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 value of the function \\( f(x) \\) at the point \\( x_{0} = 4500 \\), if \\( f(0) = 1 \\) and for any \\( x \\) the equality \\( f(x + 3) = f(x) + 2x + 3 \\) holds. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6750001"
} | {
"split": "train",
"index": 18749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 18750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 5-digit numbers $\\overline{a b c d e}$ exist such that digits $b$ and $d$ are each the sum of the digits to their immediate left and right? (That is, $b=a+c$ and $d=c+e$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "330"
} | {
"split": "train",
"index": 18751,
"solution": "Note that $a>0$, so that $b>c$, and $e \\geq 0$ so that $d \\geq c$. Conversely, for each choice of $(b, c, d)$ with $b>c$ and $d \\geq c$, there exists a unique pair $(a, e)$ such that $\\overline{a b c d e}$ is a number having the desired property. Thus, we compu... |
deepscaler | [
{
"role": "user",
"content": "On an island, there are only knights, who always tell the truth, and liars, who always lie. One fine day, 30 islanders sat around a round table. Each of them can see everyone except himself and his neighbors. Each person in turn said the phrase: \"Everyone I see is a liar.\" Ho... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 18752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Harriet ran a 1000 m course in 380 seconds. She ran the first 720 m of the course at a constant speed of 3 m/s. What was her speed for the remaining part of the course? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 18753,
"solution": "Since Harriet ran 720 m at 3 m/s, then this segment took her 720 m / 3 m/s = 240 s. In total, Harriet ran 1000 m in 380 s, so the remaining part of the course was a distance of 1000 m - 720 m = 280 m which she ran in 380 s - 240 s = 140 s. Since she ran this sectio... |
deepscaler | [
{
"role": "user",
"content": "Find the residue of the function\n\n$$\nw=z^{2} \\sin \\frac{1}{z+1}\n$$\n\nat its singular point. 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": 18754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( N \\) is a 6-digit number having base-10 representation \\( \\underline{a} \\underline{b} \\underline{c} \\underline{d} \\underline{e} \\underline{f} \\). If \\( N \\) is \\( \\frac{6}{7} \\) of the number having base-10 representation \\( \\underline{d} \\underline... | math | {
"style": "rule",
"ground_truth": "461538"
} | {
"split": "train",
"index": 18755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes $1, 10, 11, 12, \\ldots$ but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number fo... | math | {
"style": "rule",
"ground_truth": "116"
} | {
"split": "train",
"index": 18756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of the original integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "725"
} | {
"split": "train",
"index": 18757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( O \\) is located on side \\( AC \\) of triangle \\( ABC \\) such that \\( CO : CA = 2 : 3 \\). When this triangle is rotated by a certain angle around point \\( O \\), vertex \\( B \\) moves to vertex \\( C \\), and vertex \\( A \\) moves to point \\( D \\), which lie... | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"split": "train",
"index": 18758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. What ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4m}"
} | {
"split": "train",
"index": 18759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the same road, two trucks are driving in the same direction. Initially, Truck A is 4 kilometers ahead of Truck B. The speed of Truck A is 45 kilometers per hour, and the speed of Truck B is 60 kilometers per hour. How far apart are the two trucks 1 minute before Truck B catc... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 18760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A car is averaging 50 miles per hour. If the car maintains this speed, how many minutes less would a 450-mile trip take than a 475-mile trip? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 18761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone has a base with radius $600$ and height $200\\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $125$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose... | math | {
"style": "rule",
"ground_truth": "625"
} | {
"split": "train",
"index": 18762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let a $9$ -digit number be balanced if it has all numerals $1$ to $9$ . Let $S$ be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of $k$ such that any two subsequences of... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 18763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kiana has two older twin siblings, and their ages, together with her own, multiply together to equal 162. Calculate the sum of their ages. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 18764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin \\omega x\\cos \\omega x- \\sqrt{3}\\cos^2\\omega x+ \\frac{\\sqrt{3}}{2} (\\omega > 0)$, the two adjacent axes of symmetry of its graph are $\\frac{\\pi}{2}$.\n\n$(1)$ Find the equation of the axis of symmetry for the function $y=f(x)$.\n\n$(2)$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 18765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ant starts at the origin, facing in the positive $x$-direction. Each second, it moves 1 unit forward, then turns counterclockwise by $\\sin ^{-1}\\left(\\frac{3}{5}\\right)$ degrees. What is the least upper bound on the distance between the ant and the origin? (The least upp... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 18766,
"solution": "We claim that the points the ant visits lie on a circle of radius $\\frac{\\sqrt{10}}{2}$. We show this by saying that the ant stays a constant distance $\\frac{\\sqrt{10}}{2}$ from the point $\\left(\\frac{1}{2}, \\frac{3}{2}\\right)$. Suppose the ant moves on a p... |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle $\\alpha$ is in the second quadrant and intersects the unit circle at point $P(m, \\frac{\\sqrt{15}}{4})$. \n$(1)$ Find the value of the real number $m$; \n$(2)$ Let $f(\\alpha) = \\frac{\\cos(2\\pi - \\alpha) + \\tan(3\\pi + \\alpha)}{\\... | math | {
"style": "rule",
"ground_truth": "-\\frac{4 + 16\\sqrt{15}}{15}"
} | {
"split": "train",
"index": 18767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a machine that transforms a positive integer \\( N \\) based on the rule: if \\( N = 7 \\), the machine outputs \\( 3 \\times 7 + 1 = 22 \\).\n\nBy inputting the result back into the machine and repeating five times, the output sequence is:\n\\[ 7 \\rightarrow 22 \\righta... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 18768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\\sqrt{n_1}+r_2\\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the ... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 18769,
"solution": "To solve this problem, we need to consider all possible configurations of the trapezoid with sides 3, 5, 7, and 11. We will use Heron's formula to calculate the area of triangles formed by these sides and then determine the area of the trapezoid.\n\n#### Step 1: Id... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is an isosceles triangle with side lengths of 25, 25 and 48 centimeters. What is the area of triangle $ABC$, in square centimeters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 18770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all real numbers $r$ and $s$, define the mathematical operation $\\#$ such that the following conditions apply: $r\\ \\#\\ 0 = r, r\\ \\#\\ s = s\\ \\#\\ r$, and $(r + 1)\\ \\#\\ s = (r\\ \\#\\ s) + s + 1$. What is the value of $11\\ \\#\\ 5$? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 18771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ordered pair $(n,p)$ is *juicy* if $n^{2} \\equiv 1 \\pmod{p^{2}}$ and $n \\equiv -1 \\pmod{p}$ for positive integer $n$ and odd prime $p$ . How many juicy pairs exist such that $n,p \\leq 200$ ?\n\n\nProposed by Harry Chen (Extile) Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every positive integer \\( n \\), define \\( a_{n} \\) as the last digit of the sum of the digits of the number formed by writing \"2005\" \\( n \\) times consecutively. For example, \\(\\mathrm{a}_{1}=7\\) and \\(\\mathrm{a}_{2}=4\\).\n\na) What are the positive integers \... | math | {
"style": "rule",
"ground_truth": "9025"
} | {
"split": "train",
"index": 18773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Class A and Class B each send 2 students to participate in the grade math competition. The probability of each participating student passing the competition is 0.6, and the performance of the participating students does not affect each other. Find: \n(1) The probability that t... | math | {
"style": "rule",
"ground_truth": "0.9744"
} | {
"split": "train",
"index": 18774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What was the range of temperatures on Monday in Fermatville, given that the minimum temperature was $-11^{\\circ} \\mathrm{C}$ and the maximum temperature was $14^{\\circ} \\mathrm{C}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25^{\\circ} \\mathrm{C}"
} | {
"split": "train",
"index": 18775,
"solution": "Since the maximum temperature was $14^{\\circ} \\mathrm{C}$ and the minimum temperature was $-11^{\\circ} \\mathrm{C}$, then the range of temperatures was $14^{\\circ} \\mathrm{C} - (-11^{\\circ} \\mathrm{C}) = 25^{\\circ} \\mathrm{C}$."
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field ... | math | {
"style": "rule",
"ground_truth": "50000000"
} | {
"split": "train",
"index": 18776,
"solution": "1. **Convert dimensions from feet to inches**: \n - The width of the field in inches is:\n \\[\n 300 \\text{ feet} \\times \\frac{12 \\text{ inches}}{1 \\text{ foot}} = 3600 \\text{ inches}\n \\]\n - The length of the field in inches is:\n \\[\n... |
deepscaler | [
{
"role": "user",
"content": "Given a set $A_n = \\{1, 2, 3, \\ldots, n\\}$, define a mapping $f: A_n \\rightarrow A_n$ that satisfies the following conditions:\n① For any $i, j \\in A_n$ with $i \\neq j$, $f(i) \\neq f(j)$;\n② For any $x \\in A_n$, if the equation $x + f(x) = 7$ has $K$ pairs of solutions,... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 18777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f_n (x)\\ (n=1,2,\\cdots)$ is defined as follows.\r\n\r\n\\[f_1 (x)=x,\\ f_{n+1}(x)=2x^{n+1}-x^n+\\frac{1}{2}\\int_0^1 f_n(t)\\ dt\\ \\ (n=1,2,\\cdots)\\]\r\n\r\nEvaluate \r\n\r\n\\[\\lim_{n\\to\\infty} f_n \\left(1+\\frac{1}{2n}\\right)\\] Let's think step by ... | math | {
"style": "rule",
"ground_truth": "e^{1/2}"
} | {
"split": "train",
"index": 18778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the odd function $f(x)$ that is increasing on the interval $[3,7]$ and has a minimum value of $5$, determine the behavior of $f(x)$ on the interval $[-7,-3]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 18779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For example, June 17 is a relatively prime date because the greatest common factor of 6 and 17 is 1. How many relatively prime dates are in the month with the few... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mode and median of the data $9.30$, $9.05$, $9.10$, $9.40$, $9.20$, $9.10$ are ______ and ______, respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9.15"
} | {
"split": "train",
"index": 18781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows an arc \\( PQ \\) of a circle with center \\( O \\) and radius 8. Angle \\( QOP \\) is a right angle, the point \\( M \\) is the midpoint of \\( OP \\), and \\( N \\) lies on the arc \\( PQ \\) so that \\( MN \\) is perpendicular to \\( OP \\). Which of the fo... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 18782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the \"Lucky Sum\" lottery, there are a total of $N$ balls numbered from 1 to $N$. During the main draw, 10 balls are randomly selected. During the additional draw, 8 balls are randomly selected from the same set of balls. The sum of the numbers on the selected balls in each ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 18783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \\[x^3 - 8x^2 + px - q = 0\\] are three distinct positive integers. Compute $p + q.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 18784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nFrom May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangement... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 18785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that Fatima gets fewer heads than tails if she flips 10 coins? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{193}{512}"
} | {
"split": "train",
"index": 18786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many even natural-number factors does $n = 2^2 \\cdot 3^1 \\cdot 7^2$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"split": "train",
"index": 18788,
"solution": "1. **Identify the Required Sequence**: \n Debra must get two heads in a row, but she must see a second tail before seeing the second head. This means the sequence of flips must start with \"THT\" (Tail, Head, Tail).\n\n2. **Calculate the Probability of Starting w... |
deepscaler | [
{
"role": "user",
"content": "Given plane vectors $\\overrightarrow{a}=(-1,2)$ and $\\overrightarrow{b}=(1,-4)$. \n$(1)$ If $4\\overrightarrow{a}+\\overrightarrow{b}$ is perpendicular to $k\\overrightarrow{a}-\\overrightarrow{b}$, find the value of the real number $k$. \n$(2)$ If $\\theta$ is the angle be... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"split": "train",
"index": 18789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Flower Town, there are $99^{2}$ residents, some of whom are knights (who always tell the truth) and others are liars (who always lie). The houses in the town are arranged in the cells of a $99 \\times 99$ square grid (totaling $99^{2}$ houses, arranged on 99 vertical and 99 ... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 18790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\[ 1 \\cdot 1995 + 2 \\cdot 1994 + 3 \\cdot 1993 + \\dots + 1994 \\cdot 2 + 1995 \\cdot 1 = 1995 \\cdot 997 \\cdot y, \\]\ncompute the integer \\( y \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "665"
} | {
"split": "train",
"index": 18791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nWhat is the largest integer \\( k \\) such that \\( k+1 \\) divides\n\\[ k^{2020} + 2k^{2019} + 3k^{2018} + \\cdots + 2020k + 2021? \\ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 18792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $(x - y)(x + y)$ if $x = 10$ and $y = 15$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-125"
} | {
"split": "train",
"index": 18793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6 and with their centers 13 units apart. If a plane \\(\\alpha\\) is tangent to both spheres and intersects the cylindrical surface forming an ellipse, what is the length of the major axis of... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a square with side length $2$ , and let a semicircle with flat side $CD$ be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "4 - 2\\sqrt{3}"
} | {
"split": "train",
"index": 18795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\\sqrt{2}$, what is the volume of the solid?\n[asy] size(180); import three; pathpen = black+linewidth(0.65); pointp... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 18796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sets \\( A = \\{2, 0, 1, 7\\} \\) and \\( B = \\{ x \\mid x^2 - 2 \\in A, \\, x - 2 \\notin A \\} \\), the product of all elements in set \\( B \\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base six number $53_{6}$ is equal to the base $b$ number $113_{b}$. What is the positive value of $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 18798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the triangular region bounded by the $x$-axis, the $y$-axis and the line with equation $y=2x-6$ is one-quarter of the area of the triangular region bounded by the $x$-axis, the line with equation $y=2x-6$ and the line with equation $x=d$, where $d>0$. What is the va... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 18799,
"solution": "The line with equation $y=2x-6$ has $y$-intercept -6. Also, the $x$-intercept of $y=2x-6$ occurs when $y=0$, which gives $0=2x-6$ or $2x=6$ which gives $x=3$. Therefore, the triangle bounded by the $x$-axis, the $y$-axis, and the line with equation $y=2x-6$ has bas... |
deepscaler | [
{
"role": "user",
"content": "The product of the base seven numbers $24_7$ and $30_7$ is expressed in base seven. What is the base seven sum of the digits of this product? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 18800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1,$ $x_2,$ $x_3$ be positive real numbers such that $x_1 + 2x_2 + 3x_3 = 60.$ Find the smallest possible value of\n\\[x_1^2 + x_2^2 + x_3^2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1800}{7}"
} | {
"split": "train",
"index": 18801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=|\\overrightarrow{b}|=2$, and $\\overrightarrow{b} \\perp (2\\overrightarrow{a}+ \\overrightarrow{b})$, calculate the angle between vector $\\overrightarrow{a}$ and $\\overrightarrow{b}... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2\\pi}{3}"
} | {
"split": "train",
"index": 18802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCan you multiply 993 and 879 in your head? Interestingly, if we have two two-digit numbers containing the same number of tens, and the sum of the digits of their units place equals 10, then such numbers can always be multiplied mentally as follows:\nSuppose we need to multipl... | math | {
"style": "rule",
"ground_truth": "872847"
} | {
"split": "train",
"index": 18803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( A \\) and \\( B \\) are two points on the surface of a sphere with a radius of 5, and \\( AB = 8 \\). Planes \\( O_1AB \\) and \\( O_2AB \\) are perpendicular to each other and pass through \\( AB \\). The intersections of these planes with the sphere create cros... | math | {
"style": "rule",
"ground_truth": "41 \\pi"
} | {
"split": "train",
"index": 18804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A plane flies between two cities. With the tailwind, it takes 5 hours and 30 minutes, and against the wind, it takes 6 hours. Given that the wind speed is 24 kilometers per hour, and assuming the plane's flying speed is $x$ kilometers per hour, then the speed of the plane with ... | math | {
"style": "rule",
"ground_truth": "528"
} | {
"split": "train",
"index": 18805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of the third term and the constant term in the expansion of $\\left(x^3 + \\frac{2}{3x^2}\\right)^5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{80}{27}"
} | {
"split": "train",
"index": 18806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Great Eighteen Hockey League is divided into two divisions, with nine teams in each division. Each team plays each of the other teams in its own division three times and every team in the other division twice. How many league games are scheduled? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "378"
} | {
"split": "train",
"index": 18807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ones digit of $7^{35}$ when written as an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 18808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, there are points $P_0$, $P_1$, $P_2$, $P_3$, ..., $P_{n-1}$, $P_n$. Let the coordinates of point $P_k$ be $(x_k,y_k)$ $(k\\in\\mathbb{N},k\\leqslant n)$, where $x_k$, $y_k\\in\\mathbb{Z}$. Denote $\\Delta x_k=x_k-x_{k-1}$, $\\Delta y_k=y_k-y_... | math | {
"style": "rule",
"ground_truth": "4066272"
} | {
"split": "train",
"index": 18809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a sheet of graph paper, two rectangles are outlined. The first rectangle has a vertical side shorter than the horizontal side, and for the second rectangle, the opposite is true. Find the maximum possible area of their intersection if the first rectangle contains 2015 cells ... | math | {
"style": "rule",
"ground_truth": "1302"
} | {
"split": "train",
"index": 18810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $EFGH$, we have $E=(1,1)$, $F=(101,21)$, and $H=(3,y)$ for some integer $y$. What is the area of rectangle $EFGH$?\nA) 520\nB) 1040\nC) 2080\nD) 2600 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1040"
} | {
"split": "train",
"index": 18811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Into each row of a \\( 9 \\times 9 \\) grid, Nigel writes the digits \\( 1, 2, 3, 4, 5, 6, 7, 8, 9 \\) in order, starting at one of the digits and returning to 1 after 9: for example, one row might contain \\( 7, 8, 9, 1, 2, 3, 4, 5, 6 \\). The grid is gorgeous if each nine-dig... | math | {
"style": "rule",
"ground_truth": "9^8"
} | {
"split": "train",
"index": 18812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $a+b=11$. Choose one of the following two conditions as known, and find:<br/>$(Ⅰ)$ the value of $a$;<br/>$(Ⅱ)$ $\\sin C$ and the area of $\\triangle ABC$.<br/>Condition 1: $c=7$, $\\cos A=-\\frac{1}{7}$;<br/>Condition 2: $\\cos A=\\frac{1}{8}$, $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{15\\sqrt{7}}{4}"
} | {
"split": "train",
"index": 18813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 18814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x) = x(x - m)^2$ reaches its maximum value at $x = -2$. Determine the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 18815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large urn contains $100$ balls, of which $36 \\%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \\%$? (No red balls are to be removed.) Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18816,
"solution": "1. **Calculate the initial number of red and blue balls:**\n Given that $36\\%$ of the balls are red, and there are $100$ balls in total, the number of red balls is:\n \\[\n 0.36 \\times 100 = 36 \\text{ red balls}\n \\]\n The rest of the balls are blue, ... |
deepscaler | [
{
"role": "user",
"content": "If \\(a, c, d\\) are integers, \\(b\\) is a positive integer, and they satisfy \\(a + b = c\\), \\(b + c = d\\), and \\(c + d = a\\), then the maximum value of \\(a + b + c + d\\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 18817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular \\( n \\)-gon, \\( A_{1} A_{2} A_{3} \\cdots A_{n} \\), where \\( n > 6 \\), sides \\( A_{1} A_{2} \\) and \\( A_{5} A_{4} \\) are extended to meet at point \\( P \\). If \\( \\angle A_{2} P A_{4}=120^\\circ \\), determine the value of \\( n \\). Let's think step b... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 18818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, it is known that $\\overrightarrow {AB}\\cdot \\overrightarrow {AC}=9$ and $\\overrightarrow {AB}\\cdot \\overrightarrow {BC}=-16$. Find:\n 1. The value of $AB$;\n 2. The value of $\\frac {sin(A-B)}{sinC}$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"split": "train",
"index": 18819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T = \\{ 1, 2, 3, \\dots, 14, 15 \\}$ . Say that a subset $S$ of $T$ is *handy* if the sum of all the elements of $S$ is a multiple of $5$ . For example, the empty set is handy (because its sum is 0) and $T$ itself is handy (because its sum is 120). Compute the... | math | {
"style": "rule",
"ground_truth": "6560"
} | {
"split": "train",
"index": 18820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The specific heat capacity of a body with mass \\( m = 3 \\) kg depends on the temperature in the following way: \\( c = c_{0}(1 + \\alpha t) \\), where \\( c_{0} = 200 \\) J/kg·°C is the specific heat capacity at \\( 0^{\\circ} \\mathrm{C} \\), \\( \\alpha = 0.05 \\,^{\\circ} ... | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"split": "train",
"index": 18821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the sum of all the real coefficients of the expansion of ${(1+ix)}^{2009}$. What is $\\log_{2}(S)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"split": "train",
"index": 18822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral pyramid \\( S A B C D \\):\n\n- The lateral faces \\( S A B \\), \\( S B C \\), \\( S C D \\), and \\( S D A \\) have areas 9, 9, 27, 27 respectively;\n- The dihedral angles at the edges \\( A B \\), \\( B C \\), \\( C D \\), \\( D A \\) are equal;\n- The q... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 18823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?\n\n*Ray Li* Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 18824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\log_{4}3=x$. Then $\\log_{2}27=kx$. Find $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "k=6"
} | {
"split": "train",
"index": 18825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\\text{th}}$ integer in the list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31254"
} | {
"split": "train",
"index": 18826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $x$ that satisfies $\\sqrt[5]{x\\sqrt{x^3}}=3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 18827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of the following expressions without using a calculator:\n\\\\((1)\\\\lg 5^{2}+ \\frac {2}{3}\\lg 8+\\lg 5\\lg 20+(\\lg 2)^{2}\\\\)\n\\\\((2)\\\\) Let \\(2^{a}=5^{b}=m\\), and \\(\\frac {1}{a}+ \\frac {1}{b}=2\\), find \\(m\\). Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\sqrt {10}"
} | {
"split": "train",
"index": 18828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system $Ox$, the polar equation of curve $C_{1}$ is $ρ=\\frac{2\\sqrt{2}}{sin(θ+\\frac{π}{4})}$, with the pole $O$ as the origin and the polar axis $Ox$ as the x-axis. A Cartesian coordinate system $xOy$ is established with the same unit length. It is kn... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 18829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( \\cos A + \\cos B + \\cos C = \\sin A + \\sin B + \\sin C = 0 \\), find the value of \\( \\cos^4 A + \\cos^4 B + \\cos^4 C \\). 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": 18830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit positive integers are there such that the product of their digits equals 18? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Distinct ways to distribute $7$ identical balls into $4$ distinct boxes such that no box is empty. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 18832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest divisor of 360 that is smaller than 60 and also a factor of 90? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 18833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sets \\(A, B\\), and \\(C\\) satisfy \\(|A| = 92\\), \\(|B| = 35\\), \\(|C| = 63\\), \\(|A \\cap B| = 16\\), \\(|A \\cap C| = 51\\), and \\(|B \\cap C| = 19\\). Compute the number of possible values of \\(|A \\cap B \\cap C|\\). Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A = \\{1, 2, \\cdots, 2004\\} \\) and \\( f: A \\rightarrow A \\) be a bijection satisfying \\( f^{[2004]}(x) = f(x) \\), where \\( f^{[2004]}(x) \\) denotes applying \\( f \\) 2004 times to \\( x \\). How many such functions \\( f \\) are there? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "1 + 2004!"
} | {
"split": "train",
"index": 18835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{x+1}{3}+\\frac{2-3x}{2}$. Express your answer as a single fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8-7x}{6}"
} | {
"split": "train",
"index": 18836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sqrt{3}\\sin \\omega x+\\cos \\omega x (\\omega > 0)$, where the x-coordinates of the points where the graph of $f(x)$ intersects the x-axis form an arithmetic sequence with a common difference of $\\frac{\\pi}{2}$, determine the probability of the ev... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 18837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. Wha... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 18838,
"solution": "1. **Define Variables:**\n Let the length of the track be $x$ meters. Since Brenda and Sally start at diametrically opposite points, they start $\\frac{x}{2}$ meters apart.\n\n2. **Analyze the First Meeting:**\n When Brenda and Sally first meet, Brenda has run ... |
deepscaler | [
{
"role": "user",
"content": "In a grain storage facility, the following are the amounts of grain (in tons) that were received or dispatched over a period of 6 days (where \"+\" indicates received and \"-\" indicates dispatched): \n+26, -32, -15, +34, -38, -20. \n(1) After these 6 days, did the amount of ... | math | {
"style": "rule",
"ground_truth": "825"
} | {
"split": "train",
"index": 18839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A showroom has 150 lights, all of which are initially turned on. Each light has an individual switch, numbered from 1 to 150. A student first toggles all switches that are multiples of 3, and then toggles all switches that are multiples of 5. How many lights remain on in the sh... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 18840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exists a positive real number $x$ such that $\\cos (\\tan^{-1}(x))=x$. Find the value of $x^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-1+\\sqrt{5})/2"
} | {
"split": "train",
"index": 18841,
"solution": "Draw a right triangle with legs $1, x$; then the angle $\\theta$ opposite $x$ is $\\tan^{-1} x$, and we can compute $\\cos (\\theta)=1 / \\sqrt{x^{2}+1}$. Thus, we only need to solve $x=1 / \\sqrt{x^{2}+1}$. This is equivalent to $x \\sqrt{x^{2}+1}=1$. Square both ... |
deepscaler | [
{
"role": "user",
"content": "If $2n + 5 = 16$, what is the value of the expression $2n - 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 18842,
"solution": "Since $2n + 5 = 16$, then $2n - 3 = (2n + 5) - 8 = 16 - 8 = 8$. Alternatively, we could solve the equation $2n + 5 = 16$ to obtain $2n = 11$ or $n = \\frac{11}{2}$. From this, we see that $2n - 3 = 2\\left(\\frac{11}{2}\\right) - 3 = 11 - 3 = 8$."
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum angle formed by any triple among five points on the plane such that the minimum angle is greater than or equal to $36^{\\circ}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36^{\\circ}"
} | {
"split": "train",
"index": 18843,
"solution": "We will show that $36^{\\circ}$ is the desired answer for the problem. First, we observe that if the given 5 points form a regular pentagon, then the minimum of the angles formed by any triple among the five vertices is $36^{\\circ}$, and therefore, the answer we s... |
deepscaler | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\\triangle PAB$ and $\\triangle QAB$ form a $120^{\\circ}$ dihedr... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 18844,
"solution": "Draw a good diagram. Draw $CH$ as an altitude of the triangle. Scale everything down by a factor of $100\\sqrt{3}$, so that $AB=2\\sqrt{3}$. Finally, call the center of the triangle U. Draw a cross-section of the triangle via line $CH$, which of course includes $P,... |
deepscaler | [
{
"role": "user",
"content": "Given $x+m≤{e}^{\\frac{2x}{m}+n}$ holds for any $x\\in \\left(-m,+\\infty \\right)$, then the minimum value of $m\\cdot n$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{e^2}"
} | {
"split": "train",
"index": 18845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a finite sequence $B = (b_1, b_2, \\dots, b_n)$ of numbers, the Cesaro sum is defined as\n\\[\\frac{T_1 + T_2 + \\cdots + T_n}{n},\\] \nwhere $T_k = b_1 + b_2 + \\cdots + b_k$ for $1 \\leq k \\leq n$.\n\nIf the Cesaro sum of the 100-term sequence $(b_1, b_2, \\dots, b_{100}... | math | {
"style": "rule",
"ground_truth": "1191"
} | {
"split": "train",
"index": 18846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 27 cubes are randomly arranged to form a $3\\times 3 \\times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\\frac{p^a}... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 18847,
"solution": "We can consider the orientation of each of the individual cubes independent=== The unit cube at the center of our large cube has no exterior faces, so all of its orientations work.\n For the six unit cubes and the centers of the faces of the large cube, we need tha... |
deepscaler | [
{
"role": "user",
"content": "All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 18848,
"solution": ""
} |
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