data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let *Revolution* $(x) = x^3 +Ux^2 +Sx + A$ , where $U$ , $S$ , and $A$ are all integers and $U +S + A +1 = 1773$ . Given that *Revolution* has exactly two distinct nonzero integer roots $G$ and $B$ , find the minimum value of $|GB|$ .\n\n*Proposed by Jacob Xu*\n\n<deta... | math | {
"style": "rule",
"ground_truth": "392"
} | {
"split": "train",
"index": 36049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $PQRS$ is inscribed in triangle $XYZ$, with side $PS$ of the rectangle on side $XZ$. Triangle $XYZ$ has an altitude from $Y$ to side $XZ$ of 8 inches, and the side $XZ$ measures 15 inches. The length of segment $PQ$ is one-third the length of segment $PS$. Calculate t... | math | {
"style": "rule",
"ground_truth": "\\frac{4800}{169}"
} | {
"split": "train",
"index": 36050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\[f(x) = \\left\\{ \\begin{aligned} x+3 & \\quad \\text{ if } x < 2 \\\\ x^2 & \\quad \\text{ if } x \\ge 2 \\end{aligned} \\right.\\] determine the value of \\(f^{-1}(-5) + f^{-1}(-4) + \\dots + f^{-1}(2) + f^{-1}(3). \\) Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "-35 + \\sqrt{2} + \\sqrt{3}"
} | {
"split": "train",
"index": 36051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Half the people in a room left. One third of those remaining started to dance. There were then $12$ people who were not dancing. The original number of people in the room was Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36052,
"solution": "1. **Define the variable:**\n Let $x$ be the original number of people in the room.\n\n2. **Calculate the number of people left after half left:**\n Since half the people left, the number of people remaining in the room is $\\frac{x}{2}$.\n\n3. **Calculate the ... |
deepscaler | [
{
"role": "user",
"content": "Given the equations $x^2+kx+6=0$ and $x^2-kx+6=0$. If, when the roots of the equation are suitably listed, each root of the second equation is $5$ more than the corresponding root of the first equation, then $k$ equals: Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 36053,
"solution": "1. **Identify the roots of the equations**: Let the roots of the first equation $x^2 + kx + 6 = 0$ be $r$ and $s$. \n\n2. **Apply Vieta's formulas**: From Vieta's formulas, we know:\n \\[\n r + s = -k \\quad \\text{(sum of roots)}\n \\]\n \\[\n rs = 6 \\q... |
deepscaler | [
{
"role": "user",
"content": "Let $z=a+bi$ be the complex number with $\\vert z \\vert = 5$ and $b > 0$ such that the distance between $(1+2i)z^3$ and $z^5$ is maximized, and let $z^4 = c+di$. Find $c+d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 36054,
"solution": "Let's consider the maximization constraint first: we want to maximize the value of $|z^5 - (1+2i)z^3|$ Simplifying, we have\n$|z^3| * |z^2 - (1+2i)|$\n$=|z|^3 * |z^2 - (1+2i)|$\n$=125|z^2 - (1+2i)|$\n Thus we only need to maximize the value of $|z^2 - (1+2i)|$.\n T... |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integ... | math | {
"style": "rule",
"ground_truth": "937"
} | {
"split": "train",
"index": 36055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya wants to create an unusual die, which should have the shape of a cube, with dots drawn on the faces (different numbers of dots on different faces). Additionally, on each pair of adjacent faces, the number of dots must differ by at least two (it is allowed to have more tha... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 36056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $a=2$ and $\\cos C=-\\frac{1}{4}$.\n1. If $b=3$, find the value of $c$.\n2. If $c=2\\sqrt{6}$, find the value of $\\sin B$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{4}"
} | {
"split": "train",
"index": 36057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\binom{18}{6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18564"
} | {
"split": "train",
"index": 36058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1979"
} | {
"split": "train",
"index": 36059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle has its center at $(2,0)$ with a radius of 2, and another circle has its center at $(5,0)$ with a radius of 1. A line is tangent to both circles in the first quadrant. The $y$-intercept of this line is closest to: Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{2}"
} | {
"split": "train",
"index": 36060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\( \\frac{1}{2} + \\frac{2}{4} + \\frac{4}{8} + \\frac{8}{16} \\) is what? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 36061,
"solution": "In the given sum, each of the four fractions is equivalent to \\( \\frac{1}{2} \\). Therefore, the given sum is equal to \\( \\frac{1}{2} + \\frac{1}{2} + \\frac{1}{2} + \\frac{1}{2} = 2 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16.$ The set of all possible values of $r$ is an interval $[a,b).$ What is $a+b?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{15}+8"
} | {
"split": "train",
"index": 36062,
"solution": "To solve this problem, we first need to understand the properties of the ellipse and the circle described in the problem.\n\n1. **Identify the ellipse properties:**\n The given ellipse equation is \\(x^2 + 16y^2 = 16\\). We can rewrite this equation in standard f... |
deepscaler | [
{
"role": "user",
"content": "Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 36063,
"solution": "1. **Understanding the Problem**: Bernardo writes the base-5 and base-6 representations of a three-digit number $N$ on a blackboard. LeRoy, treating these numbers as base-10 integers, adds them to get $S$. We need to find how many such $N$ exist such that the two r... |
deepscaler | [
{
"role": "user",
"content": "In 2019, a team, including professor Andrew Sutherland of MIT, found three cubes of integers which sum to 42: $42=\\left(-8053873881207597 \\_\\right)^{3}+(80435758145817515)^{3}+(12602123297335631)^{3}$. One of the digits, labeled by an underscore, is missing. What is that dig... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36064,
"solution": "Let the missing digit be $x$. Then, taking the equation modulo 10, we see that $2 \\equiv-x^{3}+5^{3}+1^{3}$. This simplifies to $x^{3} \\equiv 4(\\bmod 10)$, which gives a unique solution of $x=4$."
} |
deepscaler | [
{
"role": "user",
"content": "For some complex number $\\omega$ with $|\\omega| = 2,$ there is some real $\\lambda > 1$ such that $\\omega,$ $\\omega^2,$ and $\\lambda \\omega$ form an equilateral triangle in the complex plane. Find $\\lambda.$ Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{13}}{2}"
} | {
"split": "train",
"index": 36065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a>0$ and $b>0,$ and define two operations:\n$$a \\nabla b = \\dfrac{a + b}{1 + ab}$$\n$$a \\Delta b = \\dfrac{a - b}{1 - ab}$$\nCalculate $3 \\nabla 4$ and $3 \\Delta 4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{11}"
} | {
"split": "train",
"index": 36066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a trapezoid, the lengths of the diagonals are known to be 6 and 8, and the length of the midline is 5. Find the height of the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.8"
} | {
"split": "train",
"index": 36067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"split": "train",
"index": 36068,
"solution": "\nTo solve this problem, we need to ensure that the sum of the numbers in each row and each column is odd. We can achieve this by having either three odd numbers or one odd and two even numbers in each row and column. Let's analyze the possible configurations and c... |
deepscaler | [
{
"role": "user",
"content": "Given the functions $f(x)=x+e^{x-a}$ and $g(x)=\\ln (x+2)-4e^{a-x}$, where $e$ is the base of the natural logarithm. If there exists a real number $x_{0}$ such that $f(x_{0})-g(x_{0})=3$, find the value of the real number $a$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "-\\ln 2-1"
} | {
"split": "train",
"index": 36069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, $c > 1$, and $a + b = 1$, find the minimum value of $( \\frac{a^{2}+1}{ab} - 2) \\cdot c + \\frac{\\sqrt{2}}{c - 1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 36070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( M = \\{1, 2, \\cdots, 10\\} \\) and let \\( T \\) be a collection of certain two-element subsets of \\( M \\), such that for any two different elements \\(\\{a, b\\} \\) and \\(\\{x, y\\} \\) in \\( T \\), the condition \\( 11 \\nmid (ax + by)(ay + bx) \\) is satisfied.... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 36071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a tetrahedron \\(A B C D\\), a plane is drawn through the midpoint \\(M\\) of edge \\(A D\\), vertex \\(C\\), and point \\(N\\) on edge \\(B D\\) such that \\(B N: N D = 2: 1\\). In what ratio does this plane divide the segment \\(K P\\), where \\(K\\) and \\(P\\) are the mi... | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"split": "train",
"index": 36072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number 119 has the following property:\n\n- Division by 2 leaves a remainder of 1;\n- Division by 3 leaves a remainder of 2;\n- Division by 4 leaves a remainder of 3;\n- Division by 5 leaves a remainder of 4;\n- Division by 6 leaves a remainder of 5.\n\nHow many positive in... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 36073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 36074,
"solution": "1. **Label the seats and calculate unrestricted seating for Derek and Eric:** \n Label the seats from left to right as $1$ through $5$. Without any restrictions, Derek and Eric can be seated in $5 \\times 4 = 20$ ways, since Derek has 5 choices and Eric has 4 re... |
deepscaler | [
{
"role": "user",
"content": "The vertical coordinate of the intersection point of the new graph obtained by shifting the graph of the quadratic function $y=x^{2}+2x+1$ $2$ units to the left and then $3$ units up is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum value of $\\frac{(2^t-3t)t}{4^t}$ for real values of $t?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 36076,
"solution": "1. **Substitute $2^t = x$**: We start by letting $2^t = x$, which implies that $\\log_2{x} = t$. This substitution simplifies the expression:\n \\[\n \\frac{(2^t-3t)t}{4^t} = \\frac{(x - 3\\log_2{x})\\log_2{x}}{x^2}.\n \\]\n\n2. **Rewrite the expression**: Us... |
deepscaler | [
{
"role": "user",
"content": "If $x$ is real and positive and grows beyond all bounds, then $\\log_3{(6x-5)}-\\log_3{(2x+1)}$ approaches: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 36077,
"solution": "1. **Rewrite the expression using logarithmic properties**: \n Given the expression $\\log_3{(6x-5)}-\\log_3{(2x+1)}$, we can use the logarithmic property that states $\\log_b a - \\log_b b = \\log_b \\left(\\frac{a}{b}\\right)$. Applying this property, we get:\... |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $5 - x = 8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 36078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a backpacking trip with 10 people, in how many ways can I choose 2 cooks and 1 medical helper if any of the 10 people may fulfill these roles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 36079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the triangle with vertices $(2, -3),$ $(1, 4),$ and $(-3, -2).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 36080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive real number $x > 1,$ the Riemann zeta function $\\zeta(x)$ is defined by\n\\[\\zeta(x) = \\sum_{n = 1}^\\infty \\frac{1}{n^x}.\\]Compute\n\\[\\sum_{k = 2}^\\infty \\{\\zeta(2k - 1)\\}.\\]Note: For a real number $x,$ $\\{x\\}$ denotes the fractional part of $x.$ L... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 36081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the equation of line $l_{1}$ is $y=x$, and the equation of line $l_{2}$ is $y=kx-k+1$, find the value of $k$ for which the area of triangle $OAB$ is $2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 36082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A coordinate system and parametric equations are given in the plane rectangular coordinate system $xOy$. The parametric equations of the curve $C\\_1$ are $\\begin{cases} x=\\sqrt{2}\\sin(\\alpha + \\frac{\\pi}{4}) \\\\ y=\\sin(2\\alpha) + 1 \\end{cases}$, where $\\alpha$ is th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2} - 1"
} | {
"split": "train",
"index": 36083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $7 \\cdot 12\\frac{1}{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85\\frac{3}{4}"
} | {
"split": "train",
"index": 36084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n$, let $L(n)$ be the largest factor of $n$ other than $n$ itself. Determine the number of ordered pairs of composite positive integers $(m, n)$ for which $L(m) L(n)=80$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36085,
"solution": "Let $x$ be an integer, and let $p_{x}$ be the smallest prime factor of $x$. Then, if $L(a)=x$, we note that we must have $a=p x$ for some prime $p \\leq p_{x}$. (Otherwise, if $p>p_{x}$, then $\\frac{p x}{p_{x}}>x$. If $p$ is composite, then $k x>x$ for some factor... |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number such that\n\\[x^2 + 4 \\left( \\frac{x}{x - 2} \\right)^2 = 45.\\]Find all possible values of $y = \\frac{(x - 2)^2 (x + 3)}{2x - 3}.$ Enter all possible values, separated by commas. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "2,16"
} | {
"split": "train",
"index": 36086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the probability distribution of the random variable $X$ is given by $P\\left(X=\\frac{k}{5}\\right)=ak$, where $k=1,2,3,4,5$.\n(1) Find the value of $a$.\n(2) Calculate $P\\left(X \\geq \\frac{3}{5}\\right)$.\n(3) Find $P\\left(\\frac{1}{10} < X \\leq \\frac{7}{10}\\rig... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 36087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 36088,
"solution": "1. **Identify the diameters of the circles**: Given that the circle centered at $Q$ has a diameter of $4$, and since the circles are congruent, the circles centered at $P$ and $R$ also have diameters of $4$.\n\n2. **Determine the height of the rectangle**: Consider... |
deepscaler | [
{
"role": "user",
"content": "There are two arithmetic sequences $\\\\{a_{n}\\\\}$ and $\\\\{b_{n}\\\\}$, with respective sums of the first $n$ terms denoted by $S_{n}$ and $T_{n}$. Given that $\\dfrac{S_{n}}{T_{n}} = \\dfrac{3n}{2n+1}$, find the value of $\\dfrac{a_{1}+a_{2}+a_{14}+a_{19}}{b_{1}+b_{3}+b_{1... | math | {
"style": "rule",
"ground_truth": "\\dfrac{17}{13}"
} | {
"split": "train",
"index": 36089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n$$\n\\begin{array}{c}\nA=\\left(\\binom{2010}{0}-\\binom{2010}{-1}\\right)^{2}+\\left(\\binom{2010}{1}-\\binom{2010}{0}\\right)^{2}+\\left(\\binom{2010}{2}-\\binom{2010}{1}\\right)^{2} \\\\\n+\\cdots+\\left(\\binom{2010}{1005}-\\binom{2010}{1004}\\right)^{2}\n\\end{array}\... | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"split": "train",
"index": 36090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $BC = 4$, $AC = 3 \\sqrt{2}$, and $\\angle C = 45^\\circ$. Altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. Find $AH:HD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 36091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum $$\\frac{2^1}{4^1 - 1} + \\frac{2^2}{4^2 - 1} + \\frac{2^4}{4^4 - 1} + \\frac{2^8}{4^8 - 1} + \\cdots.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 36092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In tetrahedron $ABCD,$\n\\[\\angle ADB = \\angle ADC = \\angle BDC = 90^\\circ.\\]Also, $x = \\sin \\angle CAD$ and $y = \\sin \\angle CBD.$ Express $\\cos \\angle ACB$ in terms of $x$ and $y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "xy"
} | {
"split": "train",
"index": 36093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of $\\sqrt{x^2+y^2}$ if $5x+12y=60$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{60}{13}"
} | {
"split": "train",
"index": 36094,
"solution": "\n#### Step 1: Understand the problem\nWe need to find the minimum value of $\\sqrt{x^2 + y^2}$ given the constraint $5x + 12y = 60$. This expression $\\sqrt{x^2 + y^2}$ represents the distance from the origin $(0,0)$ to the point $(x,y)$.\n\n#### Step 2: Relate th... |
deepscaler | [
{
"role": "user",
"content": "A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a $\\textit{lune}$. Determine the area of this lune. Express your answer in terms of $\\pi$ and in simpl... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4} - \\frac{1}{24}\\pi"
} | {
"split": "train",
"index": 36095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A biologist wishes to estimate the fish population in a protected area. Initially, on March 1st, she captures and tags 80 fish, then releases them back into the water. Four months later, on July 1st, she captures another 90 fish for a follow-up study, finding that 4 of these ar... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 36096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large shopping mall designed a lottery activity to reward its customers. In the lottery box, there are $8$ small balls of the same size, with $4$ red and $4$ black. The lottery method is as follows: each customer draws twice, picking two balls at a time from the lottery box e... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"split": "train",
"index": 36097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $k$ candies were distributed among seven people so that each person received the same number of candies and each person received as many candies as possible, there were 3 candies left over. If instead, $3 k$ candies were distributed among seven people in this way, then how... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 36098,
"solution": "Suppose that each of the 7 people received $q$ candies under the first distribution scheme. Then the people received a total of $7 q$ candies and 3 candies were left over. Since there were $k$ candies, then $k=7 q+3$. Multiplying both sides by 3, we obtain $3 k=21 ... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all real numbers \\(x\\) for which \\(|x^2 - 14x + 45| = 3?\\)\nA) 12\nB) 14\nC) 16\nD) 18 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 36099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\\neq 0$, find the equation of the line. Enter your answer i... | math | {
"style": "rule",
"ground_truth": "y=10x-4"
} | {
"split": "train",
"index": 36100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of an equilateral triangle inscribed in a circle is 81 cm². Find the radius of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6 \\sqrt[4]{3}"
} | {
"split": "train",
"index": 36101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person was asked how much he paid for a hundred apples and he answered the following:\n\n- If a hundred apples cost 4 cents more, then for 1 dollar and 20 cents, he would get five apples less.\n\nHow much did 100 apples cost? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 36102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\\pi$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 36103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hotelier wanted to equip the dining room with new chairs. In the catalog, he chose a type of chair. When placing the order, he learned from the manufacturer that as part of a discount event, every fourth chair is offered at half price and that, therefore, he could save the eq... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 36104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylinder of base radius 1 is cut into two equal parts along a plane passing through the center of the cylinder and tangent to the two base circles. Suppose that each piece's surface area is $m$ times its volume. Find the greatest lower bound for all possible values of $m$ as ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 36105,
"solution": "Let $h$ be the height of the cylinder. Then the volume of each piece is half the volume of the cylinder, so it is $\\frac{1}{2} \\pi h$. The base of the piece has area $\\pi$, and the ellipse formed by the cut has area $\\pi \\cdot 1 \\cdot \\sqrt{1+\\frac{h^{2}}{4... |
deepscaler | [
{
"role": "user",
"content": "A $9 \\times 9 \\times 9$ cube is composed of twenty-seven $3 \\times 3 \\times 3$ cubes. The big cube is ‘tunneled’ as follows: First, the six $3 \\times 3 \\times 3$ cubes which make up the center of each face as well as the center $3 \\times 3 \\times 3$ cube are removed. Se... | math | {
"style": "rule",
"ground_truth": "1056"
} | {
"split": "train",
"index": 36106,
"solution": "\n#### Step-by-step Analysis:\n\n1. **Understanding the Structure**:\n - The original structure is a $9 \\times 9 \\times 9$ cube, which is composed of twenty-seven $3 \\times 3 \\times 3$ cubes.\n - The center cube of each face and the very center cube of the ... |
deepscaler | [
{
"role": "user",
"content": "Suppose that $m$ and $n$ are integers with $1 \\leq m \\leq 49$ and $n \\geq 0$ such that $m$ divides $n^{n+1}+1$. What is the number of possible values of $m$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 36107,
"solution": "If $n$ is even, $n+1 \\mid n^{n+1}+1$, so we can cover all odd $m$. If $m$ is even and $m \\mid n^{n+1}+1$, then $n$ must be odd, so $n+1$ is even, and $m$ cannot be divisible by 4 or any prime congruent to $3(\\bmod 4)$. Conversely, if $m / 2$ has all factors $1(\... |
deepscaler | [
{
"role": "user",
"content": "Two cards are placed in each of three different envelopes, and cards 1 and 2 are placed in the same envelope. Calculate the total number of different arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 36108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all ordered pairs of integers $(a, b)$ such that $1 \\leq a \\leq b \\leq 100$ and $$\\frac{(a+b)(a+b+1)}{a b}$$ is an integer. Among these pairs, find the one with largest value of $b$. If multiple pairs have this maximal value of $b$, choose the one with largest $a$.... | math | {
"style": "rule",
"ground_truth": "(35,90)"
} | {
"split": "train",
"index": 36109,
"solution": "Firstly note that $\\frac{(a+b)(a+b+1)}{a b}=2+\\frac{a^{2}+b^{2}+a+b}{a b}$. Let $c$ be this fraction so that $(a+b)(a+b+1)=a b(c+2)$ for some integers $a, b, c$. Suppose $(a, b)$ with $a \\geq b$ is a solution for some $c$. Consider the quadratic $$x^{2}-(b c-1) ... |
deepscaler | [
{
"role": "user",
"content": "Each outcome on the spinner below has equal probability. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit and the third outcome is the units digit, ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 36110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a defect rate of 3%, products are drawn from the batch without replacement until a non-defective product is found or a maximum of three draws have been made. Let $X$ represent the number of products drawn, and calculate $P(X=3)$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "(0.03)^2 \\times 0.97 + (0.03)^3"
} | {
"split": "train",
"index": 36111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 36112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tour... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 36113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(x, y)$ be a point in the cartesian plane, $x, y>0$. Find a formula in terms of $x$ and $y$ for the minimal area of a right triangle with hypotenuse passing through $(x, y)$ and legs contained in the $x$ and $y$ axes. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "2 x y"
} | {
"split": "train",
"index": 36114,
"solution": "$2 x y$."
} |
deepscaler | [
{
"role": "user",
"content": "A \"clearance game\" has the following rules: in the $n$-th round, a die is rolled $n$ times. If the sum of the points from these $n$ rolls is greater than $2^n$, then the player clears the round. Questions:\n(1) What is the maximum number of rounds a player can clear in this g... | math | {
"style": "rule",
"ground_truth": "\\frac{100}{243}"
} | {
"split": "train",
"index": 36115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams).\n\nMartin and Théodore unearth a treasure consisting of precious stones that are either dia... | math | {
"style": "rule",
"ground_truth": "2000000"
} | {
"split": "train",
"index": 36116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function f is defined on the set of integers and satisfies $f(n)=\\begin{cases} n-3&\\mbox{if}\\ n\\ge 1000\\\\ f(f(n+5))&\\mbox{if}\\ n<1000\\end{cases}$\n Find $f(84)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "997"
} | {
"split": "train",
"index": 36117,
"solution": "Assume that $f(84)$ is to be performed $n+1$ times. Then we have \\[f(84)=f^{n+1}(84)=f(f^n(84+5))\\] In order to find $f(84)$, we want to know the smallest value of \\[f^n(84+5)\\ge1000\\] Because then \\[f(84)=f(f^n(84+5))=(f^n(84+5))-3\\] From which we'll get a ... |
deepscaler | [
{
"role": "user",
"content": "Given three sequences $\\{F_n\\}$, $\\{k_n\\}$, $\\{r_n\\}$ satisfying: $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$ ($n\\in\\mathbb{N}^*$), $r_n=F_n-3k_n$, $k_n\\in\\mathbb{N}$, $0\\leq r_n<3$, calculate the sum $r_1+r_3+r_5+\\ldots+r_{2011}$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "1509"
} | {
"split": "train",
"index": 36118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $(xOy)$, the parametric equation of line $l$ is given by $\\begin{cases}x=1+t\\cos a \\\\ y= \\sqrt{3}+t\\sin a\\end{cases} (t\\text{ is a parameter})$, where $0\\leqslant α < π$. In the polar coordinate system with $O$ as the pole and the pos... | math | {
"style": "rule",
"ground_truth": "\\frac{6 \\sqrt{2}}{5}"
} | {
"split": "train",
"index": 36119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest possible positive integer \\( n \\) with the following property: For all positive integers \\( x, y, \\) and \\( z \\) with \\( x \\mid y^{3} \\), \\( y \\mid z^{3} \\), and \\( z \\mid x^{3} \\), it also holds that \\( x y z \\mid (x+y+z)^{n} \\). Let's ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 36120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many functions $f:\\{0,1\\}^{3} \\rightarrow\\{0,1\\}$ satisfy the property that, for all ordered triples \\left(a_{1}, a_{2}, a_{3}\\right) and \\left(b_{1}, b_{2}, b_{3}\\right) such that $a_{i} \\geq b_{i}$ for all $i, f\\left(a_{1}, a_{2}, a_{3}\\right) \\geq f\\left(b_... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36121,
"solution": "Consider the unit cube with vertices $\\{0,1\\}^{3}$. Let $O=(0,0,0), A=(1,0,0), B=(0,1,0), C=(0,0,1)$, $D=(0,1,1), E=(1,0,1), F=(1,1,0)$, and $P=(1,1,1)$. We want to find a function $f$ on these vertices such that $f(1, y, z) \\geq f(0, y, z)$ (and symmetric repre... |
deepscaler | [
{
"role": "user",
"content": "Find the number of odd digits in the base-4 representation of $233_{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 36122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a certain hyperbola\n\\[\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1,\\]where $a > b,$ the angle between the asymptotes is $60^\\circ.$ Find $\\frac{a}{b}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 36123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some positive real $\\alpha$, the set $S$ of positive real numbers $x$ with $\\{x\\}>\\alpha x$ consists of the union of several intervals, with total length 20.2. The value of $\\alpha$ can be expressed as $\\frac{a}{b}$, where $a, b$ are relatively prime positive integers... | math | {
"style": "rule",
"ground_truth": "4633"
} | {
"split": "train",
"index": 36124,
"solution": "If we note that $x=\\{x\\}+\\lfloor x\\rfloor$, then we can rewrite our given inequality as $\\{x\\}>\\frac{\\alpha}{1-\\alpha}\\lfloor x\\rfloor$. However, since $\\{x\\}<1$, we know that we must have $\\frac{\\alpha}{1-\\alpha}\\lfloor x\\rfloor<\\{x\\}<1$, so ea... |
deepscaler | [
{
"role": "user",
"content": "Find all real values of $x$ for which $$\\frac{1}{\\sqrt{x}+\\sqrt{x-2}}+\\frac{1}{\\sqrt{x+2}+\\sqrt{x}}=\\frac{1}{4}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{257}{16}"
} | {
"split": "train",
"index": 36125,
"solution": "We note that $$\\begin{aligned} \\frac{1}{4} & =\\frac{1}{\\sqrt{x}+\\sqrt{x-2}}+\\frac{1}{\\sqrt{x+2}+\\sqrt{x}} \\\\ & =\\frac{\\sqrt{x}-\\sqrt{x-2}}{(\\sqrt{x}+\\sqrt{x-2})(\\sqrt{x}-\\sqrt{x-2})}+\\frac{\\sqrt{x+2}-\\sqrt{x}}{(\\sqrt{x+2}+\\sqrt{x})(\\sqrt{x+2}... |
deepscaler | [
{
"role": "user",
"content": "Given a square $R_1$ with an area of 25, where each side is trisected to form a smaller square $R_2$, and this process is repeated to form $R_3$ using the same trisection points strategy, calculate the area of $R_3$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{400}{81}"
} | {
"split": "train",
"index": 36126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=\\cos x+\\left(x+1\\right)\\sin x+1$ on the interval $\\left[0,2\\pi \\right]$, find the minimum and maximum values of $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}+2"
} | {
"split": "train",
"index": 36127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola C: x² = 2py (p > 0), draw a line l: y = 6x + 8, which intersects the parabola C at points A and B. Point O is the origin, and $\\overrightarrow{OA} \\cdot \\overrightarrow{OB} = 0$. A moving circle P has its center on the parabola C and passes through a fixed... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"split": "train",
"index": 36128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$H O W, B O W$, and $D A H$ are equilateral triangles in a plane such that $W O=7$ and $A H=2$. Given that $D, A, B$ are collinear in that order, find the length of $B A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 36129,
"solution": "Note that $H \\neq B$ since otherwise $D A B$ is an equilateral triangle. Let $M$ be the midpoint of $D A$, so $H B=7 \\sqrt{3}$ and $H M=\\sqrt{3}$, and $\\angle H M B=90^{\\circ}$. By the Pythagorean theorem, $$ B M=\\sqrt{(7 \\sqrt{3})^{2}-(\\sqrt{3})^{2}}=12 $$... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the digits of the square of $111111111$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 36130,
"solution": "1. **Identify the Pattern**: We start by observing the squares of numbers consisting of repeated digits of 1:\n - $1^2 = 1$\n - $11^2 = 121$\n - $111^2 = 12321$\n - $1111^2 = 1234321$\n\n From these examples, we notice a pattern in the digits of the squar... |
deepscaler | [
{
"role": "user",
"content": "Given that YQZC is a rectangle with YC = 8 and CZ = 15, and equilateral triangles ABC and PQR each with a side length of 9, where R and B are on sides YQ and CZ, respectively, determine the length of AP. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of the circle $x^{2}+y^{2}-8x+15=0$, find the minimum value of $k$ such that there exists at least one point on the line $y=kx+2$ that can serve as the center of a circle with a radius of $1$ that intersects with circle $C$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "- \\frac{4}{3}"
} | {
"split": "train",
"index": 36132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is now 3:15:20 PM, as read on a 12-hour digital clock. In 305 hours, 45 minutes, and 56 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 36133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line passes through point $Q(\\frac{1}{3}, \\frac{4}{3})$ and intersects the hyperbola $x^{2}- \\frac{y^{2}}{4}=1$ at points $A$ and $B$. Point $Q$ is the midpoint of chord $AB$.\n\n1. Find the equation of the line containing $AB$.\n2. Find the length of $|AB|$. Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 36134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\\frac{z^{}_{}}{40}$ and $\\frac{40^{}_{}}{\\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$, inclusive. Find the area of $A.$ Let's think step b... | math | {
"style": "rule",
"ground_truth": "1200 - 200 \\pi"
} | {
"split": "train",
"index": 36135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the focus of the parabola $x = -\\frac{1}{12} y^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-3,0)"
} | {
"split": "train",
"index": 36136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be the function defined by $f(x) = -2 \\sin(\\pi x)$. How many values of $x$ such that $-2 \\le x \\le 2$ satisfy the equation $f(f(f(x))) = f(x)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 36137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nLet \\(\\lfloor x\\rfloor\\) denote the greatest integer which is less than or equal to \\(x\\). For example, \\(\\lfloor\\pi\\rfloor = 3\\). \\(S\\) is the integer equal to the sum of the 100 terms shown:\n$$\nS = \\lfloor\\pi\\rfloor + \\left\\lfloor\\pi + \\frac{1}{100}\\r... | math | {
"style": "rule",
"ground_truth": "314"
} | {
"split": "train",
"index": 36138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John rode his bicycle for 45 minutes at a speed of 20 mph, then walked for 120 minutes at a speed of 3 mph. Calculate the overall average speed in miles per hour. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.64"
} | {
"split": "train",
"index": 36139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sin\\theta + \\cos\\theta = \\frac{2\\sqrt{2}-1}{3}$ ($0 < \\theta < \\pi$), then $\\tan\\theta = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2\\sqrt{2}"
} | {
"split": "train",
"index": 36140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a ship, it was decided to determine the depth of the ocean at their current location. The signal sent by the echo sounder was received on the ship after 5 seconds. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "3750"
} | {
"split": "train",
"index": 36141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane, the area of the region formed by the points \\((x, y)\\) that satisfy \\( |x| + |y| + |x - 2| \\leqslant 4 \\) is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the beach, there was a pile of apples belonging to 3 monkeys. The first monkey came, divided the apples into 3 equal piles with 1 apple remaining, then it threw the remaining apple into the sea and took one pile for itself. The second monkey came, divided the remaining apple... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 36143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of an ellipse is $\\dfrac {x^{2}}{a^{2}} + \\dfrac {y^{2}}{b^{2}} = 1 (a > b > 0)$, a line passing through the right focus of the ellipse and perpendicular to the $x$-axis intersects the ellipse at points $P$ and $Q$. The directrix of the ellipse on the right... | math | {
"style": "rule",
"ground_truth": "\\dfrac { \\sqrt {3}}{3}"
} | {
"split": "train",
"index": 36144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rectangular coordinate system, a circle centered at the point $(1,0)$ with radius $r$ intersects the parabola $y^2 = x$ at four points $A$, $B$, $C$, and $D$. If the intersection point $F$ of diagonals $AC$ and $BD$ is exactly the focus of the parabola, determine $r$. Let'... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{4}"
} | {
"split": "train",
"index": 36145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\frac {2}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))(n∈N^{})$, vector $ \\overrightarrow{j}=(0,1)$, and $θ_{n}$ is the angle between vector $ \\overrightarrow{OA_{n}}$ and $ \\overrightarrow{j}$, determine the value of $\\frac {co... | math | {
"style": "rule",
"ground_truth": "\\frac{4032}{2017}"
} | {
"split": "train",
"index": 36146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two players take turns placing Xs and Os in the cells of a $9 \\times 9$ square (the first player places Xs, and their opponent places Os). At the end of the game, the number of rows and columns where there are more Xs than Os are counted as points for the first player. The num... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the following system of equations: \\begin{align*}\n3x-5y&=-1.5,\\\\\n7x+2y&=4.7.\n\\end{align*}Express your answer as an ordered pair $(x,y)$ of decimals. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0.5,0.6)"
} | {
"split": "train",
"index": 36148,
"solution": ""
} |
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