data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A merchant buys $n$ radios for $d$ dollars, where $d$ is a positive integer. The merchant sells two radios at half the cost price to a charity sale, and the remaining radios at a profit of 8 dollars each. If the total profit is 72 dollars, what is the smallest possible value of... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "a) Vanya flips a coin 3 times, and Tanya flips a coin 2 times. What is the probability that Vanya gets more heads than Tanya?\n\nb) Vanya flips a coin $n+1$ times, and Tanya flips a coin $n$ times. What is the probability that Vanya gets more heads than Tanya? Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 27351,
"solution": "1. **Identify the chips in each bag:**\n - Bag A contains odd numbers: 1, 3, 5.\n - Bag B contains even numbers: 2, 4, 6.\n\n2. **Understand the nature of sums:**\n - The sum of an odd number (from Bag A) and an even number (from Bag B) is always odd.\n\n3. *... |
deepscaler | [
{
"role": "user",
"content": "In the figure, circle $O$ has radius 6 units. Chord $CD$ has length 8 units and is parallel to segment $KB$. If $KA$ = 12 units and points $K$, $A$, $O$ and $B$ are collinear, what is the area of triangle $KDC$? Express your answer in simplest radical form. [asy]\ndraw(Circle((... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{5}"
} | {
"split": "train",
"index": 27352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square array of dots with 10 rows and 10 columns is given. Each dot is coloured either blue or red. Whenever two dots of the same colour are adjacent in the same row or column, they are joined by a line segment of the same colour as the dots. If they are adjacent but of diffe... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 27353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The deli has four kinds of bread, six kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, chicken, cheddar cheese, and white bread are each offered at the deli. If Al never orders a sandwich with a ha... | math | {
"style": "rule",
"ground_truth": "111"
} | {
"split": "train",
"index": 27354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nikola had one three-digit number and one two-digit number. Each of these numbers was positive and made up of different digits. The difference between Nikola's numbers was 976.\n\nWhat was their sum? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "996"
} | {
"split": "train",
"index": 27355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence \\((b_n)\\) satisfies\n\\[kb_1 + kb_2 + kb_3 + \\dots + kb_n = n^2 kb_n\\] for all \\(n \\ge 2.\\) If \\(b_{70} = 2\\) and \\(k=3\\), find \\(b_1.\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4970"
} | {
"split": "train",
"index": 27356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $10^{93}-93$ is expressed as a single whole number, the sum of the digits is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "826"
} | {
"split": "train",
"index": 27357,
"solution": "1. **Observation of Pattern**: We start by observing the pattern in the expression $10^n - 93$ for small values of $n$:\n \\[\n \\begin{align*}\n 10^2 - 93 &= 100 - 93 = 7, \\\\\n 10^3 - 93 &= 1000 - 93 = 907, \\\\\n 10^4 - 93 &= 10000 - 93 = 9907.\n \\... |
deepscaler | [
{
"role": "user",
"content": "Farmer Tim starts walking from the origin following the path \\((t, \\sin t)\\) where \\(t\\) is the time in minutes. Five minutes later, Alex enters the forest and follows the path \\((m, \\cos t)\\) where \\(m\\) is the time since Alex started walking. What is the greatest di... | math | {
"style": "rule",
"ground_truth": "\\sqrt{29}"
} | {
"split": "train",
"index": 27358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[\\frac{\\sqrt{(x^2 + y^2)(3x^2 + y^2)}}{xy}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{3}"
} | {
"split": "train",
"index": 27359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns hom... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 27360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two lines $l_1: y = m$ and $l_2: y = \\frac{8}{2m+1}$ ($m > 0$), line $l_1$ intersects the graph of the function $y = |\\log_2 x|$ from left to right at points $A$ and $B$, and line $l_2$ intersects the graph of the function $y = |\\log_2 x|$ from left to right at points ... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}"
} | {
"split": "train",
"index": 27361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $1.\\overline{27}$ 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": "\\dfrac{14}{11}"
} | {
"split": "train",
"index": 27362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1)$ State the operation laws or rules used in each step of the following calculation:<br/>$(-0.4)\\times \\left(-0.8\\right)\\times \\left(-1.25\\right)\\times 2.5$<br/>$=-\\left(0.4\\times 0.8\\times 1.25\\times 2.5\\right) \\text{(Step 1)}$<br/>$=-\\left(0.4\\times 2.5\\tim... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 27363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle $C: (x-1)^2+(y-2)^2=25$, and a line $l: (2m+1)x+(m+1)y-7m-4=0$. If the chord intercepted by line $l$ on circle $C$ is the shortest, then the value of $m$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"split": "train",
"index": 27364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 27365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\\overline{AB}$ and $\\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two sol... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 27366,
"solution": "Define a coordinate system with $D = (0,0,0)$, $M = (1, \\frac{1}{2}, 0)$, $N = (0,1,\\frac{1}{2})$. The plane formed by $D$, $M$, and $N$ is $z = \\frac{y}{2} - \\frac{x}{4}$. It intersects the base of the unit cube at $y = \\frac{x}{2}$. The z-coordinate of the p... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 27367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of functions $f: \\{1, 2, 3\\} \\rightarrow \\{1, 2, 3\\}$ satisfying the property $f(f(x)) = f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for\n$\\begin{array}{ccc} X & X & X \\ & Y & X \\ + & & X \\ \\hline \\end{array}$\nhas the form Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "YYZ"
} | {
"split": "train",
"index": 27369,
"solution": "1. **Identify the structure of the sum**: The problem involves summing three numbers where the digits are represented by $X$ and $Y$. The numbers are $XXX$, $YX$, and $X$. We can rewrite these numbers in terms of their decimal values:\n - $XXX = 100X + 10X + X = ... |
deepscaler | [
{
"role": "user",
"content": "A triangular pyramid \\( S-ABC \\) has a base that is an equilateral triangle with a side length of 4. It is given that \\( AS = BS = \\sqrt{19} \\) and \\( CS = 3 \\). Find the surface area of the circumscribed sphere of the triangular pyramid \\( S-ABC \\). Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac{268\\pi}{11}"
} | {
"split": "train",
"index": 27370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the parabola consisting of the points $(x, y)$ in the real plane satisfying $(y+x)=(y-x)^{2}+3(y-x)+3$. Find the minimum possible value of $y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 27371,
"solution": "Let $w=y-x$. Adding $w$ to both sides and dividing by two gives $$y=\\frac{w^{2}+4w+3}{2}=\\frac{(w+2)^{2}-1}{2}$$ which is minimized when $w=-2$. This yields $y=-\\frac{1}{2}$."
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $a=(1,1)$ and $b=(2,t)$, find the value of $t$ such that $|a-b|=a·b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-5 - \\sqrt{13}}{2}"
} | {
"split": "train",
"index": 27372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sue owns 11 pairs of shoes: six identical black pairs, three identical brown pairs and two identical gray pairs. If she picks two shoes at random, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? Express your answer... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{33}"
} | {
"split": "train",
"index": 27373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I have four boxes arranged in a $2$ x $2$ grid, in how many distinct ways can I place the digits $1$, $2$, and $3$ in the boxes such that each box contains at most one digit? (I only have one of each digit, so one box will remain blank.) Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 27374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n$ that satisfy the following inequalities: $$ -46 \\leq \\frac{2023}{46-n} \\leq 46-n $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 27375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 50 students are selected from 2013 students using a two-step process, first eliminating 13 students through simple random sampling and then selecting 50 from the remaining 2000 using systematic sampling, determine the probability of each person being selected. Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{50}{2013}"
} | {
"split": "train",
"index": 27376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/7"
} | {
"split": "train",
"index": 27377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the measure of angle 4 if $m\\angle 1 = 76^{\\circ}, m\\angle 2 = 27^{\\circ}$ and $m\\angle 3 = 17^{\\circ}$?\n\n[asy]\ndraw((0,0)--(4,8)--(10,0)--cycle,linewidth(1));\ndraw((0,0)--(5,3)--(10,0),linewidth(1));\nlabel(\"2\",(1,1.2));\nlabel(\"1\",(4,7.75),S);\nlabel(\"4... | math | {
"style": "rule",
"ground_truth": "120^{\\circ}"
} | {
"split": "train",
"index": 27378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $\\dfrac{45}{28} \\cdot \\dfrac{49}{75} \\cdot \\dfrac{100}{63}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 27379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A factory must filter its emissions before discharging them. The relationship between the concentration of pollutants $p$ (in milligrams per liter) and the filtration time $t$ (in hours) during the filtration process is given by the equation $p(t) = p_0e^{-kt}$. Here, $e$ is th... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 27380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The seventh and tenth terms of a geometric sequence are $7$ and $21$, respectively. What is the $13$th term of this progression? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 27381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic $ax^2 + bx + c$ can be expressed in the form $2(x - 4)^2 + 8$. When the quadratic $3ax^2 + 3bx + 3c$ is expressed in the form $n(x - h)^2 + k$, what is $h$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 27382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $102012_{(3)}$ to base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "320"
} | {
"split": "train",
"index": 27383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.230"
} | {
"split": "train",
"index": 27384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$30$ same balls are put into four boxes $A$, $B$, $C$, $D$ in such a way that sum of number of balls in $A$ and $B$ is greater than sum of in $C$ and $D$. How many possible ways are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2600"
} | {
"split": "train",
"index": 27385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected to serve. Calculate the number of ways to select exactly one person to serve for both days. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 27386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The difference between two numbers is 9, and the sum of the squares of each number is 153. What is the value of the product of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 27387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $8!=40320$, what is the value of $8!\\div3!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6720"
} | {
"split": "train",
"index": 27388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five volunteers and two elderly people are taking a photo, and they need to be arranged in a row. The two elderly people must stand next to each other but cannot be at either end of the row. How many different arrangements are possible? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 27389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Working 22 hours in the second week of June, Xenia was able to earn $\\$$47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answer to the nearest... | math | {
"style": "rule",
"ground_truth": "\\$ 251.60"
} | {
"split": "train",
"index": 27390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function f(x) (x ∈ R) satisfies f(x + π) = f(x) + sin(x), and f(x) = 0 when 0 ≤ x ≤ π. Find f(23π/6). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many distinct ways can I arrange my five keys on a keychain, if I want to put my house key next to my car key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order without taking the keys off the chain--t... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 27392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are six students with unique integer scores in a mathematics exam. The average score is 92.5, the highest score is 99, and the lowest score is 76. What is the minimum score of the student who ranks 3rd from the highest? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 27393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the squares of the solutions to\n\\[\\left| x^2 - x + \\frac{1}{2008} \\right| = \\frac{1}{2008}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1003}{502}"
} | {
"split": "train",
"index": 27394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the positive difference between the two solutions of $|x + 5| = 20$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 27395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the three-digit numbers composed of the digits $0$ to $9$, the number of numbers where the digits are arranged in strictly increasing or strictly decreasing order, calculate the total. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "204"
} | {
"split": "train",
"index": 27396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a function defined on $\\mathbb{R}$ with a period of $2$, and for any real number $x$, it always holds that $f(x)-f(-x)=0$. When $x \\in [0,1]$, $f(x)=-\\sqrt{1-x^{2}}$. Determine the number of zeros of the function $g(x)=f(x)-e^{x}+1$ in the interval $[-2018,2018... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 27397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of \\( a \\) for which the equation \\( x^{2} + 2ax = 8a \\) has two distinct integer roots. In your answer, record the product of all such values of \\( a \\), rounding to two decimal places if necessary. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "506.25"
} | {
"split": "train",
"index": 27398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ be a real number between 0 and 1. Jocelin has a coin that lands heads with probability $p$ and tails with probability $1-p$; she also has a number written on a blackboard. Each minute, she flips the coin, and if it lands heads, she replaces the number $x$ on the blackbo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 27399,
"solution": "If the blackboard has the value $x$ written on it, then the expected value of the value after one flip is $$f(x)=p(3 x-1)+(1-p) x / 2$$ Because this expression is linear, we can say the same even if we only know the blackboard's initial expected value is $x$. There... |
deepscaler | [
{
"role": "user",
"content": "Set $A$ has $20$ elements, and set $B$ has $15$ elements. What is the smallest possible number of elements in $A \\cup B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 27400,
"solution": "1. **Understanding the Problem**: We are given two sets, $A$ and $B$, with $|A| = 20$ and $|B| = 15$. We need to find the smallest possible number of elements in the union of $A$ and $B$, denoted as $A \\cup B$.\n\n2. **Using the Principle of Inclusion-Exclusion**:... |
deepscaler | [
{
"role": "user",
"content": "Given a geometric progression of five terms, each a positive integer less than $100$. The sum of the five terms is $211$. If $S$ is the sum of those terms in the progression which are squares of integers, then $S$ is: Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "133"
} | {
"split": "train",
"index": 27401,
"solution": "1. **Identify the terms of the geometric progression**: Let the first term be $a$ and the common ratio be $r$. The terms of the progression are $a, ar, ar^2, ar^3, ar^4$. The sum of these terms is given by:\n \\[\n a + ar + ar^2 + ar^3 + ar^4 = a(1 + r + r^2 + ... |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a\\_n\\}(n=1,2,3,...,2016)$, circle $C\\_1$: $x^{2}+y^{2}-4x-4y=0$, circle $C\\_2$: $x^{2}+y^{2}-2a_{n}x-2a_{2017-n}y=0$. If circle $C\\_2$ bisects the circumference of circle $C\\_1$, then the sum of all terms in the sequence $\\{a\\_n\\}$ is $\\_\\_\\_\... | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"split": "train",
"index": 27402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence of integers $\\{a_i\\}_{i = 0}^{\\infty}$ satisfies $a_0 = 3$ , $a_1 = 4$ , and\n\\[a_{n+2} = a_{n+1} a_n + \\left\\lceil \\sqrt{a_{n+1}^2 - 1} \\sqrt{a_n^2 - 1}\\right\\rceil\\]\nfor $n \\ge 0$ . Evaluate the sum\n\\[\\sum_{n = 0}^{\\infty} \\left(\\frac{a_{n... | math | {
"style": "rule",
"ground_truth": "0.518"
} | {
"split": "train",
"index": 27403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number greater than 1 is called \"good\" if it is equal to the product of its distinct proper divisors (excluding 1 and the number itself). Find the sum of the first ten \"good\" natural numbers. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 27404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square \\(ABCD\\) is inscribed in circle \\(\\omega\\) with radius 10. Four additional squares are drawn inside \\(\\omega\\) but outside \\(ABCD\\) such that the lengths of their diagonals are as large as possible. A sixth square is drawn by connecting the centers of the four ... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 27405,
"solution": "Let \\(DEGF\\) denote the small square that shares a side with \\(AB\\), where \\(D\\) and \\(E\\) lie on \\(AB\\). Let \\(O\\) denote the center of \\(\\omega, K\\) denote the midpoint of \\(FG\\), and \\(H\\) denote the center of \\(DEGF\\). The area of the sixth... |
deepscaler | [
{
"role": "user",
"content": "From the integers 1 to 2020, there are a total of 1616 integers that are not multiples of 5. These 1616 numbers need to be divided into groups (each group may have a different number of elements), such that the difference (larger number minus smaller number) between any two num... | math | {
"style": "rule",
"ground_truth": "404"
} | {
"split": "train",
"index": 27406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest value of \\(x\\) that satisfies the equation \\(\\sqrt{3x} = 5x^2\\). Express your answer in simplest fractional form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{3}{25}\\right)^{\\frac{1}{3}}"
} | {
"split": "train",
"index": 27407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle PQR with PQ = 60 and PR = 20, the area is 240. Let M be the midpoint of PQ and N be the midpoint of PR. An altitude from P to side QR intersects MN and QR at X and Y, respectively. Find the area of quadrilateral XYMR. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 27408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of \\(10 \\cdot \\operatorname{ctg}(\\operatorname{arcctg} 3 + \\operatorname{arcctg} 7 + \\operatorname{arcctg} 13 + \\operatorname{arcctg} 21)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the data whose frequency histogram is shown, by how many days is the mean number of days missed per student greater than the median number of days missed per student for the 15 students? Express your answer as a common fraction.\n\n[asy]\ndraw((0,0)--(0,6),linewidth(2));\nd... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{15}\\text{ days}"
} | {
"split": "train",
"index": 27410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the length of the segment tangent from the origin to the circle that passes through the points $(3,4),$ $(6,8),$ and $(5,13).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{2}"
} | {
"split": "train",
"index": 27411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16:3"
} | {
"split": "train",
"index": 27412,
"solution": "1. **Set up the given ratios**:\n - The ratio of $w$ to $x$ is given as $\\frac{w}{x} = \\frac{4}{3}$.\n - The ratio of $y$ to $z$ is given as $\\frac{y}{z} = \\frac{3}{2}$.\n - The ratio of $z$ to $x$ is given as $\\frac{z}{x} = \\frac{1}{6}$.\n\n2. **Expres... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, if the terminal side of angle α passes through point P(sin (5π/3), cos (5π/3)), evaluate sin(π+α). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 27413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function \\( f \\), defined on the set of integers, satisfies the following conditions:\n\n1) \\( f(1) + 1 > 0 \\)\n2) \\( f(x + y) - x f(y) - y f(x) = f(x)f(y) - x - y + xy \\) for any \\( x, y \\in \\mathbb{Z} \\)\n3) \\( 2f(x) = f(x + 1) - x + 1 \\) for any \\( x \\in \\ma... | math | {
"style": "rule",
"ground_truth": "1014"
} | {
"split": "train",
"index": 27414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\\plus{}5^q.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 3), (2, 5), (3, 2), (5, 2), (5, 5), (5, 313), (313, 5)"
} | {
"split": "train",
"index": 27415,
"solution": "\nTo find all pairs of prime numbers \\((p, q)\\) such that \\(pq \\mid 5^p + 5^q\\), we consider several cases:\n\n1. **Case 1: Either \\(p\\) or \\(q\\) is 2.**\n - Assume \\(p = 2\\). Then \\(2q \\mid 5^2 + 5^q\\).\n - This simplifies to \\(2q \\mid 25 + 5^q... |
deepscaler | [
{
"role": "user",
"content": "When $\\{a,0,-1\\} = \\{4,b,0\\}$, find the values of $a$ and $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 27416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line segment is divided into four parts by three randomly selected points. What is the probability that these four parts can form the four sides of a quadrilateral? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 27417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the coefficients of the expansion of $(\\frac{3}{x}-\\sqrt{x})^n$ is $512$. Find:<br/>\n$(1)$ The coefficient of the term containing $x^{3}$ in the expansion;<br/>\n$(2)$ The constant term in the expansion of $(1+\\frac{1}{x})(2x-1)^n$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 27418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $7^{-1} \\equiv 55 \\pmod{101}$, find $49^{-1} \\pmod{101}$, as a residue modulo 101 (Give an answer between 0 and 100, inclusive). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 27419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\\triangle$ . Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by $\\triangle '$ . Similarly, let $T'' = (... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 27420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $91.234$ to $42.7689$ and round your answer to the nearest hundredth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "134.00"
} | {
"split": "train",
"index": 27421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $sinα-\\sqrt{3}cosα=1$, then the value of $sin({\\frac{{7π}}{6}-2α})$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be distinct real numbers for which\n\\[\\frac{a}{b} + \\frac{a+10b}{b+10a} = 2.\\]Find $\\frac{a}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 27423,
"solution": "1. **Assumption and Equation Setup**: Assume without loss of generality (WLOG) that $b = 1$. This simplifies the problem since $\\frac{a}{b} = a$. The given equation becomes:\n \\[\n \\frac{a}{1} + \\frac{a + 10 \\cdot 1}{1 + 10a} = 2.\n \\]\n Simplifying t... |
deepscaler | [
{
"role": "user",
"content": "The circle centered at $(3,-2)$ and with radius $5$ intersects the circle centered at $(3,4)$ and with radius $\\sqrt{13}$ at two points $C$ and $D$. Find $(CD)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 27424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The stem and leaf plot represents the heights, in inches, of the players on the Pine Ridge Middle School boys' basketball team. Calculate the mean height of the players on the team. (Note: $5|3$ represents 53 inches.)\n\nHeight of the Players on the Basketball Team (inches)\n\n... | math | {
"style": "rule",
"ground_truth": "61.44"
} | {
"split": "train",
"index": 27425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all quadruples $(x,y,z,t)$ of positive integers such that\n\\[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 1, 3, 1)"
} | {
"split": "train",
"index": 27426,
"solution": "\nTo solve the problem of determining all quadruples \\((x, y, z, t)\\) of positive integers such that:\n\n\\[\n20^x + 14^{2y} = (x + 2y + z)^{zt},\n\\]\n\nwe begin by analyzing the constraints in the given equation with small values for \\(x\\), \\(y\\), \\(z\\), ... |
deepscaler | [
{
"role": "user",
"content": "Solve the equations: \n(1) $x(x+4)=-5(x+4)$ \n(2) $(x+2)^2=(2x-1)^2$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 27427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $120_4\\times13_4\\div2_4$. Express your answer in base 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1110_4"
} | {
"split": "train",
"index": 27428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α \\in (0, \\frac{π}{2})$ and $\\sin α - \\cos α = \\frac{1}{2}$, find the value of $\\frac{\\cos 2α}{\\sin (α + \\frac{π}{4})}$.\n\nA) $-\\frac{\\sqrt{2}}{2}$\nB) $\\frac{\\sqrt{2}}{2}$\nC) $-1$\nD) $1$ Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 27429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $\\csc y + \\cot y = \\frac{25}{7}$ and $\\sec y + \\tan y = \\frac{p}{q}$, where $\\frac{p}{q}$ is in lowest terms. Find $p+q$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29517"
} | {
"split": "train",
"index": 27430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ a... | math | {
"style": "rule",
"ground_truth": "515"
} | {
"split": "train",
"index": 27431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y = ax^2 + bx + c$ is shown, where $a$, $b$, and $c$ are integers. The vertex of the parabola is at $(-2, 3)$, and the point $(1, 6)$ lies on the graph. Determine the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 27432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the shortest distance from the origin to the circle defined by $x^2-24x +y^2+10y +160=0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side length of square \\(ABCD\\) is 4. Point \\(E\\) is the midpoint of \\(AB\\), and point \\(F\\) is a moving point on side \\(BC\\). Triangles \\(\\triangle ADE\\) and \\(\\triangle DCF\\) are folded up along \\(DE\\) and \\(DF\\) respectively, making points \\(A\\) and ... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 27434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ denote the value of the sum\\[\\sum_{n=0}^{668} (-1)^{n} {2004 \\choose 3n}\\]Determine the remainder obtained when $S$ is divided by $1000$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 27435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In what ratio does the line $TH$ divide the side $BC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"split": "train",
"index": 27436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kaashish has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 27437,
"solution": "1. Let the first number be $x$ and the second number be $y$. According to the problem, Kaashish has written down one integer ($x$) two times and another integer ($y$) three times. The sum of these five numbers is given by:\n \\[\n 2x + 3y = 100\n \\]\n\n2. We... |
deepscaler | [
{
"role": "user",
"content": "Find $(-2)^{3}+(-2)^{2}+(-2)^{1}+2^{1}+2^{2}+2^{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 27438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two consecutive page numbers is $20{,}412$. What is the sum of these two page numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "285"
} | {
"split": "train",
"index": 27439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid \\( S A B C D \\) has a trapezoid \\( A B C D \\) as its base, with bases \\( B C \\) and \\( A D \\). Points \\( P_1, P_2, P_3 \\) lie on side \\( B C \\) such that \\( B P_1 < B P_2 < B P_3 < B C \\). Points \\( Q_1, Q_2, Q_3 \\) lie on side \\( A D \\) such that \\... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"split": "train",
"index": 27440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two 2-digit numbers is $5488$. What is the smaller of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 27441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Replace the symbols $\\$ last\\$ with the same numbers so that the equation becomes true: $\\$ \\$ |\\frac{20}{x} - \\frac{x}{15} = \\frac{20}{15} \\$ \\$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 27442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonals of a rhombus are 6 inches and 8 inches. What is the perimeter, in inches, of the rhombus below? [asy]\nimport geometry; size(150); defaultpen(linewidth(0.8));\npair A = (0,0),B = (8,6),C = (16,0),D = (8,-6), E = (8,0);\ndraw(A--B--C--D--cycle);\ndraw(A--C,StickInt... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 27443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\((-1)^{3}+(-1)^{2}+(-1)\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 27444,
"solution": "Since -1 raised to an even exponent equals 1 and -1 raised to an odd exponent equals -1, then \\((-1)^{3}+(-1)^{2}+(-1)=-1+1-1=-1\\)."
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Charlie, and Dave, refuse to sit in four consecutive seats? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3507840"
} | {
"split": "train",
"index": 27445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many times does the digit 9 appear in the list of all integers from 1 to 1000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 27446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $1357 + 3571 + 5713 - 7135$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3506"
} | {
"split": "train",
"index": 27447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\"Modulo $m$ graph paper\" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\\le x, y <m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\\equiv... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 27448,
"solution": ""
} |
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