data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If there are exactly three points on the circle $x^{2}+y^{2}-2x-6y+1=0$ that are at a distance of $2$ from the line $y=kx$, determine the possible values of $k$. 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": 23849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is half of the absolute value of the difference of the squares of 21 and 15 added to the absolute value of the difference of their cubes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3051"
} | {
"split": "train",
"index": 23850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the projection of the vector $\\begin{pmatrix} 4 \\\\ 5 \\end{pmatrix}$ onto the vector $\\begin{pmatrix} 2 \\\\ 0 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 4 \\\\ 0 \\end{pmatrix}"
} | {
"split": "train",
"index": 23851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example, $3333$ ), two values (for example, $2666$ ), three values (for example, $5215$ ), or four values (for example, $4236$ ). The mean number of values that show is $... | math | {
"style": "rule",
"ground_truth": "887"
} | {
"split": "train",
"index": 23852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A passenger car traveling at a speed of 66 km/h arrives at its destination at 6:53, while a truck traveling at a speed of 42 km/h arrives at the same destination via the same route at 7:11. How many kilometers before the destination did the passenger car overtake the truck? Let... | math | {
"style": "rule",
"ground_truth": "34.65"
} | {
"split": "train",
"index": 23853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\( J L M R \\) and \\( J K Q R \\) are rectangles.\n\nAlso, \\( J R = 2 \\), \\( R Q = 3 \\), and \\( J L = 8 \\). What is the area of rectangle \\( K L M Q \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The prime factorization of 1386 is $2 \\times 3 \\times 3 \\times 7 \\times 11$. How many ordered pairs of positive integers $(x,y)$ satisfy the equation $xy = 1386$, and both $x$ and $y$ are even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\\angle DAC = 30^\\circ$ and $m\\angle BAC = 45^\\circ$, the ratio of the area of $ABCD$ to the area of the circle can be expressed as a common fraction in simplest radical form in te... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 23856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum: \n\\[\n\\left(\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{2016}\\right)+\\left(\\frac{2}{3}+\\frac{2}{4}+\\cdots+\\frac{2}{2016}\\right)+\\left(\\frac{3}{4}+\\frac{3}{5}+\\cdots+\\frac{3}{2016}\\right)+\\cdots+\\left(\\frac{2014}{2015}+\\frac{2014}{2016}\\rig... | math | {
"style": "rule",
"ground_truth": "1015560"
} | {
"split": "train",
"index": 23857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quantity $\\tan 7.5^\\circ$ can be expressed in the form\n\\[\\tan 7.5^\\circ = \\sqrt{a} - \\sqrt{b} + \\sqrt{c} - d,\\]where $a \\ge b \\ge c \\ge d$ are positive integers. Find $a + b + c + d.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 23858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A school is arranging chairs in rows for an assembly. $11$ chairs make a complete row, and right now there are $110$ chairs total. The school wants to have as few empty seats as possible, but all rows of chairs must be complete. If $70$ students will attend the assembly, how ma... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 23859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c,$ $d$ be distinct integers, and let $\\omega$ be a complex number such that $\\omega^4 = 1$ and $\\omega \\neq 1.$ Find the smallest possible value of\n\\[|a + b \\omega + c \\omega^2 + d \\omega^3|.\\] Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\sqrt{4.5}"
} | {
"split": "train",
"index": 23860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle $PQR$, medians are drawn from $P$ and $Q$ to divide segments $\\overline{QR}$ and $\\overline{PR}$ in half, respectively. If the length of the median from $P$ to the midpoint of $QR$ is $8$ units, and the median from $Q$ to the midpoint of $PR$ is $4\\sqrt{5... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bacteria in a lab dish double in number every four hours. If 500 bacteria cells are in the dish now, in how many hours will there be exactly 32,000 bacteria? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 23862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume the function $f(x) = 2\\sin x \\cos^2\\left(\\frac{\\varphi}{2}\\right) + \\cos x \\sin\\varphi - \\sin x$, where $(0 < \\varphi < \\pi)$, takes its minimum value at $x = \\pi$. \n(i) Find the value of $\\varphi$ and simplify $f(x)$. \n(ii) In triangle $ABC$, $a$, $b$,... | math | {
"style": "rule",
"ground_truth": "\\frac{7\\pi}{12}"
} | {
"split": "train",
"index": 23863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $C(0,p)$ lies on the $y$-axis between $Q(0,15)$ and $O(0,0)$. Point $B$ has coordinates $(15,0)$. Determine an expression for the area of $\\triangle COB$ in terms of $p$, and compute the length of segment $QB$. Your answer should be simplified as much as possible. Let's ... | math | {
"style": "rule",
"ground_truth": "15\\sqrt{2}"
} | {
"split": "train",
"index": 23864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of distinct residues of the number $2012^n+m^2$ on $\\mod 11$ where $m$ and $n$ are positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 23865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four positive integers $A$, $B$, $C$ and $D$ have a sum of 36. If $A+2 = B-2 = C \\times 2 = D \\div 2$, what is the value of the product $A \\times B \\times C \\times D$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3840"
} | {
"split": "train",
"index": 23866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A root of unity is a complex number that is a solution to $z^{n}=1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^{2}+a z+b=0$ for some integers $a$ 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": 23867,
"solution": "The only real roots of unity are 1 and -1. If $\\zeta$ is a complex root of unity that is also a root of the equation $z^{2}+a z+b$, then its conjugate $\\bar{\\zeta}$ must also be a root. In this case, $|a|=|\\zeta+\\bar{\\zeta}| \\leq|\\zeta|+|\\bar{\\zeta}|=2$ a... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\ln x-mx (m\\in R)$.\n(I) Discuss the monotonic intervals of the function $f(x)$;\n(II) When $m\\geqslant \\frac{ 3 \\sqrt {2}}{2}$, let $g(x)=2f(x)+x^{2}$ and its two extreme points $x\\_1$, $x\\_2 (x\\_1 < x\\_2)$ are exactly the zeros of $h(x)=\\ln ... | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3} + \\ln 2"
} | {
"split": "train",
"index": 23868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $7.\\overline{123}$ as a common fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2372}{333}"
} | {
"split": "train",
"index": 23869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touc... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 23870,
"solution": "The path is a circle with radius equal to the slant height of the cone, which is $\\sqrt {r^{2} + h^{2}}$. Thus, the length of the path is $2\\pi\\sqrt {r^{2} + h^{2}}$.\n Also, the length of the path is 17 times the circumference of the base, which is $34r\\pi$. S... |
deepscaler | [
{
"role": "user",
"content": "How many equilateral hexagons of side length $\\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.) Let's think step b... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 23871,
"solution": "We perform casework on the point three vertices away from $(0,0)$. By inspection, that point can be $( \\pm 8, \\pm 3),( \\pm 7, \\pm 2),( \\pm 4, \\pm 3),( \\pm 3, \\pm 2),( \\pm 2, \\pm 1)$ or their reflections across the line $y=x$. The cases are as follows: If ... |
deepscaler | [
{
"role": "user",
"content": "Write a twelve-digit number that is not a perfect cube. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100000000000"
} | {
"split": "train",
"index": 23872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A three-digit number has a remainder of 2 when divided by 4, 5, and 6. If three digits are appended to this three-digit number to make it a six-digit number divisible by 4, 5, and 6, what is the smallest six-digit number that meets this condition? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "122040"
} | {
"split": "train",
"index": 23873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 13 balls of each of 4 different colors. How many balls must be taken out to ensure that among the balls taken out, there are at least 3 balls of different colors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 23874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $ . Find the maximum value of\n\\[ (a-2bc)(b-2ca)(c-2ab) \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 23875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha$ and $\\beta$ be acute angles, and $\\cos \\alpha = \\frac{\\sqrt{5}}{5}$, $\\sin (\\alpha + \\beta) = \\frac{3}{5}$. Find $\\cos \\beta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{25}"
} | {
"split": "train",
"index": 23876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer \\( n \\), \\( -180 \\le n \\le 180 \\), such that \\( \\cos n^\\circ = \\cos 745^\\circ \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-25"
} | {
"split": "train",
"index": 23877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A community organization begins with twenty members, among which five are leaders. The leaders are replaced annually. Each remaining member persuades three new members to join the organization every year. Additionally, five new leaders are elected from outside the community eac... | math | {
"style": "rule",
"ground_truth": "15365"
} | {
"split": "train",
"index": 23878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles trapezoid has legs of length 30 cm each, two diagonals of length 40 cm each and the longer base is 50 cm. What is the trapezoid's area in sq cm? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 23879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose you have a Viennese pretzel lying on the table. What is the maximum number of parts you can cut it into with one straight swing of the knife? In which direction should this cut be made? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\{a_k\\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,\n\\[a_k=\\begin{cases}(0.\\underbrace{20101\\cdots 0101}_{k+2\\text{ digits}})^{a_{k-1}}\\qquad\\text{if }k\... | math | {
"style": "rule",
"ground_truth": "1341"
} | {
"split": "train",
"index": 23881,
"solution": "To solve this problem, we need to understand the behavior of the sequence $\\{a_k\\}_{k=1}^{2011}$ and determine when $a_k = b_k$, where $\\{b_k\\}_{k=1}^{2011}$ is the sequence $\\{a_k\\}_{k=1}^{2011}$ rearranged in decreasing order.\n\n#### Step 1: Analyze the se... |
deepscaler | [
{
"role": "user",
"content": "If $y$ varies directly as $x$, and if $y=8$ when $x=4$, the value of $y$ when $x=-8$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 23882,
"solution": "1. **Understanding Direct Variation**:\n Given that $y$ varies directly as $x$, we can express this relationship using the equation:\n \\[\n y = kx\n \\]\n where $k$ is the constant of proportionality.\n\n2. **Finding the Constant of Proportionality**:\n ... |
deepscaler | [
{
"role": "user",
"content": "What is the largest four-digit number that is divisible by 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9960"
} | {
"split": "train",
"index": 23883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the arithmetic mean of the reciprocals of the first three prime numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31}{90}"
} | {
"split": "train",
"index": 23884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has an area of $256$ square units. Point $E$ lies on side $\\overline{BC}$ and divides it in the ratio $3:1$. Points $F$ and $G$ are the midpoints of $\\overline{AE}$ and $\\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has an area of $48$ square un... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 23885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the product of all positive integers less than $5!$ which are invertible modulo $5!$. Find the remainder when $m$ is divided by $5!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "119"
} | {
"split": "train",
"index": 23886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $[ x ]$ denote the greatest integer less than or equal to $x$. For example, $[10.2] = 10$. Calculate the value of $\\left[\\frac{2017 \\times 3}{11}\\right] + \\left[\\frac{2017 \\times 4}{11}\\right] + \\left[\\frac{2017 \\times 5}{11}\\right] + \\left[\\frac{2017 \\times ... | math | {
"style": "rule",
"ground_truth": "6048"
} | {
"split": "train",
"index": 23887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers $u$ and $v$ are each chosen independently and uniformly at random from the interval $(0, 2)$. What is the probability that $\\lfloor \\log_3 u \\rfloor = \\lfloor \\log_3 v \\rfloor$?\nA) $\\frac{1}{9}$\nB) $\\frac{1}{3}$\nC) $\\frac{4}{9}$\nD) $\\frac{5}{9}$\nE) $... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 23888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Steve says to Jon, \"I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x) = 2x^3-2ax^2+(a^2-81)x-c$ for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?\"\n After some calculations, Jon says, \"Ther... | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 23889,
"solution": "Since each of the roots is positive, the local maximum of the function must occur at a positive value of $x$. Taking $\\frac{d}{dx}$ of the polynomial yields $6x^2-4ax+a^2-81$, which is equal to $0$ at the local maximum. Since this is a quadratic in $a$, we can fin... |
deepscaler | [
{
"role": "user",
"content": "Square \\(ABCD\\) has side length 2, and \\(X\\) is a point outside the square such that \\(AX = XB = \\sqrt{2}\\). What is the length of the longest diagonal of pentagon \\(AXB\\)?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 23890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 cities in a state, and some pairs of cities are connected by roads. There are 40 roads altogether. A city is called a \"hub\" if it is directly connected to every other city. What is the largest possible number of hubs? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 23891,
"solution": "If there are $h$ hubs, then $\\binom{h}{2}$ roads connect the hubs to each other, and each hub is connected to the other $10-h$ cities; we thus get $\\binom{h}{2}+h(10-h)$ distinct roads. So, $40 \\geq\\binom{ h}{2}+h(10-h)=-h^{2} / 2+19 h / 2$, or $80 \\geq h(19-h... |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$, $y$, $z$ satisfying $2x-y-2z-6=0$, and $x^2+y^2+z^2\\leq4$, calculate the value of $2x+y+z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 23892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Thirty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \\%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\\frac{p}{q},$ where $p$ and $q$ are relat... | math | {
"style": "rule",
"ground_truth": "409"
} | {
"split": "train",
"index": 23893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left( x \\right)=\\frac{{{x}^{2}}}{1+{{x}^{2}}}$.\n\n(1) Find the values of $f\\left( 2 \\right)+f\\left( \\frac{1}{2} \\right),f\\left( 3 \\right)+f\\left( \\frac{1}{3} \\right),f\\left( 4 \\right)+f\\left( \\frac{1}{4} \\right)$ and conjecture a general... | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"split": "train",
"index": 23894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the MP Donut Hole Factory, Niraek, Theo, and Akshaj are coating spherical donut holes in powdered sugar. Niraek's donut holes have radius 6 mm, Theo's donut holes have radius 8 mm, and Akshaj's donut holes have radius 10 mm. All three workers coat the surface of the donut ... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 23895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is given that $A B = B C = 4$ and $A A_{1} = 2$. Point $P$ lies on the plane $A_{1} B C$, and it holds that $\\overrightarrow{D P} \\cdot \\overrightarrow{P B} = 0$. Find the area of the plane region formed by all s... | math | {
"style": "rule",
"ground_truth": "\\frac{36\\pi}{5}"
} | {
"split": "train",
"index": 23896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $n$ satisfy the following condition:\n$(130n)^{50} > n^{100} > 2^{200}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 23897,
"solution": "We are given the inequality $(130n)^{50} > n^{100} > 2^{200}$ and need to find the number of positive integers $n$ that satisfy this condition.\n\n1. **First Inequality Analysis:**\n \\[\n (130n)^{50} > n^{100}\n \\]\n Simplifying, we get:\n \\[\n 130^{... |
deepscaler | [
{
"role": "user",
"content": "Given that the direction vector of line $l$ is $(4,2,m)$, the normal vector of plane $\\alpha$ is $(2,1,-1)$, and $l \\perp \\alpha$, find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 23898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circular sheet of paper with a radius of $12$ cm is cut into four congruent sectors. A cone is formed by rolling one of these sectors until the edges meet. Calculate both the height and the volume of this cone. Express the height in simplest radical form and the volume in ter... | math | {
"style": "rule",
"ground_truth": "9\\pi\\sqrt{15}"
} | {
"split": "train",
"index": 23899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the range of\n\\[f(A)=\\frac{\\sin A(3\\cos^{2}A+\\cos^{4}A+3\\sin^{2}A+\\sin^{2}A\\cos^{2}A)}{\\tan A (\\sec A-\\sin A\\tan A)}\\]if $A\\neq \\dfrac{n\\pi}{2}$ for any integer $n.$ Enter your answer using interval notation. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "(3,4)"
} | {
"split": "train",
"index": 23900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence of positive integers $a_1, a_2, a_3, \\dots$ follows the rule:\n\\[a_{n+2} = a_{n+1} + a_n\\]\nfor all $n \\geq 1$. If $a_6 = 50$, find $a_7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 23901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hyperbolas \\[\\frac{x^2}{4} - \\frac{y^2}{9} = 1\\]and \\[\\frac{y^2}{18} - \\frac{x^2}{N} = 1\\]have the same asymptotes. Find $N.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 23902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A, B, C, \\) and \\( D \\) be positive real numbers such that\n\\[\n\\log_{10} (AB) + \\log_{10} (AC) = 3, \\\\\n\\log_{10} (CD) + \\log_{10} (CB) = 4, \\\\\n\\log_{10} (DA) + \\log_{10} (DB) = 5.\n\\]\nCompute the value of the product \\( ABCD \\). Let's think step by ... | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 23903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all solutions to the equation $\\displaystyle\\sqrt[3]{3 - x} = -\\frac{3}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{51}{8}"
} | {
"split": "train",
"index": 23904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 23905,
"solution": "1. **Identify the dimensions of the rectangle and squares**: \n - Each square $ABCD$ and $PQRS$ has a side length of $15$.\n - The rectangle $AQRD$ formed by the overlap of these squares has dimensions $15 \\times 25$.\n\n2. **Calculate the area of rectangle $A... |
deepscaler | [
{
"role": "user",
"content": "On a plate, there are different candies of three types: 2 lollipops, 3 chocolate candies, and 5 jelly candies. Sveta ate all of them one by one, choosing each next candy at random. Find the probability that the first and last candies she ate were of the same type. Let's think s... | math | {
"style": "rule",
"ground_truth": "14/45"
} | {
"split": "train",
"index": 23906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be complex numbers such that\n\\[a + b + c = ab + ac + bc = abc = 1.\\]Enter the values $a,$ $b,$ $c,$ separated by commas, in any order. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1,i,-i"
} | {
"split": "train",
"index": 23907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, $E$ is the midpoint of the leg $AB$ of trapezoid $ABCD$. $DF \\perp EC$, $DF=10$, and $EC=24$. Find the area of trapezoid $ABCD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 23908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\(0.001 + 1.01 + 0.11\\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.121"
} | {
"split": "train",
"index": 23909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An eight-sided die is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 23910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the integers that are both greater than 3 and less than 12? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 23911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x=2y$ and $y \\neq 0$, what is the value of $(x+2y)-(2x+y)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-y"
} | {
"split": "train",
"index": 23912,
"solution": "We simplify first, then substitute $x=2y$: $(x+2y)-(2x+y)=x+2y-2x-y=y-x=y-2y=-y$. Alternatively, we could substitute first, then simplify: $(x+2y)-(2x+y)=(2y+2y)-(2(2y)+y)=4y-5y=-y$."
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability, expressed as a fraction, of drawing a marble that is either green or white from a bag containing 4 green, 3 white, 5 red, and 6 blue marbles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"split": "train",
"index": 23913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four positive integers $A$, $B$, $C$ and $D$ have a sum of 64. If $A+3 = B-3 = C \\times 3 = D \\div 3$, what is the value of the product $A \\times B \\times C \\times D$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19440"
} | {
"split": "train",
"index": 23914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, h... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 23915,
"solution": "1. **Set up the problem**: Michael walks at $5$ feet per second and trash pails are located every $200$ feet. The garbage truck moves at $10$ feet per second and stops for $30$ seconds at each pail. Michael sees the truck leaving a pail $200$ feet ahead as he passe... |
deepscaler | [
{
"role": "user",
"content": "Two three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a^{}_{}$ when it should have been a $b^{}_{}$, or... | math | {
"style": "rule",
"ground_truth": "532"
} | {
"split": "train",
"index": 23916,
"solution": "Consider $n$ letter strings instead. If the first letters all get transmitted correctly, then the $a$ string will be first. Otherwise, the only way is for both of the first letters to be the same, and then we consider the next $n-1$ letter string following the firs... |
deepscaler | [
{
"role": "user",
"content": "A point is randomly thrown onto the segment $[11, 18]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}+2k-99\\right)x^{2}+(3k-7)x+2=0$ satisfy the condition $x_{1} \\leq 2x_{2}$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 23917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Grain warehouses A and B each originally stored a certain number of full bags of grain. If 90 bags are transferred from warehouse A to warehouse B, the number of bags in warehouse B will be twice the number in warehouse A. If an unspecified number of bags are transferred from w... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 23918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distribution of populations in a group of counties is shown in this pie chart. What percent of the counties have fewer than 100,000 residents?\n\n[asy]\ndraw(Circle((0,0),25),linewidth(1));\ndraw((-24,-7)--(0,0)--(24,-7),linewidth(1));\ndraw((0,0)--(7,-24),linewidth(1));\nl... | math | {
"style": "rule",
"ground_truth": "84\\%"
} | {
"split": "train",
"index": 23919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $|\\overrightarrow{a}-\\overrightarrow{b}|=\\sqrt{7}$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 23920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cuboctahedron is a polyhedron whose faces are squares and equilateral triangles such that two squares and two triangles alternate around each vertex. What is the volume of a cuboctahedron of side length 1? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{2} / 3"
} | {
"split": "train",
"index": 23921,
"solution": "We can construct a cube such that the vertices of the cuboctahedron are the midpoints of the edges of the cube. Let $s$ be the side length of this cube. Now, the cuboctahedron is obtained from the cube by cutting a tetrahedron from each corner. Each such tetrahedro... |
deepscaler | [
{
"role": "user",
"content": "A sphere is cut into three equal wedges. The circumference of the sphere is $18\\pi$ inches. What is the volume of the intersection between one wedge and the top half of the sphere? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "162\\pi"
} | {
"split": "train",
"index": 23922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\(a, b,\\) and \\(c\\) (not necessarily integers) satisfy the conditions\n\n\\[\na + b + c = 0 \\quad \\text{and} \\quad \\frac{a}{b} + \\frac{b}{c} + \\frac{c}{a} = 100\n\\]\n\nWhat is the value of \\(\\frac{b}{a} + \\frac{c}{b} + \\frac{a}{c}\\)? Let's think step... | math | {
"style": "rule",
"ground_truth": "-101"
} | {
"split": "train",
"index": 23923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the volume of the region given by the inequality\n\\[|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \\le 4.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"split": "train",
"index": 23924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many graphs are there on 10 vertices labeled \\(1,2, \\ldots, 10\\) such that there are exactly 23 edges and no triangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42840"
} | {
"split": "train",
"index": 23925,
"solution": "Note that the sum of the degrees of the graph is \\(23 \\cdot 2=46\\), so at least one vertex has degree 5 or more. We casework on the maximal degree \\(n\\). Case 1: \\(n \\geq 7\\), then none of the \\(n\\) neighbors can have an edge between each other, for \\(\\... |
deepscaler | [
{
"role": "user",
"content": "In a regular triangle $ABC$ with side length $3$, $D$ is a point on side $BC$ such that $\\overrightarrow{CD}=2\\overrightarrow{DB}$. Calculate the dot product $\\overrightarrow{AB} \\cdot \\overrightarrow{AD}$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{2}"
} | {
"split": "train",
"index": 23926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an equilateral triangle $DEF$ with $DE = DF = EF = 8$ units and a circle with radius $4$ units tangent to line $DE$ at $E$ and line $DF$ at $F$, calculate the area of the circle passing through vertices $D$, $E$, and $F$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{64\\pi}{3}"
} | {
"split": "train",
"index": 23927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a conference, the 2016 participants were registered from P1 to P2016. Each participant from P1 to P2015 shook hands with exactly the same number of participants as the number on their registration form. How many hands did the 2016th participant shake? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 23928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)$ that satisfies $f\\left(\\frac{\\pi}{2} - x\\right) + f(x) = 0$ and $f(\\pi + x) = f(-x)$, calculate the value of $f\\left(\\frac{79\\pi}{24}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2} - \\sqrt{6}}{4}"
} | {
"split": "train",
"index": 23929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a corridor that is 100 meters long, there are 20 rugs with a total length of 1 kilometer. Each rug is as wide as the corridor. What is the maximum possible total length of the sections of the corridor that are not covered by the rugs? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 23930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the range of the function $$G(x) = |x+1|-|x-1|~?$$Express your answer in interval notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "[-2,2]"
} | {
"split": "train",
"index": 23931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 30$ and \\[\\frac{1}{p} + \\frac{1}{q} + \\frac{1}{r} + \\frac{240}{pqr} = 1,\\] compute $pqr$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1080"
} | {
"split": "train",
"index": 23932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find four-thirds of $\\frac{9}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 23933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of \\(\\log _{2}\\left[2^{3} 4^{4} 8^{5} \\cdots\\left(2^{20}\\right)^{22}\\right]\\). \nChoose one of the following options:\n(a) 3290\n(b) 3500\n(c) 3710\n(d) 4172 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5950"
} | {
"split": "train",
"index": 23934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the sequence of positive integers $a_n$ recursively by $a_1=7$ and $a_n=7^{a_{n-1}}$ for all $n\\geq 2$ . Determine the last two digits of $a_{2007}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 23935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quantity\n\\[\\frac{\\tan \\frac{\\pi}{5} + i}{\\tan \\frac{\\pi}{5} - i}\\]is a tenth root of unity. In other words, it is equal to $\\cos \\frac{2n \\pi}{10} + i \\sin \\frac{2n \\pi}{10}$ for some integer $n$ between 0 and 9 inclusive. Which value of $n$? Let's think s... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 23936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $2x^6-x^4+4x^2-7$ is divided by $x^2+4x+3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-704x-706"
} | {
"split": "train",
"index": 23937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify \\[\\frac{1}{\\dfrac{1}{\\sqrt{2}+1} + \\dfrac{2}{\\sqrt{3}-1}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt3-\\sqrt2"
} | {
"split": "train",
"index": 23938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-similar regular 720-pointed stars are there, given that a regular $n$-pointed star requires its vertices to not all align with vertices of a smaller regular polygon due to common divisors other than 1 between the step size and $n$? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 23939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function be $$f(x)= \\sqrt {3}\\sin 2x+2\\cos^{2}x+2$$.\n(I) Find the smallest positive period and the range of $f(x)$;\n(II) In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $$A= \\frac {\\pi }{3}$$ and the are... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 23940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of paper with dimensions 8 cm by 6 cm is folded in half horizontally. After folding, the paper is cut vertically at 3 cm and 5 cm from one edge, forming three distinct rectangles. Calculate the ratio of the perimeter of the smallest rectangle to the perimete... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 23941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $AB = 30$ and $\\angle ADB = 90^\\circ$. If $\\sin A = \\frac{3}{5}$ and $\\sin C = \\frac{1}{4}$, what is the length of $DC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18\\sqrt{15}"
} | {
"split": "train",
"index": 23942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the integers from \\(2^{10}\\) to \\(2^{18}\\) inclusive are divisible by \\(2^{9}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "511"
} | {
"split": "train",
"index": 23943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jane can walk any distance in half the time it takes Hector to walk the same distance. They set off in opposite directions around the outside of the 18-block area as shown. When they meet for the first time, they will be closest to Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "D"
} | {
"split": "train",
"index": 23944,
"solution": "1. **Understanding the Problem**: Jane and Hector start at the same point and walk in opposite directions around an 18-block area. Jane walks twice as fast as Hector.\n\n2. **Setting Up the Problem**: Let's denote the speed of Hector as $s$ blocks per unit time. Th... |
deepscaler | [
{
"role": "user",
"content": "In the figure below, if the area of $\\triangle ABC$ is 27, what is the value of $p$? [asy]\nsize(5cm);defaultpen(fontsize(9));\npair o = (0, 0); pair q = (0, 12); pair b = (12, 0);\npair a = (2, 12); pair t = (2, 0); pair c = (0, 9);\n\ndraw((-2, 0)--(15, 0), Arrow);\ndraw((0,... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 23945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 23946,
"solution": "1. **Identify the minimum and maximum scores**: The player can score a minimum of 2 points per basket and a maximum of 3 points per basket. If all baskets were 2-pointers, the minimum score would be:\n \\[\n 5 \\times 2 = 10\n \\]\n If all baskets were 3-po... |
deepscaler | [
{
"role": "user",
"content": "A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$ . Heracle d... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $p=1, 2, \\cdots, 10$ let $S_p$ be the sum of the first $40$ terms of the arithmetic progression whose first term is $p$ and whose common difference is $2p-1$; then $S_1+S_2+\\cdots+S_{10}$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80200"
} | {
"split": "train",
"index": 23948,
"solution": "1. **Identify the $40$th term of the sequence**: \n For an arithmetic progression (AP) with first term $a = p$ and common difference $d = 2p - 1$, the $n$th term of the AP is given by:\n \\[\n a_n = a + (n-1)d = p + (n-1)(2p-1).\n \\]\n Substituting $n = ... |
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