data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Five integers are written on a board. By summing them in pairs, the following set of 10 numbers was obtained: $-1, 5, 8, 9, 11, 12, 14, 18, 20, 24$. Determine which numbers are written on the board. Provide their product as the answer. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "-2002"
} | {
"split": "train",
"index": 29849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $y=f(x)+\\sin \\frac {π}{6}x$ is an even function, and $f(\\log _{ \\sqrt {2}}2)= \\sqrt {3}$, determine $f(\\log _{2} \\frac {1}{4})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {3}"
} | {
"split": "train",
"index": 29850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ with an angle of $\\frac {2\\pi}{3}$ between them, $|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=3$, let $\\overrightarrow {m}=3\\overrightarrow {a}-2\\overrightarrow {b}$ and $\\overrightarrow {n}=2\\ove... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 29851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A charity sells $140$ benefit tickets for a total of $2001$ dollars. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "$782"
} | {
"split": "train",
"index": 29852,
"solution": "\nLet $f$ be the number of full-price tickets and $h$ be the number of half-price tickets. Let $p$ be the price of a full-price ticket. We know that the total number of tickets sold is $140$, and the total revenue is $2001$. Therefore, we can set up the following e... |
deepscaler | [
{
"role": "user",
"content": "Given positive integers $a$ and $b$ are members of a set where $a \\in \\{2, 3, 5, 7\\}$ and $b \\in \\{2, 4, 6, 8\\}$, and the sum of $a$ and $b$ must be even, determine the smallest possible value for the expression $2 \\cdot a - a \\cdot b$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 29853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three types of snacks for the kitten. It eats a stick of cat food every 1 day, an egg yolk every 2 days, and nutritional cream every 3 days. The kitten ate cat stick and nutritional cream on March 23, and ate cat stick and egg yolk on March 25. Which day in March does... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 29854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\\frac{1}{2}$ point if the game was a tie. After the completion of the tournament,... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 29855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $P$ is any point on the circle $C$: $(x-2)^{2}+(y-2)^{2}=1$, and $Q$ is any point on the line $l$: $x+y=1$, find the minimum value of $| \\overrightarrow{OP}+ \\overrightarrow{OQ}|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{2}-2}{2}"
} | {
"split": "train",
"index": 29856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the average of a set of sample data 4, 5, 7, 9, $a$ is 6, then the variance $s^2$ of this set of data is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{5}"
} | {
"split": "train",
"index": 29857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + z = 0$ are $n^{\\text{th}}$ roots of unity? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 29858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, vertex $B$ is connected to the point that divides side $AC$ into thirds, closer to $A$. The intersection of this segment and the median from $A$ is denoted as $P$, and the centroid of the triangle as $S$. In what ratio does $P$ divide the segment $AS$? Let's ... | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"split": "train",
"index": 29859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is inscribed in a convex quadrilateral \\(ABCD\\) with its center at point \\(O\\), and \\(AO=OC\\). Additionally, \\(BC=5\\), \\(CD=12\\), and \\(\\angle DAB\\) is a right angle.\n\nFind the area of the quadrilateral \\(ABCD\\). Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 29860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two particles move along the sides of a right $\\triangle ABC$ with $\\angle B = 90^\\circ$ in the direction \\[A\\Rightarrow B\\Rightarrow C\\Rightarrow A,\\] starting simultaneously. One starts at $A$ moving at speed $v$, the other starts at $C$ moving at speed $2v$. The midp... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 29861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Between A and B, there are 6 parallel network cables, with their maximum information capacities being 1, 1, 2, 2, 3, and 4, respectively. Now, if we randomly select 3 of these network cables, in how many ways can we ensure that the sum of the maximum information capacities of t... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 29862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an opaque bag, there are 2 red balls and 5 black balls, all identical in size and material. Balls are drawn one by one without replacement until all red balls are drawn. Calculate the expected number of draws. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\dfrac{16}{3}"
} | {
"split": "train",
"index": 29863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the differential, calculate to an accuracy of 0.01 the increment of the function \\( y = x \\sqrt{x^{2} + 5} \\) at \\( x = 2 \\) and \\( \\Delta x = 0.2 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.87"
} | {
"split": "train",
"index": 29864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the polynomial $Q(x) = 3x^3 + dx^2 + ex + f$. The polynomial has the property that the reciprocal of the sum of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = Q(x)$ is 9. Find the value of $e$.... | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 29865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $326_{13} + 4C9_{14}$ as a base 10 integer, where $C = 12$ in base 14. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 29866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube has six faces. Each face has some dots on it. The numbers of dots on the six faces are 2, 3, 4, 5, 6, and 7. Harry removes one of the dots at random, with each dot equally likely to be removed. When the cube is rolled, each face is equally likely to be the top face. What... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{27}"
} | {
"split": "train",
"index": 29867,
"solution": "When a dot is removed from a face with an even number of dots, that face then has an odd number of dots. When a dot is removed from a face with an odd number of dots, that face then has an even number of dots. Initially, there are 3 faces with an even number of dot... |
deepscaler | [
{
"role": "user",
"content": "Find an integer $n$ such that the decimal representation of the number $5^{n}$ contains at least 1968 consecutive zeros. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1968"
} | {
"split": "train",
"index": 29868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 29869,
"solution": "1. **Identify the total sales**: The problem provides the sales for each month from January through April as $100$, $60$, $40$, and $120$ dollars respectively.\n\n2. **Calculate the total sales**: Add the sales for each month to find the total sales over the four m... |
deepscaler | [
{
"role": "user",
"content": "Given that one air conditioner sells for a 10% profit and the other for a 10% loss, and the two air conditioners have the same selling price, determine the percentage change in the shopping mall's overall revenue. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "1\\%"
} | {
"split": "train",
"index": 29870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x+2 y-3 z=7$ and $2 x-y+2 z=6$, determine $8 x+y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 29871,
"solution": "$8 x+y=2(x+2 y-3 z)+3(2 x-y+2 z)=2(7)+3(6)=32$"
} |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x) = x^2$ has a domain $D$ and its range is $\\{0, 1, 2, 3, 4, 5\\}$, how many such functions $f(x)$ exist? (Please answer with a number). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 29872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diameter, in inches, of a sphere with twice the volume of a sphere of radius 9 inches can be expressed in the form $a\\sqrt[3]{b}$ where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 29873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 2x - 3$ and $g(x) = x + 1$. What is the value of $f(1 + g(2))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 29874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z} \\ | \\ z \\in R\\right\\rbrace$.... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3} + 2 \\pi"
} | {
"split": "train",
"index": 29875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Radford and Peter ran a race, during which they both ran at a constant speed. Radford began the race 30 m ahead of Peter. After 3 minutes, Peter was 18 m ahead of Radford. Peter won the race exactly 7 minutes after it began. How far from the finish line was Radford when Peter w... | math | {
"style": "rule",
"ground_truth": "82 \\mathrm{~m}"
} | {
"split": "train",
"index": 29876,
"solution": "Over the first 3 minutes of the race, Peter ran 48 m farther than Radford. Here is why: We note that at a time of 0 minutes, Radford was at the 30 m mark. If Radford ran $d \\mathrm{~m}$ over these 3 minutes, then he will be at the $(d+30) \\mathrm{m}$ mark after 3... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a=5$, $b=8$, $C=60^{\\circ}$, the value of $\\overrightarrow{BC}\\cdot \\overrightarrow{CA}$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 29877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the three lines $3y-2x=1$, $x+2y=2$ and $4x-6y=5$ are drawn in the plane, how many points will lie at the intersection of at least two of the three lines? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 29878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On graph paper, two right triangles are drawn. Find the sum of the angles BCA and \\(\\mathrm{B}_{1} \\mathrm{C}_{1} \\mathrm{~A}_{1}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 29879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many triangles can be formed using the vertices of a regular pentadecagon (a 15-sided polygon), if no side of the triangle can be a side of the pentadecagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 29880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\\sqrt7$ .\n\nNotes: ''Using a ruler construct points $\\ldots$ '' means: N... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 29881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?\n\n*2019 CCA Math Bonanza Individual Round #5* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 29882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area enclosed by the graph of \\( |x| + |3y| = 9 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 29883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the sum of fractions $\\frac{3}{8} + \\frac{5}{32}$ as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.53125"
} | {
"split": "train",
"index": 29884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Acute-angled $\\triangle ABC$ is inscribed in a circle with center at $O$. The measures of arcs are $\\stackrel \\frown {AB} = 80^\\circ$ and $\\stackrel \\frown {BC} = 100^\\circ$. A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Find the ratio o... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $7 \\cdot 9\\frac{2}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65 \\frac{4}{5}"
} | {
"split": "train",
"index": 29886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a particular right triangle, the lengths of the two legs are 30 inches and 24 inches. Calculate both the area of the triangle and the length of its hypotenuse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{1476}"
} | {
"split": "train",
"index": 29887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 2x^4-17x^3+26x^2-24x-60$. Find $f(7)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 29888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $sin(x+\\frac{π}{12})=-\\frac{1}{4}$, find the value of $cos(\\frac{5π}{6}-2x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{8}"
} | {
"split": "train",
"index": 29889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the sum of the divisors of $200$. What is the largest prime factor of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 29890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $|\\vec{a}|=1$, $|\\vec{b}|=2$, and the angle between $\\vec{a}$ and $\\vec{b}$ is $60^{\\circ}$.\n$(1)$ Find $\\vec{a}\\cdot \\vec{b}$, $(\\vec{a}- \\vec{b})\\cdot(\\vec{a}+ \\vec{b})$;\n$(2)$ Find $|\\vec{a}- \\vec{b}|$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 29891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between $1000$ and $9999$ have four distinct digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4536"
} | {
"split": "train",
"index": 29892,
"solution": "To find the number of integers between $1000$ and $9999$ that have four distinct digits, we need to consider the constraints on each digit of a four-digit number.\n\n1. **First Digit**: The first digit of a four-digit number ranges from $1$ to $9$ (it cannot be $0$... |
deepscaler | [
{
"role": "user",
"content": "The spinner shown is divided into 6 sections of equal size. Determine the probability of landing on a section that contains the letter Q using this spinner. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{6}"
} | {
"split": "train",
"index": 29893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area, in square units, of the square with the four vertices at $A\\ (0, 0)$, $B\\ (-5, -1)$, $C\\ (-4, -6)$ and $D\\ (1, -5)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 29894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right square pyramid with volume $54$ has a base with side length $6.$ The five vertices of the pyramid all lie on a sphere with radius $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 29895,
"solution": "Although I can't draw the exact picture of this problem, but it is quite easy to imagine that four vertices of the base of this pyramid is on a circle (Radius $\\frac{6}{\\sqrt{2}} = 3\\sqrt{2}$). Since all five vertices are on the sphere, the distances of the sphe... |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) \\):\n\\[ f(x) = \\begin{cases} \n\\ln x & \\text{if } x > 1, \\\\ \n\\frac{1}{2} x + \\frac{1}{2} & \\text{if } x \\leq 1 \n\\end{cases} \\]\nIf \\( m < n \\) and \\( f(m) = f(n) \\), what is the minimum value of \\( n - m \\)? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "3 - 2\\ln 2"
} | {
"split": "train",
"index": 29896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x, y$, and $N$ be real numbers, with $y$ nonzero, such that the sets $\\left\\{(x+y)^{2},(x-y)^{2}, x y, x / y\\right\\}$ and $\\{4,12.8,28.8, N\\}$ are equal. Compute the sum of the possible values of $N$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "85.2"
} | {
"split": "train",
"index": 29897,
"solution": "First, suppose that $x$ and $y$ were of different signs. Then $x y<0$ and $x / y<0$, but the set has at most one negative value, a contradiction. Hence, $x$ and $y$ have the same sign; without loss of generality, we say $x$ and $y$ are both positive. Let $(s, d):=(... |
deepscaler | [
{
"role": "user",
"content": "A regular polygon has exterior angles each measuring 15 degrees. How many sides does the polygon have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 29898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Farmer Bill's 1000 animals - ducks, cows, and rabbits - are standing in a circle. In order to feel safe, every duck must either be standing next to at least one cow or between two rabbits. If there are 600 ducks, what is the least number of cows there can be for this to be poss... | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 29899,
"solution": "Suppose Bill has $r$ rabbits and $c$ cows. At most $r-1$ ducks can be between two rabbits: each rabbit can serve up to two such ducks, so at most $2 r / 2=r$ ducks will each be served by two rabbits, but we cannot have equality, since this would require alternating... |
deepscaler | [
{
"role": "user",
"content": "Each day, John ate 30% of the chocolates that were in his box at the beginning of that day. At the end of the third day, 28 chocolates remained. How many chocolates were in the box originally? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "82"
} | {
"split": "train",
"index": 29900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcirc... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 29901,
"solution": "\nGiven an acute triangle \\( ABC \\), let \\( M \\) be the midpoint of \\( AC \\). A circle \\( \\omega \\) that passes through points \\( B \\) and \\( M \\) intersects side \\( AB \\) at point \\( P \\) and side \\( BC \\) at point \\( Q \\). Point \\( T \\) is ... |
deepscaler | [
{
"role": "user",
"content": "Approximate the number $0.00356$ to the nearest ten-thousandth: $0.00356 \\approx$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.0036"
} | {
"split": "train",
"index": 29902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 29903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the following sum as a simple fraction in lowest terms.\n\n$$\\frac{1}{1\\times2} + \\frac{1}{2\\times3} + \\frac{1}{3\\times4} + \\frac{1}{4\\times5} + \\frac{1}{5\\times6}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 29904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xoy$, given point $A(0,-2)$, point $B(1,-1)$, and $P$ is a moving point on the circle $x^{2}+y^{2}=2$, then the maximum value of $\\dfrac{|\\overrightarrow{PB}|}{|\\overrightarrow{PA}|}$ is ______. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 29905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor the following expression: $37a^2 +111a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37a(a+3)"
} | {
"split": "train",
"index": 29906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the height of a cylinder is $2$, and the circumferences of its two bases lie on the surface of the same sphere with a diameter of $2\\sqrt{6}$, calculate the surface area of the cylinder. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(10+4\\sqrt{5})\\pi"
} | {
"split": "train",
"index": 29907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the numbe... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 29908,
"solution": "1. **Understanding the Problem**: We need to arrange the numbers $1$ through $9$ in a $3 \\times 3$ grid such that consecutive numbers are adjacent (share an edge), and the sum of the numbers in the four corners is $18$.\n\n2. **Finding Corner Numbers**: Since cons... |
deepscaler | [
{
"role": "user",
"content": "Use the bisection method to find an approximate zero of the function $f(x) = \\log x + x - 3$, given that approximate solutions (accurate to 0.1) are $\\log 2.5 \\approx 0.398$, $\\log 2.75 \\approx 0.439$, and $\\log 2.5625 \\approx 0.409$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "2.6"
} | {
"split": "train",
"index": 29909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers \\( n \\) for which \\( n^2 + 20n + 11 \\) is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 29910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two people, A and B, play a guessing game. First, A thinks of a number denoted as $a$, then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\\{0,1,2,…,9\\}$. If $|a-b|=1$, then A and B are said to have a \"telepathic connection\... | math | {
"style": "rule",
"ground_truth": "\\dfrac {9}{50}"
} | {
"split": "train",
"index": 29911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \\ldots, 20$ on its sides). He conceals the results but tells you that at least half of the rolls are 20. Assuming that Steven is truthful, what is the probability that all three remaining... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{58}"
} | {
"split": "train",
"index": 29912,
"solution": "The given information is equivalent to the first two dice being 20 and 19 and there being at least two 20's among the last three dice. Thus, we need to find the probability that given at least two of the last three dice are 20's, all three are. Since there is only ... |
deepscaler | [
{
"role": "user",
"content": "Let $a_1, a_2, \\dots, a_k$ be a finite arithmetic sequence with $a_4 + a_7 + a_{10} = 17$ and $a_4 + a_5 + \\dots + a_{13} + a_{14} = 77$. If $a_k = 13$, then $k = $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 29913,
"solution": "1. **Identify the common difference and terms in terms of $a_7$:**\n Given the arithmetic sequence $a_1, a_2, \\ldots, a_k$, we know that $a_n = a_1 + (n-1)d$ where $d$ is the common difference. For the terms $a_4, a_7, a_{10}$, we can express them in terms of $a... |
deepscaler | [
{
"role": "user",
"content": "What are the last two digits in the sum of the factorials of the first 15 positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 29914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, every number beginning at 30 equals twice the sum of the two numbers to its immediate left. The value of \\( c \\) is:\n\n| 10 | \\(a\\) | 30 | \\(b\\) | \\(c\\) |\n| :--- | :--- | :--- | :--- | :--- | Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 29915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the parabola passing through points $A(2-3b, m)$ and $B(4b+c-1, m)$ is $y=-\\frac{1}{2}x^{2}+bx-b^{2}+2c$, if the parabola intersects the $x$-axis, calculate the length of segment $AB$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 29916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $x^{44} + x^{33} + x^{22} + x^{11} + 1$ is divided by $x^4 + x^3 + x^2 + x + 1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 29917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of an ellipse is $\\dfrac {x^{2}}{a^{2}} + \\dfrac {y^{2}}{b^{2}} = 1 \\; (a > b > 0)$, one of its vertices is $M(0,1)$, and its eccentricity $e = \\dfrac {\\sqrt {6}}{3}$. \n$(1)$ Find the equation of the ellipse; \n$(2)$ Suppose a line $l$ intersects the ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {\\sqrt {3}}{2}"
} | {
"split": "train",
"index": 29918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle has a radius of 6. What is the area of the smallest square that can entirely contain this circle, and what is the circumference of the circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"split": "train",
"index": 29919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered quadruples $(a, b, c, d)$ of four distinct numbers chosen from the set $\\{1,2,3, \\ldots, 9\\}$ satisfy $b<a, b<c$, and $d<c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "630"
} | {
"split": "train",
"index": 29920,
"solution": "Given any 4 elements $p<q<r<s$ of $\\{1,2, \\ldots, 9\\}$, there are 5 ways of rearranging them to satisfy the inequality: prqs, psqr, $q s p r, q r p s$, and $r s p q$. This gives a total of $\\binom{9}{4} \\cdot 5=630$ quadruples."
} |
deepscaler | [
{
"role": "user",
"content": "How many positive factors does 30 have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 29921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six consecutive numbers were written on a board. When one of them was crossed out and the remaining were summed, the result was 10085. What number could have been crossed out? Specify all possible options. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"split": "train",
"index": 29922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in quadrilateral $ABCD$, $m\\angle B = m \\angle C = 120^{\\circ}, AB=3, BC=4,$ and $CD=5$, calculate the area of $ABCD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"split": "train",
"index": 29923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Abby, Bernardo, Carl, and Debra play a revised game where each starts with five coins and there are five rounds. In each round, five balls are placed in an urn—two green, two red, and one blue. Each player draws a ball at random without replacement. If a player draws a green ba... | math | {
"style": "rule",
"ground_truth": "\\frac{64}{15625}"
} | {
"split": "train",
"index": 29924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the ellipse $\\frac{x^2}{16} + \\frac{y^2}{12} = 1$ whose left and right intersection points are $F_1$ and $F_2$, respectively. Let point $P$ be on the ellipse and satisfy $\\vec{PF_1} \\cdot \\vec{PF_2} = 9$. Find the value of $|\\vec{PF_1}| \\cdot |\\vec{PF_2}|$. Let... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 29925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the parallelogram generated by the vectors $\\mathbf{a}$ and $\\mathbf{b}$ is 8. Find the area of the parallelogram generated by the vectors $2 \\mathbf{a} + 3 \\mathbf{b}$ and $\\mathbf{a} - 5 \\mathbf{b}.$ Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 29926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If five people are selected at random from a group of ten men and five women, what is the probability that at least one woman is selected? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{917}{1001}"
} | {
"split": "train",
"index": 29927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A big $L$ is formed as shown. What is its area? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 29928,
"solution": "1. **Identify the dimensions of the big rectangle**: The problem states that the shape is a big \"L\" formed within a rectangle. Assume the dimensions of the big rectangle are $8$ units by $5$ units based on the typical dimensions used in such problems.\n\n2. **Cal... |
deepscaler | [
{
"role": "user",
"content": "The volume of a sphere is increased to $72\\pi$ cubic inches. What is the new surface area of the sphere? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi \\cdot 2^{2/3}"
} | {
"split": "train",
"index": 29929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{P}$ be a parallelepiped with side lengths $x$ , $y$ , and $z$ . Suppose that the four space diagonals of $\\mathcal{P}$ have lengths $15$ , $17$ , $21$ , and $23$ . Compute $x^2+y^2+z^2$ . Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "371"
} | {
"split": "train",
"index": 29930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P_1$ be a regular $r~\\mbox{gon}$ and $P_2$ be a regular $s~\\mbox{gon}$ $(r\\geq s\\geq 3)$ such that each interior angle of $P_1$ is $\\frac{59}{58}$ as large as each interior angle of $P_2$. What's the largest possible value of $s$?\n Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 29931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ that satisfies the equation: ${a_{n+1}}+{({-1})^n}{a_n}=3n-1$ ($n∈{N^*}$), calculate the sum of the first $60$ terms of the sequence $\\{a_{n}\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2760"
} | {
"split": "train",
"index": 29932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Carlos divides his marbles into groups of six, seven, or eight, he has one marble left over. Assuming Carlos has more than one marble, what is the smallest possible number of marbles in Carlos's collection? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 29933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parabola \\( y^2 = 6x \\) with two variable points \\( A(x_1, y_1) \\) and \\( B(x_2, y_2) \\), where \\( x_1 \\neq x_2 \\) and \\( x_1 + x_2 = 4 \\). The perpendicular bisector of segment \\( AB \\) intersects the x-axis at point \\( C \\). Find the maximum area of tri... | math | {
"style": "rule",
"ground_truth": "\\frac{14}{3}\\sqrt{7}"
} | {
"split": "train",
"index": 29934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\\sqrt 2$, and $\\angle BAC = 45^\\circ$. Point $D$ is $20$ meters due north of point $C$. The distance $AD$ is between which two integers? Let's think step by... | math | {
"style": "rule",
"ground_truth": "31 \\text{ and } 32"
} | {
"split": "train",
"index": 29935,
"solution": "1. **Identify the geometric relationships**: \n - Point $B$ is due east of point $A$.\n - Point $C$ is due north of point $B$.\n - Therefore, $\\angle CBA = 90^\\circ$ (since east to north is a right angle).\n\n2. **Analyze $\\triangle ABC$**:\n - Given $\\... |
deepscaler | [
{
"role": "user",
"content": "In a small town, the police are looking for a wanderer. There is a four in five chance that he is in one of the eight bars in the town, with no preference for any particular one. Two officers visited seven bars but did not find the wanderer. What are the chances of finding him ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 29936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\\otimes$ is continuous, commutative $(a \\otimes b=b \\otimes a)$, distributive across multiplication $(a \\otimes(b c)=(a \\otimes b)(a \\otimes... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 29937,
"solution": "We note that $\\left(a \\otimes b^{k}\\right)=(a \\otimes b)^{k}$ for all positive integers $k$. Then for all rational numbers $\\frac{p}{q}$ we have $a \\otimes b^{\\frac{p}{q}}=\\left(a \\otimes b^{\\frac{1}{q}}\\right)^{p}=(a \\otimes b)^{\\frac{p}{q}}$. So by c... |
deepscaler | [
{
"role": "user",
"content": "The numbers assigned to 100 athletes range from 1 to 100. If each athlete writes down the largest odd factor of their number on a blackboard, what is the sum of all the numbers written by the athletes? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "3344"
} | {
"split": "train",
"index": 29938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\mathbb{N} denote the natural numbers. Compute the number of functions $f: \\mathbb{N} \\rightarrow\\{0,1, \\ldots, 16\\}$ such that $$f(x+17)=f(x) \\quad \\text { and } \\quad f\\left(x^{2}\\right) \\equiv f(x)^{2}+15 \\quad(\\bmod 17)$$ for all integers $x \\geq 1$ Let's... | math | {
"style": "rule",
"ground_truth": "12066"
} | {
"split": "train",
"index": 29939,
"solution": "By plugging in $x=0$, we get that $f(0)$ can be either $-1,2$. As $f(0)$ is unrelated to all other values, we need to remember to multiply our answer by 2 at the end. Similarly, $f(1)=-1$ or 2 . Consider the graph $x \\rightarrow x^{2}$. It is a binary tree rooted ... |
deepscaler | [
{
"role": "user",
"content": "Let $n\\ge 3$ be a fixed integer. There are $m\\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\\emph{not}$ ... | math | {
"style": "rule",
"ground_truth": "n^2-n-1"
} | {
"split": "train",
"index": 29940,
"solution": "\nLet \\( n \\geq 3 \\) be a fixed integer. We need to find the largest number \\( m \\) for which it is not possible to paint \\( m \\) beads on a circular necklace using \\( n \\) colors such that each color appears at least once among any \\( n+1 \\) consecutive... |
deepscaler | [
{
"role": "user",
"content": "Jeff rotates spinners $P$, $Q$ and $R$ and adds the resulting numbers. What is the probability that his sum is an odd number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 29941,
"solution": "1. **Identify the possible outcomes for each spinner:**\n - Spinner $P$ has numbers 1, 2, 3. Thus, it has 1 even number (2) and 2 odd numbers (1, 3).\n - Spinner $Q$ has numbers 2, 4, 6. All numbers are even.\n - Spinner $R$ has numbers 1, 3, 5. All numbers a... |
deepscaler | [
{
"role": "user",
"content": "In a privately-owned company in Wenzhou manufacturing a product, it is known from past data that the fixed daily cost of producing the product is 14,000 RMB. The variable cost increases by 210 RMB for each additional unit produced. The relationship between the daily sales volum... | math | {
"style": "rule",
"ground_truth": "30000"
} | {
"split": "train",
"index": 29942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a math competition, a team of $8$ students has $2$ hours to solve $30$ problems. If each problem needs to be solved by $2$ students, on average how many minutes can a student spend on a problem? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 29943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that points $P$ and $Q$ are moving points on the curve $y=xe^{-2x}$ and the line $y=x+2$ respectively, find the minimum distance between points $P$ and $Q$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 29944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The planning committee at school has 10 members. Exactly four of these members are teachers. A four-person subcommittee with at least one member who is a teacher must be formed from the members of the planning committee. How many distinct subcommittees are possible? Let's think... | math | {
"style": "rule",
"ground_truth": "195"
} | {
"split": "train",
"index": 29945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\[\n\\sum_{k=1}^{70} \\frac{k}{x-k} \\geq \\frac{5}{4}\n\\]\nis a union of disjoint intervals the sum of whose lengths is 1988. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1988"
} | {
"split": "train",
"index": 29946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many orderings $(a_{1}, \\ldots, a_{8})$ of $(1,2, \\ldots, 8)$ exist such that $a_{1}-a_{2}+a_{3}-a_{4}+a_{5}-a_{6}+a_{7}-a_{8}=0$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4608"
} | {
"split": "train",
"index": 29947,
"solution": "We can divide the numbers up based on whether they have a + or - before them. Both the numbers following +'s and -'s must add up to 18. Without loss of generality, we can assume the + 's contain the number 1 (and add a factor of 2 at the end to account for this). T... |
deepscaler | [
{
"role": "user",
"content": "Two arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the 51st term of sequence $... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 29948,
"solution": ""
} |
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