosceles triangles. The side \\( AB \\) has a length of 5. Find \\( XC \\cdot BX \\). Let's... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 5550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin \\left( \\frac {5\\pi}{6}-2x\\right)-2\\sin \\left(x- \\frac {\\pi}{4}\\right)\\cos \\left(x+ \\frac {3\\pi}{4}\\right).$ \n$(1)$ Find the minimum positive period and the intervals of monotonic increase for the function $f(x)$; \n$(2)$ If $x_{0}... | math | {
"style": "rule",
"ground_truth": "- \\frac {2 \\sqrt {6}+1}{6}"
} | {
"split": "train",
"index": 5551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $1.\\overline{03}$ as a reduced fraction, given that $0.\\overline{01}$ is $\\frac{1}{99}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{34}{33}"
} | {
"split": "train",
"index": 5552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What was Tony's average speed, in miles per hour, during the 3-hour period when his odometer increased from 12321 to the next higher palindrome? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33.33"
} | {
"split": "train",
"index": 5553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If four consecutive natural numbers are all composite numbers, find the smallest sum of these four numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 5554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $503, 1509, 3015, 6021, \\dots$, determine how many of the first $1500$ numbers in this sequence are divisible by $503$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 5555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some nodes (vertices of unit squares) in a $6 \\times 6$ grid are painted red such that on the boundary of any $k \\times k$ subgrid ($1 \\leqslant k \\leqslant 6$) there is at least one red point. Find the minimum number of red points required to satisfy this condition. Let's ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 5556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first eighty positive odd integers subtracted from the sum of the first eighty positive even integers is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 5557,
"solution": "1. **Identify the pattern in the sequence of odd and even integers:**\n - The $n$-th odd integer can be represented as $2n - 1$.\n - The $n$-th even integer can be represented as $2n$.\n\n2. **Calculate the difference between the $n$-th even and $n$-th odd integ... |
deepscaler | [
{
"role": "user",
"content": "In unit square $A B C D$, points $E, F, G$ are chosen on side $B C, C D, D A$ respectively such that $A E$ is perpendicular to $E F$ and $E F$ is perpendicular to $F G$. Given that $G A=\\frac{404}{1331}$, find all possible values of the length of $B E$. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{11}"
} | {
"split": "train",
"index": 5558,
"solution": "Let $B E=x$, then since triangles $A B E, E C F, F D G$ are all similar, we have $C E=1-x, C F=$ $x(1-x), F D=1-x(1-x), D G=x-x^{2}(1-x), G A=1-x+x^{2}(1-x)=(1-x)\\left(x^{2}+1\\right)$, therefore it remains to solve the equation $$(1-x)\\left(x^{2}+1\\right)=\\frac... |
deepscaler | [
{
"role": "user",
"content": "How many numbers are in the list $-58, -51, -44, \\ldots, 71, 78$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of a circle $(x-1)^{2}+(y-1)^{2}=9$, point $P(2,2)$ lies inside the circle. The longest and shortest chords passing through point $P$ are $AC$ and $BD$ respectively. Determine the product $AC \\cdot BD$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "12\\sqrt{7}"
} | {
"split": "train",
"index": 5560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kymbrea's comic book collection currently has $30$ comic books in it, and she is adding to her collection at the rate of $2$ comic books per month. LaShawn's collection currently has $10$ comic books in it, and he is adding to his collection at the rate of $6$ comic books per m... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 5561,
"solution": "1. **Define the number of comic books over time for both Kymbrea and LaShawn:**\n - Kymbrea's collection: Initially, she has 30 comic books and adds 2 each month. Thus, after $x$ months, she will have:\n \\[\n 30 + 2x\n \\]\n - LaShawn's collection: ... |
deepscaler | [
{
"role": "user",
"content": "Let ellipse M be defined by the equation $$\\frac {y^{2}}{a^{2}}+ \\frac {x^{2}}{b^{2}}=1$$ where $a>b>0$. The eccentricity of ellipse M and the eccentricity of the hyperbola defined by $x^{2}-y^{2}=1$ are reciprocals of each other, and ellipse M is inscribed in the circle defi... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 5562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"split": "train",
"index": 5563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company organizes its employees into 7 distinct teams for a cycling event. The employee count is between 200 and 300. If one employee takes a day off, the teams are still equally divided among all present employees. Determine the total number of possible employees the company... | math | {
"style": "rule",
"ground_truth": "3493"
} | {
"split": "train",
"index": 5564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The terms of an arithmetic sequence add to $715$. The first term of the sequence is increased by $1$, the second term is increased by $3$, the third term is increased by $5$, and in general, the $k$th term is increased by the $k$th odd positive integer. The terms of the new seq... | math | {
"style": "rule",
"ground_truth": "195"
} | {
"split": "train",
"index": 5565,
"solution": "After the adding of the odd numbers, the total of the sequence increases by $836 - 715 = 121 = 11^2$. Since the sum of the first $n$ positive odd numbers is $n^2$, there must be $11$ terms in the sequence, so the mean of the sequence is $\\dfrac{715}{11} = 65$. Sinc... |
deepscaler | [
{
"role": "user",
"content": "Given the set of $n$ numbers; $n > 1$, of which one is $1 - \\frac {1}{n}$ and all the others are $1$. The arithmetic mean of the $n$ numbers is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\frac{1}{n^2}"
} | {
"split": "train",
"index": 5566,
"solution": "1. **Identify the numbers in the set**: We are given a set of $n$ numbers where $n > 1$. One of these numbers is $1 - \\frac{1}{n}$, and the remaining $n-1$ numbers are all $1$.\n\n2. **Calculate the sum of all numbers in the set**: \n - The sum of the $n-1$ numbe... |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the foci of the ellipse and its eccentricity when defined by the equation:\n\\[\\frac{x^2}{16} + \\frac{y^2}{9} = 8.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{4}"
} | {
"split": "train",
"index": 5567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two vertical towers, \\( AB \\) and \\( CD \\), are located \\( 16 \\mathrm{~m} \\) apart on flat ground. Tower \\( AB \\) is \\( 18 \\mathrm{~m} \\) tall and tower \\( CD \\) is \\( 30 \\mathrm{~m} \\) tall. Ropes are tied from \\( A \\) to \\( C \\) and from \\( B \\) to \\( ... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 5568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the total number of real solutions for the equation\n\\[\n\\frac{x}{50} = \\sin x.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 5569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six friends ate at a restaurant and agreed to share the bill equally. Because Luxmi forgot her money, each of her five friends paid an extra \\$3 to cover her portion of the total bill. What was the total bill? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\$90"
} | {
"split": "train",
"index": 5570,
"solution": "Since each of five friends paid an extra \\$3 to cover Luxmi's portion of the bill, then Luxmi's share was $5 \\times \\$3=\\$15$. Since each of the six friends had an equal share, then the total bill is $6 \\times \\$15=\\$90$."
} |
deepscaler | [
{
"role": "user",
"content": "In the dihedral angle $\\alpha - E F - \\beta $, $AE \\subset \\alpha, BF \\subset \\beta$, and $AE \\perp EF, BF \\perp EF$. Given $EF = 1$, $AE = 2$, and $AB = \\sqrt{2}$, find the maximum volume of the tetrahedron $ABEF$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 5571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the six-digit integer $3A6,\\!792$, what is the largest digit $A$ so that the six-digit integer will be divisible by 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 5572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an obtuse triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a=7$, $b=3$, and $\\cos C=\\frac{11}{14}$,\n\n$(1)$ Find the length of side $c$ and the measure of angle $A$;\n\n$(2)$ Calculate the valu... | math | {
"style": "rule",
"ground_truth": "\\frac{71}{98}"
} | {
"split": "train",
"index": 5573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Given that $\\tan \\alpha = -2$, calculate the value of $\\dfrac {3\\sin \\alpha + 2\\cos \\alpha}{5\\cos \\alpha - \\sin \\alpha}$.\n(2) Given that $\\sin \\alpha = \\dfrac {2\\sqrt{5}}{5}$, calculate the value of $\\tan (\\alpha + \\pi) + \\dfrac {\\sin \\left( \\dfrac {5... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{5}{2}"
} | {
"split": "train",
"index": 5574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{8}"
} | {
"split": "train",
"index": 5575,
"solution": "\nLet \\(a, b, c, d\\) be real numbers such that \\(a^2 + b^2 + c^2 + d^2 = 1\\). We want to determine the minimum value of the expression \\((a-b)(b-c)(c-d)(d-a)\\).\n\nTo find the minimum value of \\((a-b)(b-c)(c-d)(d-a)\\), we first recognize the symmetry and pot... |
deepscaler | [
{
"role": "user",
"content": "Suppose that all four of the numbers \\[2 - \\sqrt{5}, \\;4+\\sqrt{10}, \\;14 - 2\\sqrt{7}, \\;-\\sqrt{2}\\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 5576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\\dfrac{x}{t}+\\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$ . Find the area of $S$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"split": "train",
"index": 5577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the random variable $\\xi$ follows a binomial distribution $B(5,0.5)$, and $\\eta=5\\xi$, calculate the values of $E\\eta$ and $D\\eta$ respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{125}{4}"
} | {
"split": "train",
"index": 5578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=x+\\sin \\pi x-3$, calculate the value of $f\\left( \\dfrac {1}{2015}\\right)+f\\left( \\dfrac {2}{2015}\\right)+f\\left( \\dfrac {3}{2015}\\right)+\\ldots+f\\left( \\dfrac {4029}{2015}\\right)$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "-8058"
} | {
"split": "train",
"index": 5579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line passes through $A\\ (1,1)$ and $B\\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 5580,
"solution": "1. **Identify the slope of the line passing through points $A$ and $B$:**\n Given points $A(1,1)$ and $B(100,1000)$, the slope of the line can be calculated using the formula:\n \\[\n \\text{slope} = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{1000 - 1}{100 - 1} = \... |
deepscaler | [
{
"role": "user",
"content": "The circle C is tangent to the y-axis and the line l: y = (√3/3)x, and the circle C passes through the point P(2, √3). Determine the diameter of circle C. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{14}{3}"
} | {
"split": "train",
"index": 5581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the altitude \\( BH \\) of triangle \\( ABC \\) a certain point \\( D \\) is marked. Line \\( AD \\) intersects side \\( BC \\) at point \\( E \\), and line \\( CD \\) intersects side \\( AB \\) at point \\( F \\). Points \\( G \\) and \\( J \\) are the projections of points... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} : 1"
} | {
"split": "train",
"index": 5582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the equation \\[ x^2 + 4y^2 - 10x + 56y = k\\]is a non-degenerate ellipse if and only if $k > a.$ What is $a?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-221"
} | {
"split": "train",
"index": 5583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $$\\sum_{1 \\leq a<b<c} \\frac{1}{2^{a} 3^{b} 5^{c}}$$ (i.e. the sum of $\\frac{1}{2^{a} 3^{b} 5^{c}}$ over all triples of positive integers $(a, b, c)$ satisfying $a<b<c$). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1624}"
} | {
"split": "train",
"index": 5584,
"solution": "Let $x=b-a$ and $y=c-b$ so that $b=a+x$ and $c=a+x+y$. Then $$2^{a} 3^{b} 5^{c}=2^{a} 3^{a+x} 5^{a+x+y}=30^{a} 15^{x} 5^{y}$$ and $a, x, y$ are any positive integers. Thus $$\\begin{aligned} \\sum_{1 \\leq a \\leq b<c} \\frac{1}{2^{a} 3^{b} 5^{c}} & =\\sum_{1 \\leq ... |
deepscaler | [
{
"role": "user",
"content": "On a highway, there are checkpoints D, A, C, and B arranged in sequence. A motorcyclist and a cyclist started simultaneously from A and B heading towards C and D, respectively. After meeting at point E, they exchanged vehicles and each continued to their destinations. As a resu... | math | {
"style": "rule",
"ground_truth": "340"
} | {
"split": "train",
"index": 5585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three spheres and a cube. The first sphere is tangent to each face of the cube, the second sphere is tangent to each edge of the cube, and the third sphere passes through each vertex of the cube. Then, the ratio of the surface areas of these three spheres is ______. L... | math | {
"style": "rule",
"ground_truth": "1:2:3"
} | {
"split": "train",
"index": 5586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 5587,
"solution": "1. **Understanding the Cube Configuration**:\n - A 3x3x3 cube consists of 27 smaller cubes.\n - The smaller cubes on the corners have three faces visible.\n - The smaller cubes on the edges have two faces visible.\n - The smaller cubes in the center of each ... |
deepscaler | [
{
"role": "user",
"content": "Given that $x = \\frac{3}{5}$ is a solution to the equation $30x^2 + 13 = 47x - 2$, find the other value of $x$ that will solve the equation. Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 5588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive whole numbers have cube roots that are less than $10$? For example, $20$ would count since $\\sqrt[3]{20}<10.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"split": "train",
"index": 5589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T = \\{9^k : k ~ \\mbox{is an integer}, 0 \\le k \\le 4000\\}$. Given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T$ have 9 as their leftmost digit?\n Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "184"
} | {
"split": "train",
"index": 5590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Car A and Car B are traveling in opposite directions on a road parallel to a railway. A 180-meter-long train is moving in the same direction as Car A at a speed of 60 km/h. The time from when the train catches up with Car A until it meets Car B is 5 minutes. If it takes the tra... | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"split": "train",
"index": 5591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "|r_1+r_2|>4\\sqrt{2}"
} | {
"split": "train",
"index": 5592,
"solution": "1. **Identify the coefficients and the discriminant condition**: \n Given the quadratic equation $x^2 + px + 8 = 0$, we know that the sum of the roots $r_1 + r_2 = -p$ and the product of the roots $r_1r_2 = 8$, according to Vieta's formulas. Since $r_1$ and $r_2$ ... |
deepscaler | [
{
"role": "user",
"content": "Let $O$ be the origin. There exists a scalar $m$ such that for any points $E,$ $F,$ $G,$ and $H$ satisfying the vector equation:\n\\[4 \\overrightarrow{OE} - 3 \\overrightarrow{OF} + 2 \\overrightarrow{OG} + m \\overrightarrow{OH} = \\mathbf{0},\\] \nthe four points $E,$ $F,$ $... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 5593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $y=\\left(m-1\\right)x^{|m|}$ is a direct proportion function, then the value of $m$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 4 red, 3 blue, and 6 yellow marbles. One marble is drawn and removed from the bag but is only considered in the new count if it is yellow. What is the probability, expressed as a fraction, of then drawing one marble which is either red or blue from the updated co... | math | {
"style": "rule",
"ground_truth": "\\frac{91}{169}"
} | {
"split": "train",
"index": 5595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two concentric circles have radii $1$ and $4$ . Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three da... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 5596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone is cut into five pieces by four planes parallel to its base, each piece having equal height. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{61}"
} | {
"split": "train",
"index": 5597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $4x + 14 = 8x - 48$, what is the value of $2x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 5598,
"solution": "Since $4x + 14 = 8x - 48$, then $14 + 48 = 8x - 4x$ or $62 = 4x$. Dividing both sides of this equation by 2, we obtain $\\frac{4x}{2} = \\frac{62}{2}$ which gives $2x = 31$."
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( \\frac{\\pi}{4} < \\theta < \\frac{\\pi}{2} \\), find the maximum value of \\( S = \\sin 2\\theta - \\cos^2 \\theta \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"split": "train",
"index": 5599,
"solution": ""
} |
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