data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find the time, in seconds, after 12 o'clock, when the area of $\\triangle OAB$ will reach its maximum for the first time. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{59}"
} | {
"split": "train",
"index": 800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than or equal to 240 can be expressed as a sum of distinct factorials? Consider 0 ! and 1 ! to be distinct. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 801,
"solution": "Note that $1=0$ !, $2=0$ ! +1 !, $3=0$ ! +2 !, and $4=0!+1$ ! +2 !. These are the only numbers less than 6 that can be written as the sum of factorials. The only other factorials less than 240 are $3!=6,4!=24$, and $5!=120$. So a positive integer less than or equal t... |
deepscaler | [
{
"role": "user",
"content": "Let $s$ be the set of all rational numbers $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be represented as a repeating decimal of the form $0.abcabcabc\\cdots = 0.\\dot{a}b\\dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. How many d... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"split": "train",
"index": 802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fifth number, $n$, is added to the set $\\{ 3,6,9,10 \\}$ to make the mean of the set of five numbers equal to its median. The number of possible values of $n$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 803,
"solution": "We are given the set $\\{3, 6, 9, 10\\}$ and we need to add a number $n$ such that the mean of the new set equals its median. Let's analyze the problem step by step.\n\n#### Step 1: Understand the properties of mean and median\n- **Mean** of a set is the average of a... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane $(xOy)$, the sum of the distances from point $P$ to the two points $(0,-\\sqrt{3})$ and $(0,\\sqrt{3})$ is equal to $4$. Let the trajectory of point $P$ be denoted as $C$.\n\n(I) Write the equation of $C$;\n\n(II) If the line $y=kx+1$ intersect... | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{65}}{17}"
} | {
"split": "train",
"index": 804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the midpoint of side $BC$ of triangle $\\triangle ABC$ is $D$, point $E$ lies in the plane of $\\triangle ABC$, and $\\overrightarrow{CD}=3\\overrightarrow{CE}-2\\overrightarrow{CA}$, if $\\overrightarrow{AC}=x\\overrightarrow{AB}+y\\overrightarrow{BE}$, then determi... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use all digits from 1 to 9 to form three three-digit numbers such that their product is:\n\na) the smallest; \nb) the largest. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "941 \\times 852 \\times 763"
} | {
"split": "train",
"index": 806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider $n$ disks $C_{1}, C_{2}, \\ldots, C_{n}$ in a plane such that for each $1 \\leq i<n$, the center of $C_{i}$ is on the circumference of $C_{i+1}$, and the center of $C_{n}$ is on the circumference of $C_{1}$. Define the score of such an arrangement of $n$ disks to be th... | math | {
"style": "rule",
"ground_truth": "(n-1)(n-2)/2"
} | {
"split": "train",
"index": 807,
"solution": "The answer is $(n-1)(n-2) / 2$. Let's call a set of $n$ disks satisfying the given conditions an $n$-configuration. For an $n$ configuration $\\mathcal{C}=\\left\\{C_{1}, \\ldots, C_{n}\\right\\}$, let $S_{\\mathcal{C}}=\\left\\{(i, j) \\mid C_{i}\\right.$ properly c... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has a perimeter of 2007 units. The sides have lengths that are all integer values with $AB< BC \\leq AC$. What is the smallest possible value of $BC - AB$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\( BL \\) is the angle bisector of triangle \\( ABC \\). Find its area if it is known that \\( |AL| = 3 \\), \\( |BL| = 6\\sqrt{5} \\), and \\( |CL| = 4 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21 \\sqrt{55}}{4}"
} | {
"split": "train",
"index": 809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of a triangle if it is known that its medians \\(CM\\) and \\(BN\\) are 6 and 4.5 respectively, and \\(\\angle BKM = 45^\\circ\\), where \\(K\\) is the point of intersection of the medians. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "9\\sqrt{2}"
} | {
"split": "train",
"index": 810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anna Lisa bought two dozen apples for $\\$$15.60. At that same rate, how much would three dozen of these apples cost? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$23.40"
} | {
"split": "train",
"index": 811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a pentagon ABCDE. If the vertices A, B, C, D, E are colored with one of three colors: red, yellow, green, such that adjacent vertices are of different colors, then there are a total of different coloring methods. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the parts processed by a worker, on average, $4 \\%$ are non-standard. Find the probability that among the 30 parts taken for testing, two parts will be non-standard. What is the most probable number of non-standard parts in the considered sample of 30 parts, and what is ... | math | {
"style": "rule",
"ground_truth": "0.305"
} | {
"split": "train",
"index": 813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $$\\frac {x^{2}}{a^{2}}- \\frac {y^{2}}{b^{2}}=1(a＞0，b＞0)$$, the sum of the two line segments that are perpendicular to the two asymptotes and pass through one of its foci is $a$. Find the eccentricity of the hyperbola. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\cos^2\\frac{x}{2}+\\sin x-1$. Find:\n- $(Ⅰ)$ The minimum positive period, monotonic decreasing interval, and symmetry center of $f(x)$.\n- $(Ⅱ)$ When $x\\in \\left[-\\pi ,0\\right]$, find the minimum value of $f(x)$ and the corresponding value of $x$... | math | {
"style": "rule",
"ground_truth": "-\\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2x^{2}-3x-\\ln x+e^{x-a}+4e^{a-x}$, where $e$ is the base of the natural logarithm, if there exists a real number $x_{0}$ such that $f(x_{0})=3$ holds, then the value of the real number $a$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "1-\\ln 2"
} | {
"split": "train",
"index": 816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle $ABC$, points $M$ and $N$ are on sides $AB$ and $AC$, respectively, such that $MB = BC = CN$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $ABC$, respectively. Express the ratio $MN/BC$ in terms of $R$ and $r$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{1 - \\frac{2r}{R}}"
} | {
"split": "train",
"index": 817,
"solution": "\nIn this problem, we analyze a triangle \\( \\triangle ABC \\) where points \\( M \\) and \\( N \\) lie on sides \\( AB \\) and \\( AC \\) respectively, satisfying \\( MB = BC = CN \\).\n\n### Step 1: Set up the problem using geometric principles\n\nConsider the giv... |
deepscaler | [
{
"role": "user",
"content": "A square flag has a green cross of uniform width with a yellow square in the center on a white background. The cross is symmetric with respect to each of the diagonals of the square. If the entire cross (both the green arms and the yellow center) occupies 49% of the area of the... | math | {
"style": "rule",
"ground_truth": "25.14\\%"
} | {
"split": "train",
"index": 818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be an acute triangle with $A$-excircle $\\Gamma$. Let the line through $A$ perpendicular to $B C$ intersect $B C$ at $D$ and intersect $\\Gamma$ at $E$ and $F$. Suppose that $A D=D E=E F$. If the maximum value of $\\sin B$ can be expressed as $\\frac{\\sqrt{a}+\\sqr... | math | {
"style": "rule",
"ground_truth": "705"
} | {
"split": "train",
"index": 819,
"solution": "First note that we can assume $A B<A C$. Suppose $\\Gamma$ is tangent to $B C$ at $T$. Let $A D=D E=E F=x$. Then, by Power of a Point, we have $D T^{2}=D E \\cdot D F=x \\cdot 2 x=2 x^{2} \\Longrightarrow D T=x \\sqrt{2}$. Note that $C T=s-b$, and since the length of... |
deepscaler | [
{
"role": "user",
"content": "Convert $BD4_{16}$ to base 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "233110_4"
} | {
"split": "train",
"index": 820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=( \\sqrt {2},m)(m > 0)$, $\\overrightarrow{b}=(\\sin x,\\cos x)$, and the maximum value of the function $f(x)= \\overrightarrow{a} \\cdot \\overrightarrow{b}$ is $2$.\n1. Find $m$ and the smallest positive period of the function $f(x)$;\n2. In $\\tria... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}}{4}"
} | {
"split": "train",
"index": 821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sam spends his days walking around the following $2 \\times 2$ grid of squares. Say that two squares are adjacent if they share a side. He starts at the square labeled 1 and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on ... | math | {
"style": "rule",
"ground_truth": "167"
} | {
"split": "train",
"index": 822,
"solution": "Note that on the first step, Sam can either step on 2 or 4. On the second step, Sam can either step on 1 or 3, regardless of whether he is on 2 or 4. Now, for example, say that Sam takes 8 steps. His total sum will be $2+1+2+1+2+1+2+1+2 a$, where $a$ is the number of... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $C= \\dfrac {\\pi}{6}$, $a=1$, $b= \\sqrt {3}$, find the measure of $B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {2\\pi}{3}"
} | {
"split": "train",
"index": 823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Is there a number in which every digit from 1 to 9 is used exactly once, and for which the number formed by the first $k$ digits is divisible by $k$ for every $k$ from 1 to 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "381654729"
} | {
"split": "train",
"index": 824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(2x^2 + 3xy + 2y^2 = 1\\), find the minimum value of \\(f(x, y) = x + y + xy\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{8}"
} | {
"split": "train",
"index": 825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$ .\n\n*Proposed by Andy Xu* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Jane lists the whole numbers $1$ through $50$ once and Tom copies Jane's numbers, replacing each occurrence of the digit $3$ by the digit $2$, calculate how much larger Jane's sum is than Tom's sum. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of three different positive integers is equal to $7^3$. What is the sum of the three integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist integers $a,$ $b,$ and $c$ such that\n\\[(x - a)(x - 10) + 1 = (x + b)(x + c).\\]Enter all possible values of $a,$ separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8,12"
} | {
"split": "train",
"index": 829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line with slope equal to $1$ and a line with slope equal to $2$ intersect at the point $P(1,6),$ as shown. [asy]\nunitsize(0.5 cm);\ndraw((0,-1)--(0,10),EndArrow);\ndraw((-10,0)--(5,0),EndArrow);\ndraw((-6,-1)--(5,10),linewidth(0.8));\ndraw((-2.5,-1)--(3,10),linewidth(0.8));\... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number's cube is a five-digit number; the first digit is 4, and the fifth digit is 6. What is this number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number which when divided by $10$ leaves a remainder of $9$, when divided by $9$ leaves a remainder of $8$, by $8$ leaves a remainder of $7$, etc., down to where, when divided by $2$, it leaves a remainder of $1$, is: Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "2519"
} | {
"split": "train",
"index": 832,
"solution": "1. **Understanding the Problem:**\n The problem states that a number $n$ has specific remainders when divided by integers from $10$ down to $2$. Specifically, when $n$ is divided by any integer $k$ (where $10 \\geq k \\geq 2$), it leaves a remainder of $k-1$.\n\n2.... |
deepscaler | [
{
"role": "user",
"content": "How many integers are there from 1 to 1,000,000 that are neither perfect squares, nor perfect cubes, nor fourth powers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "998910"
} | {
"split": "train",
"index": 833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the price of Product A was set at 70 yuan per piece in the first year, with an annual sales volume of 118,000 pieces, starting from the second year, the price per piece increased by $$\\frac {70\\cdot x\\%}{1-x\\%}$$ yuan due to a management fee, and the annual sales volu... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the ratio of $AE:EC$ in $\\triangle ABC$ given that $AB=6$, $BC=8$, $AC=10$, and $E$ is on $\\overline{AC}$ with $BE=6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18}{7}"
} | {
"split": "train",
"index": 835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rectangular grid measuring 8 by 6, there are $48$ grid points, including those on the edges. Point $P$ is placed at the center of the rectangle. Find the probability that the line $PQ$ is a line of symmetry of the rectangle, given that point $Q$ is randomly selected fro... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{47}"
} | {
"split": "train",
"index": 836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of the triangle formed by the line $\\frac{x}{3} + \\frac{y}{4} = 1$ and the two coordinate axes is $14$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original n... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 838,
"solution": "Let's denote the number picked by person $i$ as $a_i$. According to the problem, each person announces the average of the numbers picked by their two immediate neighbors. Therefore, if person $i$ announces $i$, the equation relating the numbers picked by their neighb... |
deepscaler | [
{
"role": "user",
"content": "How many three-digit numbers exist that are 5 times the product of their digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $1 \\le n \\le 200$, how many integers are there such that $\\frac{n}{n+1}$ is a repeating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $|\\overrightarrow{a}| = 1$, $|\\overrightarrow{b}| = \\sqrt{3}$, and $(3\\overrightarrow{a} - 2\\overrightarrow{b}) \\perp \\overrightarrow{a}$, find the angle between $\\overrightarrow{a}$ and $\\overr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top $\\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "4000"
} | {
"split": "train",
"index": 842,
"solution": "1. **Understanding the Problem**: We are given a cone-shaped mountain with a total height of 8000 feet. The top $\\frac{1}{8}$ of the volume of the mountain is above water. We need to find the depth of the ocean at the base of the mountain.\n\n2. **Volume of a Cone**... |
deepscaler | [
{
"role": "user",
"content": "\nOleg drew an empty $50 \\times 50$ table and wrote a non-zero number above each column and to the left of each row. It turned out that all 100 numbers written were different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he w... | math | {
"style": "rule",
"ground_truth": "1250"
} | {
"split": "train",
"index": 843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles triangle $ABC$ with $AB = AC = 40$ units and $BC = 24$ units, let $CX$ be the angle bisector of $\\angle BCA$. Find the ratio of the area of $\\triangle BCX$ to the area of $\\triangle ACX$. Provide your answer as a simplified fraction. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the area of a regular hexagon inscribed in an equilateral triangle with side length $s$ to the area of a regular hexagon inscribed in a circle with radius $r$? Assume the height of the equilateral triangle equals the diameter of the circle, thus $s = r \\sq... | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{16}"
} | {
"split": "train",
"index": 845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function \\( f(x) \\) defined on \\( (0, +\\infty) \\) satisfy \\( f(x) > -\\frac{3}{x} \\) for any \\( x \\in (0, +\\infty) \\) and \\( f\\left(f(x) + \\frac{3}{x}\\right) = 2 \\). Find \\( f(5) \\). Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{5}"
} | {
"split": "train",
"index": 846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $DEF$ is similar to triangle $ABC$. If $DE=6$, $EF=12$, and $BC=18$ units, what is the length of segment $AB$?\n\n[asy]draw((0,0)--(7,0));\ndraw((0,0)--(0,4));\ndraw((0,4)--(7,0));\nlabel(\"E\",(0,0),W);\nlabel(\"F\",(7,0),E);\nlabel(\"D\",(0,4),W);\ndraw((15,0)--(25.... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x^2 + 6x + c$ for all real numbers $x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11 - \\sqrt{13}}{2}"
} | {
"split": "train",
"index": 848,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polar equation of a line is $ρ\\sin(θ+ \\frac{π}{4})= \\frac{\\sqrt{2}}{2}$, and the parametric equation of the circle $M$ is $\\begin{cases} x = 2\\cosθ \\\\ y = -2 + 2\\sinθ \\end{cases}$, where $θ$ is the parameter.\n(I) Convert the line's polar equation into a Car... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2} - 2"
} | {
"split": "train",
"index": 849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The KML airline operates shuttle services between some cities such that from any city, you cannot directly reach more than three other cities. However, with at most one transfer, you can travel from any city to any other city. What is the maximum number of cities between which ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$ and non-negative integer $k$, define $W(n, k)$ recursively by $$ W(n, k)= \\begin{cases}n^{n} & k=0 \\\\ W(W(n, k-1), k-1) & k>0\\end{cases} $$ Find the last three digits in the decimal representation of $W(555,2)$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "875"
} | {
"split": "train",
"index": 851,
"solution": "For any $n$, we have $$ W(n, 1)=W(W(n, 0), 0)=\\left(n^{n}\\right)^{n^{n}}=n^{n^{n+1}} $$ Thus, $$ W(555,1)=555^{555^{556}} $$ Let $N=W(555,1)$ for brevity, and note that $N \\equiv 0(\\bmod 125)$, and $N \\equiv 3(\\bmod 8)$. Then, $$ W(555,2)=W(N, 1)=N^{N^{N+1}} $$... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $a_n=\\log_n(n+1)$ $(n\\geqslant 2,n\\in \\mathbb{N}^*)$, define: $k$ $(k\\in \\mathbb{N}^*)$ that makes the product $a_1\\cdot a_2\\cdot \\cdots \\cdot a_k$ an integer as \"simple number\". Then, the sum of all \"simple numbers\... | math | {
"style": "rule",
"ground_truth": "2035"
} | {
"split": "train",
"index": 852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less th... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 853,
"solution": "To solve for the number of positive integers less than or equal to $97$ for which $D(n) = 2$, we analyze the binary representations of numbers and count those with exactly two transitions between adjacent digits (from 0 to 1 or from 1 to 0).\n\n#### Case Analysis:\nF... |
deepscaler | [
{
"role": "user",
"content": "Given that a new kitchen mixer is listed in a store for $\\textdollar 129.99$ and an online advertisement offers the same mixer for four easy payments of $\\textdollar 29.99$ and a one-time shipping and handling fee of $\\textdollar 19.99$, calculate how many cents are saved by... | math | {
"style": "rule",
"ground_truth": "996"
} | {
"split": "train",
"index": 854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of rationals $\\frac{m}{n}$ such that \n\n(i) $0 < \\frac{m}{n} < 1$ ;\n\n(ii) $m$ and $n$ are relatively prime;\n\n(iii) $mn = 25!$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c$ be non-negative real numbers such that $ab+bc+ca=3$. Suppose that $a^{3}b+b^{3}c+c^{3}a+2abc(a+b+c)=\\frac{9}{2}$. What is the minimum possible value of $ab^{3}+bc^{3}+ca^{3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 856,
"solution": "Expanding the inequality $\\sum_{\\text {cyc }} ab(b+c-2a)^{2} \\geq 0$ gives $\\left(\\sum_{\\text {cyc }} ab^{3}\\right)+4\\left(\\sum_{\\text {cyc }} a^{3}b\\right)-4\\left(\\sum_{\\text {cyc }} a^{2}b^{2}\\right)-abc(a+b+c) \\geq 0$. Using $\\left(\\sum_{\\text {... |
deepscaler | [
{
"role": "user",
"content": "A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps. After a total of $n$ jumps, the position of the grasshopper is 162 cm to the west and 15... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 857,
"solution": "Each group of four jumps takes the grasshopper 1 cm to the east and 3 cm to the west, which is a net movement of 2 cm to the west, and 2 cm to the north and 4 cm to the south, which is a net movement of 2 cm to the south. We note that $158=2 \\times 79$. Thus, after ... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all the four-digit numbers without repeated digits, how many numbers have the digit in the thousandth place 2 greater than the digit in the unit place? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "448"
} | {
"split": "train",
"index": 859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(r(x)\\) have a domain of \\(\\{-2,-1,0,1\\}\\) and a range of \\(\\{-1,0,2,3\\}\\). Let \\(t(x)\\) have a domain of \\(\\{-1,0,1,2,3\\}\\) and be defined as \\(t(x) = 2x + 1\\). Furthermore, \\(s(x)\\) is defined on the domain \\(\\{1, 2, 3, 4, 5, 6\\}\\) by \\(s(x) = x ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin \\alpha - 2\\cos \\alpha = \\frac{\\sqrt{10}}{2}$, find $\\tan 2\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\left\\{a_{n}\\right\\}$ that satisfies\n$$\na_{n+1} = -\\frac{1}{2} a_{n} + \\frac{1}{3^{n}}\\quad (n \\in \\mathbf{Z}_{+}),\n$$\nfind all values of $a_{1}$ such that the sequence $\\left\\{a_{n}\\right\\}$ is monotonic, i.e., $\\left\\{a_{n}\\right\\}$ is... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alec wishes to construct a string of 6 letters using the letters A, C, G, and N, such that: - The first three letters are pairwise distinct, and so are the last three letters; - The first, second, fourth, and fifth letters are pairwise distinct. In how many ways can he construc... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 863,
"solution": "There are $4!=24$ ways to decide the first, second, fourth, and fifth letters because these letters can be selected sequentially without replacement from the four possible letters. Once these four letters are selected, there are 2 ways to select the third letter beca... |
deepscaler | [
{
"role": "user",
"content": "A Senate committee has 5 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions? (Two seatings are considered the same if one is a rotation of the other.) Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "362,\\!880"
} | {
"split": "train",
"index": 864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$, $a_{2}=5$, $S_{n+1}=S_{n}+a_{n}+4$; $\\{b_{n}\\}$ is a geometric sequence, $b_{2}=9$, $b_{1}+b_{3}=30$, with a common ratio $q \\gt 1$. \n$(1)$ Find the general formulas for sequences $\\{a_{n}\\}$ an... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"split": "train",
"index": 865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the alphametic $W E \\times E Y E=S C E N E$, each different letter stands for a different digit, and no word begins with a 0. The $W$ in this problem has the same value as the $W$ in problem 31. Find $S$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 866,
"solution": "Problems 31-33 go together. See below."
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we have 12 dogs and need to divide them into three groups: one with 4 dogs, one with 5 dogs, and one with 3 dogs. Determine how many ways the groups can be formed if Rocky, a notably aggressive dog, must be in the 4-dog group, and Bella must be in the 5-dog group. Let's... | math | {
"style": "rule",
"ground_truth": "4200"
} | {
"split": "train",
"index": 867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangular pyramid $A B C D$ with a base $A B C$, the lateral edges are pairwise perpendicular, $D A=D B=5$, and $D C=1$. From a point on the base, a light ray is emitted. After reflecting exactly once from each of the lateral faces (without reflecting from the edges), t... | math | {
"style": "rule",
"ground_truth": "\\frac{10\\sqrt{3}}{9}"
} | {
"split": "train",
"index": 868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers $\\,a,b,c\\,$ with $\\,1<a<b<c\\,$ such that \\[ (a-1)(b-1)(c-1) \\] is a divisor of $abc-1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 4, 8) \\text{ and } (3, 5, 15)"
} | {
"split": "train",
"index": 869,
"solution": "\nWe are tasked with finding all integers \\( a, b, c \\) with \\( 1 < a < b < c \\) such that \n\n\\[\n(a-1)(b-1)(c-1)\n\\]\n\nis a divisor of \n\n\\[\nabc - 1.\n\\]\n\nLet's first express \\( abc - 1 \\) in terms of potential divisors' expressions:\n\n1. We want \\... |
deepscaler | [
{
"role": "user",
"content": "Find all solutions to $(m^2+n)(m + n^2)= (m - n)^3$ , where m and n are non-zero integers.\nDo it Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{(-1,-1), (8,-10), (9,-6), (9,-21)\\}"
} | {
"split": "train",
"index": 870,
"solution": "Expanding both sides, \\[m^3+mn+m^2n^2+n^3=m^3-3m^2n+3mn^2-n^3\\] Note that $m^3$ can be canceled and as $n \\neq 0$ , $n$ can be factored out.\nWriting this as a quadratic equation in $n$ : \\[2n^2+(m^2-3m)n+(3m^2+m)=0\\] .\nThe discriminant $b^2-4ac$ equals \\[(m^2... |
deepscaler | [
{
"role": "user",
"content": "The fractional equation $\\dfrac{x-5}{x+2}=\\dfrac{m}{x+2}$ has a root, determine the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two 24-sided dice have the following configurations: 5 purple sides, 8 blue sides, 10 red sides, and 1 gold side. What is the probability that when both dice are rolled, they will show the same color? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\dfrac{95}{288}"
} | {
"split": "train",
"index": 872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In response to the national policy of expanding domestic demand, a manufacturer plans to hold a promotional event at the beginning of 2015. After investigation and estimation, the annual sales volume (i.e., the annual production volume $x$ in ten thousand units) and the annual ... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt {2}"
} | {
"split": "train",
"index": 873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence ${a_{n}}$, $S_{1} < 0$, $2S_{21}+S_{25}=0$, find the value of $n$ when $S_{n}$ is minimized. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. The sequence $(x_k)$ is defined by $x_0 = 0,$ $x_1 = 1,$ and\n\\[x_{k + 2} = \\frac{(n - 1) x_{k + 1} - (n - k) x_k}{k + 1}\\]for $k \\ge 0.$ Find $x_0 + x_1 + x_2 + \\dotsb$ as a function of $n.$ Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "2^{n - 1}"
} | {
"split": "train",
"index": 875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct numbers can you get by multiplying two or more distinct members of the set $\\{1, 2, 3, 7, 13\\}$, or taking any number to the power of another (excluding power 1) from the same set? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\\omega$ in the plane. Suppose that the tangent lines to $\\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\\omega$? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{85\\pi}{8}"
} | {
"split": "train",
"index": 877,
"solution": "1. **Identify the midpoint of segment $AB$**:\n Given points $A=(6,13)$ and $B=(12,11)$, the midpoint $D$ of $AB$ is calculated as:\n \\[\n D = \\left(\\frac{6+12}{2}, \\frac{13+11}{2}\\right) = (9, 12).\n \\]\n\n2. **Determine the slope of line $AB$ and the ... |
deepscaler | [
{
"role": "user",
"content": "The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first tw... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As $n$ ranges over the positive integers, what is the sum of all possible values of the greatest common divisor of $3n+4$ and $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A random simulation method is used to estimate the probability of a shooter hitting the target at least 3 times out of 4 shots. A calculator generates random integers between 0 and 9, where 0 and 1 represent missing the target, and 2 through 9 represent hitting the target. Grou... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"split": "train",
"index": 880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a pocket, there are eight cards of the same size, among which three are marked with the number $1$, three are marked with the number $2$, and two are marked with the number $3$. The first time, a card is randomly drawn from the pocket and then put back. After that, a second ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {15}{4}"
} | {
"split": "train",
"index": 881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expansion of $\\left(x-\\frac{a}{x}\\right)^{5}$, find the maximum value among the coefficients in the expansion. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the side \\( BC \\) of triangle \\( ABC \\), point \\( A_1 \\) is taken such that \\( BA_1 : A_1C = 2:1 \\). In what ratio does median \\( CC_1 \\) divide segment \\( AA_1 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"split": "train",
"index": 883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $97^2$ in your head. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9409"
} | {
"split": "train",
"index": 884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 6 married couples at a party. At the start of the party, every person shakes hands once with every other person except his or her spouse. How many handshakes are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a mathematical operator \"$\\odot$\" that satisfies the following equations: $2 \\odot 4=8, 4 \\odot 6=14, 5 \\odot 3=13, 8 \\odot 7=23$. According to this rule, what is $9 \\odot 3=?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Charlie and Dana play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where \n\\[g(n) = \\left\\{\n\\begin{array}{cl} \n7 & \\text{if } n \\text{ is a multiple of 3 and 5}, \\\\\n3 & \\text{if } n \\text{ is only a mult... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let all possible $2023$ -degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\\cdots+a_{2022}x+a_{2023}$ ,\nwhere $P(0)+P(1)=0$ , and the polynomial has 2023 real roots $r_1, r_2,\\cdots r_{2023}$ [not necessarily distinct] so that $0\\leq r_1,r_2,\\cdots r_{2... | math | {
"style": "rule",
"ground_truth": "2^{-2023}"
} | {
"split": "train",
"index": 888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The entire contents of a jug can exactly fill 9 small glasses and 4 large glasses of juice. The entire contents of the jug could instead fill 6 small glasses and 6 large glasses. If the entire contents of the jug is used to fill only large glasses, the maximum number of large g... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\( X \\) and \\( Y \\) are angles with \\( \\tan X = \\frac{1}{m} \\) and \\( \\tan Y = \\frac{a}{n} \\) for some positive integers \\( a, m, \\) and \\( n \\). Determine the number of positive integers \\( a \\leq 50 \\) for which there are exactly 6 pairs of pos... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The four points $A(-4,0), B(0,-4), X(0,8),$ and $Y(14,k)$ are grouped on the Cartesian plane. If segment $AB$ is parallel to segment $XY$ what is the value of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large number \\( y \\) is defined by \\( 2^33^54^45^76^57^38^69^{10} \\). Determine the smallest positive integer that, when multiplied with \\( y \\), results in a product that is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what smallest natural $n$ is each of the fractions\n\n$$\n\\frac{7}{n+9}, \\frac{8}{n+10}, \\ldots, \\frac{31}{n+33}\n$$\n\nin its simplest form? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the circumferences of the two bases of a cylinder lie on the surface of a sphere $O$ with a volume of $\\frac{{32π}}{3}$, the maximum value of the lateral surface area of the cylinder is ______. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"split": "train",
"index": 894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $A=\\{4, a^2\\}$, $B=\\{a-6, a+1, 9\\}$, if $A \\cap B = \\{9\\}$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When five students are lining up to take a photo, and two teachers join in, with the order of the five students being fixed, calculate the total number of ways for the two teachers to stand in line with the students for the photo. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The larger root minus the smaller root of the equation $(7+4\\sqrt{3})x^2+(2+\\sqrt{3})x-2=0$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6-3\\sqrt{3}"
} | {
"split": "train",
"index": 897,
"solution": "1. **Simplify the given quadratic equation**: The given quadratic equation is \\[(7+4\\sqrt{3})x^2+(2+\\sqrt{3})x-2=0.\\] We start by making the quadratic equation monic (i.e., the coefficient of $x^2$ is 1). This is done by dividing the entire equation by $7 + 4\\sq... |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\dfrac{30}{45} \\cdot \\dfrac{75}{128} \\cdot \\dfrac{256}{150}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a,b,c>0$, find the smallest possible value of\n\\[\\left\\lfloor{\\frac{a+b}{c}}\\right\\rfloor+\\left\\lfloor{\\frac{b+c}{a}}\\right\\rfloor+\\left\\lfloor{\\frac{c+a}{b}}\\right\\rfloor.\\](Note that $\\lfloor{x}\\rfloor$ denotes the greatest integer less than or equal to... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 899,
"solution": ""
} |
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