data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ defined on $\\mathbb{R}$ is an odd function and satisfies $f(1+x)=f(3+x)$. When $0\\leq x\\leq 1$, $f(x)=x^{3}-x$. Find $f(\\frac{11}{2})+f(6)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 21749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability that a randomly chosen divisor of $25!$ is odd. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{23}"
} | {
"split": "train",
"index": 21750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=x^{2005}+ax^{3}- \\frac {b}{x}-8$, and $f(-2)=10$, find $f(2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-26"
} | {
"split": "train",
"index": 21751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures\nFour friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.\n$\\circ$ Art's cookies are trapezoids: \n\n$\\circ$ ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 21752,
"solution": "1. **Calculate the area of one of Art's cookies**: \n Art's cookies are trapezoids. Assuming the dimensions given are for the bases and the height, the area \\( A \\) of a trapezoid can be calculated using the formula:\n \\[\n A = \\frac{1}{2} \\times (\\text... |
deepscaler | [
{
"role": "user",
"content": "2019 points are chosen independently and uniformly at random on the interval $[0,1]$. Tairitsu picks 1000 of them randomly and colors them black, leaving the remaining ones white. Hikari then computes the sum of the positions of the leftmost white point and the rightmost black ... | math | {
"style": "rule",
"ground_truth": "\\frac{1019}{2019}"
} | {
"split": "train",
"index": 21753,
"solution": "Note that each point is chosen uniformly and independently from 0 to 1, so we can apply symmetry. Given any coloring, suppose that we flip all the positions of the black points: then the problem becomes computing the probability that the leftmost white point is to ... |
deepscaler | [
{
"role": "user",
"content": "5 students stand in a row for a photo, where students A and B must stand next to each other, and A cannot stand at either end. Calculate the total number of possible arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 21754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A train took $X$ minutes ($0 < X < 60$) to travel from platform A to platform B. Find $X$ if it's known that at both the moment of departure from A and the moment of arrival at B, the angle between the hour and minute hands of the clock was $X$ degrees. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 21755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a \\gt 0$, $b\\in R$, if the inequality $\\left(ax-2\\right)(-x^{2}-bx+4)\\leqslant 0$ holds for all $x \\gt 0$, then the minimum value of $b+\\frac{3}{a}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 21756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place each of the digits 4, 5, 6, and 7 in exactly one square to make the smallest possible product. What is this product? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2622"
} | {
"split": "train",
"index": 21757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Almondine has a bag with $N$ balls, each of which is red, white, or blue. If Almondine picks three balls from the bag without replacement, the probability that she picks one ball of each color is larger than 23 percent. Compute the largest possible value of $\\left\\lfloor\\fra... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 21758,
"solution": "If $k=\\left\\lfloor\\frac{N}{3}\\right\\rfloor$, then the maximum possible probability is $\\frac{6 k^{3}}{(3 k)(3 k-1)(3 k-2)}$. with equality when there are $k$ balls of each of the three colors. Going from $3 k \\rightarrow 3 k+1$ replaces $\\frac{k}{3 k-2} \\r... |
deepscaler | [
{
"role": "user",
"content": "Cyclic pentagon $ABCDE$ has a right angle $\\angle ABC=90^{\\circ}$ and side lengths $AB=15$ and $BC=20$. Supposing that $AB=DE=EA$, find $CD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 21759,
"solution": "By Pythagoras, $AC=25$. Since $\\overline{AC}$ is a diameter, angles $\\angle ADC$ and $\\angle AEC$ are also right, so that $CE=20$ and $AD^{2}+CD^{2}=AC^{2}$ as well. Beginning with Ptolemy's theorem, $$\\begin{aligned} & (AE \\cdot CD+AC \\cdot DE)^{2}=AD^{2} \\... |
deepscaler | [
{
"role": "user",
"content": "When $n$ is divided by 3, the remainder is 2. What is the remainder when $5n$ is divided by 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 21760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On Monday, I worked $t-6$ hours and earned $2t-5$ dollars per hour. My friend Andrew worked $2t-8$ hours but only earned $t-5$ dollars an hour. As it turns out, we earned the same amount. What is the value of $t$? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\\overline{BC}$ and $\\overline{CD},$ respectively, so that $\\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\\overline{AE}.$ The length of a... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each segment with endpoints at the vertices of a regular 100-gon is colored red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers were placed at the vertices such that the sum of their squ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 21763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct three-digit positive integers have only odd digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 21764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the real-coefficient polynomial \\( f(x) = x^4 + a x^3 + b x^2 + c x + d \\) that satisfies \\( f(1) = 2 \\), \\( f(2) = 4 \\), and \\( f(3) = 6 \\), find the set of all possible values of \\( f(0) + f(4) \\). Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 21765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain place, four people, $A$, $B$, $C$, and $D$, successively contracted the novel coronavirus, with only $A$ having visited an epidemic area.\n1. If the probabilities of $B$, $C$, and $D$ being infected by $A$ are $\\frac{1}{2}$ each, what is the probability that exact... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{6}"
} | {
"split": "train",
"index": 21766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( 3^x = 5 \\), what is the value of \\( 3^{x+2} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 21767,
"solution": "Using exponent laws, \\( 3^{x+2} = 3^x \\cdot 3^2 = 3^x \\cdot 9 \\). Since \\( 3^x = 5 \\), then \\( 3^{x+2} = 3^x \\cdot 9 = 5 \\cdot 9 = 45 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the first six terms in the geometric sequence $\\frac12,\\frac14,\\frac18,\\dots$. 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{63}{64}"
} | {
"split": "train",
"index": 21768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the smallest square that can contain a circle of radius 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 21769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Molly flips a fair coin five times, and she is very surprised to flip a head each time. What is the probability she will flip a tail on her next flip of the coin? 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{1}{2}"
} | {
"split": "train",
"index": 21770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C$: $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ $(a>0, b>0)$, with left and right foci $F_{1}$, $F_{2}$, and the origin $O$, a perpendicular line is drawn from $F_{1}$ to a asymptote of $C$, with the foot of the perpendicular being $D$, and $|DF_{2}|=2\\sqrt{2}|O... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 21771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the set of integers $\\{1,2,3,\\dots,2009\\}$, choose $k$ pairs $\\{a_i,b_i\\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2009$. Find the maximum possible value of $k$. Let's think... | math | {
"style": "rule",
"ground_truth": "803"
} | {
"split": "train",
"index": 21772,
"solution": "Suppose that we have a valid solution with $k$ pairs. As all $a_i$ and $b_i$ are distinct, their sum is at least $1+2+3+\\cdots+2k=k(2k+1)$. On the other hand, as the sum of each pair is distinct and at most equal to $2009$, the sum of all $a_i$ and $b_i$ is at mos... |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1$ and the parabola $y^{2} = 4cx$, where $c = \\sqrt{a^{2} + b^{2}}$, find the eccentricity of the hyperbola given that $|AB| = 4c$. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"split": "train",
"index": 21773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest number $a$ such that a square of side $a$ can contain five disks of radius $1$ , so that no two of the disks have a common interior point. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 21774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To estimate the consumption of disposable wooden chopsticks, in 1999, a sample of 10 restaurants from a total of 600 high, medium, and low-grade restaurants in a certain county was taken. The daily consumption of disposable chopstick boxes in these restaurants was as follows: ... | math | {
"style": "rule",
"ground_truth": "7260"
} | {
"split": "train",
"index": 21775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f: \\mathbb{Z}^{2} \\rightarrow \\mathbb{Z}$ be a function such that, for all positive integers $a$ and $b$, $$f(a, b)= \\begin{cases}b & \\text { if } a>b \\\\ f(2 a, b) & \\text { if } a \\leq b \\text { and } f(2 a, b)<a \\\\ f(2 a, b)-a & \\text { otherwise }\\end{case... | math | {
"style": "rule",
"ground_truth": "203"
} | {
"split": "train",
"index": 21776,
"solution": "Note that $f(a, b)$ is the remainder of $b$ when divided by $a$. If $a>b$ then $f(a, b)$ is exactly $b$ $\\bmod a$. If instead $a \\leq b$, our \"algorithm\" doubles our $a$ by $n$ times until we have $a \\times 2^{n}>b$. At this point, we subtract $a^{\\overline{2... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^{2}}{m+1}+y^{2}=1$ $(m > 0)$ with two foci $F_{1}$ and $F_{2}$, and $E$ is a common point of the line $y=x+2$ and the ellipse, calculate the eccentricity of the ellipse when the sum of distances $|EF_{1}|+|EF_{2}|$ reaches its minimum value. Let's th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 21777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $b$ that satisfies the equation $295_{b} + 467_{b} = 762_{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3x+7\\equiv 2\\pmod{16}$, then $2x+11$ is congruent $\\pmod{16}$ to what integer between $0$ and $15$, inclusive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 21779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Tenth Kingdom, there are 17 islands, each with 119 inhabitants. The inhabitants are divided into two castes: knights, who always tell the truth, and liars, who always lie. During a population census, each person was first asked, \"Not including yourself, are there an equ... | math | {
"style": "rule",
"ground_truth": "1013"
} | {
"split": "train",
"index": 21780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the Olympic torch relay, it is planned to select 6 cities from 8 in a certain province to establish the relay route, satisfying the following conditions. How many methods are there for each condition?\n(1) Only one of the two cities, A and B, is selected. How many methods a... | math | {
"style": "rule",
"ground_truth": "19440"
} | {
"split": "train",
"index": 21781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(AD\\), \\(BE\\), and \\(CF\\) are the altitudes of the acute triangle \\(\\triangle ABC\\). If \\(AB = 26\\) and \\(\\frac{EF}{BC} = \\frac{5}{13}\\), what is the length of \\(BE\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 21782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence \\\\(\\{a_n\\}\\) satisfies \\\\(a_{n+1}= \\dfrac {2016a_n}{2014a_n+2016}(n\\in N_+)\\), and \\\\(a_1=1\\), find \\\\(a_{2017}= \\) ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {1008}{1007\\times 2017+1}"
} | {
"split": "train",
"index": 21783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function f(x) = $\\frac{1}{3}$x^3^ + $\\frac{1−a}{2}$x^2^ - ax - a, x ∈ R, where a > 0.\n(1) Find the monotonic intervals of the function f(x);\n(2) If the function f(x) has exactly two zeros in the interval (-3, 0), find the range of values for a;\n(3) When a = 1, le... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 21784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of five consecutive odd integers is 125. What is the smallest of these integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 21785,
"solution": "Suppose that the smallest of the five odd integers is $x$. Since consecutive odd integers differ by 2, the other four odd integers are $x+2, x+4, x+6$, and $x+8$. Therefore, $x + (x+2) + (x+4) + (x+6) + (x+8) = 125$. From this, we obtain $5x + 20 = 125$ and so $5x ... |
deepscaler | [
{
"role": "user",
"content": "Chelsea goes to La Verde's at MIT and buys 100 coconuts, each weighing 4 pounds, and 100 honeydews, each weighing 5 pounds. She wants to distribute them among \\( n \\) bags, so that each bag contains at most 13 pounds of fruit. What is the minimum \\( n \\) for which this is p... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 21786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\ln x+8x$, find the value of $\\lim_{n\\to\\infty} \\frac{f(1-2\\Delta x)-f(1)}{\\Delta x}$ ( ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 21787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The glass gauge on a cylindrical coffee maker shows that there are $45$ cups left when the coffee maker is $36\\%$ full. How many cups of coffee does it hold when it is full? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 21788,
"solution": "1. Let $x$ be the total capacity of the coffee maker in cups when it is full.\n2. According to the problem, when the coffee maker is $36\\%$ full, it contains $45$ cups of coffee. This can be expressed as:\n \\[\n 0.36x = 45\n \\]\n3. To find the full capacit... |
deepscaler | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ such that\n\\[\\sum_{k = 0}^n \\log_2 \\left( 1 + \\frac{1}{2^{2^k}} \\right) \\ge 1 + \\log_2 \\frac{2014}{2015}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 21789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a > 1$, then the sum of the real solutions of \n$\\sqrt{a - \\sqrt{a + x}} = x$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{4a- 3} - 1}{2}"
} | {
"split": "train",
"index": 21790,
"solution": "We start by squaring both sides of the given equation:\n\\[\n\\sqrt{a - \\sqrt{a + x}} = x \\implies a - \\sqrt{a + x} = x^2.\n\\]\nRearranging terms, we get:\n\\[\na = \\sqrt{a + x} + x^2.\n\\]\nThis equation implies that $\\sqrt{a + x} = a - x^2$. Squaring both s... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $E$: $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, its eccentricity is $\\frac{\\sqrt{2}}{2}$, point $F$ is the left focus of the ellipse, point $A$ is the right vertex, and point $B$ is the upper vertex. Additionally, $S_{\\triangle ABF} = \\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{3}"
} | {
"split": "train",
"index": 21791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If exactly three of the balls match the numbers of their boxes, calculate the number of different ways to place the balls. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the following propositions, the true one is numbered \\_\\_\\_\\_\\_\\_. \n(1) The negation of the proposition \"For all $x>0$, $x^2-x\\leq0$\" is \"There exists an $x>0$ such that $x^2-x>0$.\" \n(2) If $A>B$, then $\\sin A > \\sin B$. \n(3) Given a sequence $\\{a_n\\}... | math | {
"style": "rule",
"ground_truth": "(1)"
} | {
"split": "train",
"index": 21793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw 3 squares $\\square_{1}, \\square_{2}, \\square_{3}$ on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set $\\{1,2,3,4,5\\}$ such that no ... | math | {
"style": "rule",
"ground_truth": "541"
} | {
"split": "train",
"index": 21794,
"solution": "Relabel $a_{1}, a_{2}, a_{3}$ as $a, b, c$. This is minimized at $x=\\frac{-b}{2 a}$, so $M=c-\\frac{b^{2}}{4 a}$. If in the end $a=5$ or $b \\in\\{1,2\\}$, then $\\frac{b^{2}}{4 a} \\leq 1$ and $M \\geq 0$. The only way for Ana to block this is to set $b=5$, which... |
deepscaler | [
{
"role": "user",
"content": "From the numbers 1 to 200, one or more numbers were selected to form a group with the following property: if the group contains at least two numbers, then the sum of any two numbers in this group is divisible by 5. What is the maximum number of numbers that can be in the group ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 21795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chris received a mark of $50 \\%$ on a recent test. Chris answered 13 of the first 20 questions correctly. Chris also answered $25 \\%$ of the remaining questions on the test correctly. If each question on the test was worth one mark, how many questions in total were on the tes... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 21796,
"solution": "Suppose that there were $n$ questions on the test. Since Chris received a mark of $50 \\%$ on the test, then he answered $\\frac{1}{2} n$ of the questions correctly. We know that Chris answered 13 of the first 20 questions correctly and then $25 \\%$ of the remaini... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{v}$ be a vector such that\n\\[\\left\\| \\mathbf{v} + \\begin{pmatrix} 4 \\\\ -2 \\end{pmatrix} \\right\\| = 10.\\]\nFind the smallest possible value of $\\|\\mathbf{v}\\|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 - 2\\sqrt{5}"
} | {
"split": "train",
"index": 21797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb{Q}$ be the set of rational numbers. A function $f: \\mathbb{Q} \\to \\mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y \\in \\mathbb{Q}$,\n\\[ f(x+f(y)) = f(x) + y \\quad \\text{or} \\quad f(f(x)+y) = x + f(y). \\]\nShow that there ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 21798,
"solution": "\nLet \\( \\mathbb{Q} \\) be the set of rational numbers. We have a function \\( f: \\mathbb{Q} \\to \\mathbb{Q} \\) that satisfies the property such that for every \\( x, y \\in \\mathbb{Q} \\):\n\n\\[ f(x+f(y)) = f(x) + y \\quad \\text{or} \\quad f(f(x)+y) = x + ... |
deepscaler | [
{
"role": "user",
"content": "Calculate the probability that the line $y=kx+k$ intersects with the circle ${{\\left( x-1 \\right)}^{2}}+{{y}^{2}}=1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"split": "train",
"index": 21799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle $\\triangle ABC$, $\\angle A = 45^\\circ$ and $\\angle B = 60^\\circ$. A line segment $DE$, with $D$ on $AB$ and $\\angle ADE = 30^\\circ$, divides $\\triangle ABC$ into two pieces of equal area. Determine the ratio $\\frac{AD}{AB}$.\nA) $\\frac{\\sqrt{3}}{4}$\nB)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 21800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe area of triangle \\(ABC\\) is 1. Points \\(B'\\), \\(C'\\), and \\(A'\\) are placed respectively on the rays \\(AB\\), \\(BC\\), and \\(CA\\) such that:\n\n\\[ BB' = 2 AB, \\quad CC' = 3 BC, \\quad AA' = 4 CA. \\]\n\nCalculate the area of triangle \\(A'B'C'\\). Let's thin... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 21801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\[\\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\csc x = 7,\\]then find $\\sin 2x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22 - 8 \\sqrt{7}"
} | {
"split": "train",
"index": 21802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α∈(0,π)$, and $\\sin α + \\cos α = \\frac{\\sqrt{2}}{2}$, find the value of $\\sin α - \\cos α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 21803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{n = 2}^\\infty \\frac{4n^3 - n^2 - n + 1}{n^6 - n^5 + n^4 - n^3 + n^2 - n}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 21804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Below is a portion of the graph of an invertible function, $y=f(x)$:\n\n[asy]\nimport graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=3.25,ymin=-6.25,ymax=7.25;\n\npen cqcqcq=rgb(0.75,0.75,0.75);\n\n/*grid... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 21805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a box, there are 10 balls of the same size, among which 3 are labeled with 1, 4 are labeled with 2, and 3 are labeled with 5. First, a ball is randomly drawn from the box and then put back. After that, another ball is randomly drawn (assuming the probability of drawing each ... | math | {
"style": "rule",
"ground_truth": "5.2"
} | {
"split": "train",
"index": 21806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, with the origin as the pole and the x-axis as the positive semi-axis, a polar coordinate system is established. The polar equation of circle C is $\\rho=6\\cos\\theta$, and the parametric equation of line $l$ is \n$$\n\\begin{cases} \nx=3+ \\... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 21807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ and $n$ satisfy $mn = 6$ and $m+n = 7$. Additionally, suppose $m^2 - n^2 = 13$. Find the value of $|m-n|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{7}"
} | {
"split": "train",
"index": 21808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $n$ if $2^{n}=8^{20}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 21809,
"solution": "Since $8=2 \\times 2 \\times 2=2^{3}$, then $8^{20}=\\left(2^{3}\\right)^{20}=2^{3 \\times 20}=2^{60}$. Thus, if $2^{n}=8^{20}$, then $n=60$."
} |
deepscaler | [
{
"role": "user",
"content": "A food factory regularly purchases flour. It is known that the factory needs 6 tons of flour per day, the price of each ton of flour is 1800 yuan, and the storage and other costs for flour are an average of 3 yuan per ton per day. Each time flour is purchased, a shipping fee of... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two lines touch a circle with center \\(O\\) at points \\(A\\) and \\(B\\) and intersect at point \\(C\\). Find the angle between these lines if \\(\\angle ABO = 40^{\\circ}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 21811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of points equidistant from a circle and two parallel tangents to the circle is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 21812,
"solution": "To solve this problem, we need to understand the geometric configuration and the properties of points equidistant from a circle and two parallel tangents.\n\n1. **Understanding the Configuration**:\n - Consider a circle with center $O$ and radius $r$.\n - Let t... |
deepscaler | [
{
"role": "user",
"content": "Lines $l$ and $m$ are parallel to each other. $m\\angle A = 120^\\circ$, and $m\\angle B = 150^\\circ$. What is the number of degrees in $m\\angle C$?\n\n[asy]\nsize(100); real h = 1.2; currentpen = fontsize(10pt);\ndraw(Label(\"$l$\",Relative(1)),(0,0)--(1,0),E);\ndraw(Label(\... | math | {
"style": "rule",
"ground_truth": "90^\\circ"
} | {
"split": "train",
"index": 21813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $1-0.\\overline{9}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 21814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $64x^6-729y^6$ can be factored as $(ax+by)(cx^2+dxy+ey^2)(fx+gy)(hx^2+jxy+ky^2)$. If $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, $j$, and $k$ are all integers, find their sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 21815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ has right angles at $B$ and $D$. The length of the diagonal $AC$ is $5$. If two sides of $ABCD$ have integer lengths and one of these lengths is an odd integer, determine the area of $ABCD$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the largest real solution to the equation\n\\[\\dfrac{3}{x-3} + \\dfrac{5}{x-5} + \\dfrac{17}{x-17} + \\dfrac{19}{x-19} = x^2 - 11x - 4\\]There are positive integers $a, b,$ and $c$ such that $m = a + \\sqrt{b + \\sqrt{c}}$. Find $a+b+c$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "263"
} | {
"split": "train",
"index": 21817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? 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": "\\dfrac{7}{64}"
} | {
"split": "train",
"index": 21818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1, 3, 4, 5, 6,$ and $9$. What is the sum of the possible values for $w$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 21819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hypotenuse of a right triangle is $10$ inches and the radius of the inscribed circle is $1$ inch. The perimeter of the triangle in inches is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 21820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)\\left(-12\\right)-5+\\left(-14\\right)-\\left(-39\\right)$;<br/>$(2)-2^{2}\\times 5-\\left(-12\\right)\\div 4-4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-21"
} | {
"split": "train",
"index": 21821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3, or 7? Express your answer as a common fraction. Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{107}{150}"
} | {
"split": "train",
"index": 21822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y^{2}=2x$ with focus $F$, a line passing through $F$ intersects the parabola at points $A$ and $B$. If $|AB|= \\frac{25}{12}$ and $|AF| < |BF|$, determine the value of $|AF|$. 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": 21823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the squares of the coefficients of $4(x^4 + 3x^2 + 1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "176"
} | {
"split": "train",
"index": 21824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of a rectangle is three times its width. Given that its perimeter and area are both numerically equal to $k>0$, find $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{64}{3}"
} | {
"split": "train",
"index": 21825,
"solution": "Let $a$ be the width of the rectangle. Then the length of the rectangle is $3 a$, so the perimeter is $2(a+3 a)=8 a$, and the area is $3 a^{2}$. Since the length is numerically equal to the width, we know that $$8 a=3 a^{2}=k$$ Because $k>0$, the rectangle is non-d... |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of\n$$\n\\sqrt{2018 \\times 2021 \\times 2022 \\times 2023 + 2024^{2}} - 2024^{2}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-12138"
} | {
"split": "train",
"index": 21826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find one third of 5.4, expressed as a simplified fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{5}"
} | {
"split": "train",
"index": 21827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)=\\sin^{2}x+ \\sqrt {3}\\sin x\\cos x+2\\cos^{2}x，x∈R$$.\n(I) Find the smallest positive period and the interval where the function is monotonically increasing;\n(II) Find the maximum value of the function on the interval $$[- \\frac {π}{3}， \\frac {π}{... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}+3}{2}"
} | {
"split": "train",
"index": 21828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a set of data 3, 4, 5, a, b with an average of 4 and a median of m, where the probability of selecting the number 4 from the set 3, 4, 5, a, b, m is $\\frac{2}{3}$, calculate the variance of the set 3, 4, 5, a, b. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 21829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify \\[\\frac{1}{\\dfrac{2}{\\sqrt{5}+2} + \\dfrac{3}{\\sqrt{7}-2}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5} + \\sqrt{7} + 2}{23 + 4\\sqrt{35}}"
} | {
"split": "train",
"index": 21830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A = \\{1, 2, 3, 4, 5, 6\\} \\). Find the number of distinct functions \\( f: A \\rightarrow A \\) such that \\( f(f(f(n))) = n \\) for all \\( n \\in A \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 21831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A volleyball net is in the shape of a rectangle with dimensions of $50 \\times 600$ cells.\n\nWhat is the maximum number of strings that can be cut so that the net does not fall apart into pieces? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30000"
} | {
"split": "train",
"index": 21832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four students are admitted to three universities. Find the probability that each university admits at least one student. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 21833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many solutions does the system have: $ \\{\\begin{matrix}&(3x+2y) *(\\frac{3}{x}+\\frac{1}{y})=2 & x^2+y^2\\leq 2012 \\end{matrix} $ where $ x,y $ are non-zero integers Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 21834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the third quadrant, and $f(\\alpha)=\\frac{{\\sin({\\frac{\\pi}{2}-\\alpha})\\cos(-\\alpha)\\tan(\\pi+\\alpha)}}{{\\cos(\\pi-\\alpha)}}$. \n$(1)$ Simplify $f(\\alpha)$; \n$(2)$ If $f(\\alpha)=\\frac{{2\\sqrt{5}}}{5}$, find the value of $\\c... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 21835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. What is the sum of all possible values of $|a-d|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 21836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the New-Vasyuki currency exchange, 11 tugriks are traded for 14 dinars, 22 rupees for 21 dinars, 10 rupees for 3 thalers, and 5 crowns for 2 thalers. How many tugriks can be exchanged for 13 crowns? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 21837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(x^2+1)(2x+1)^9=a_0+a_1(x+2)+a_2(x+2)^2+\\ldots+a_{11}(x+2)^{11}, calculate the sum of the coefficients a_0 through a_11. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 21838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rhombus \\(ABCD\\), the angle \\(BCD\\) is \\(135^{\\circ}\\), and the sides are 8. A circle touches the line \\(CD\\) and intersects side \\(AB\\) at two points located 1 unit away from \\(A\\) and \\(B\\). Find the radius of this circle. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{41 \\sqrt{2}}{16}"
} | {
"split": "train",
"index": 21839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\log p+\\log q=\\log(p+q)$ only if: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "p=\\frac{q}{q-1}"
} | {
"split": "train",
"index": 21840,
"solution": "1. **Start with the given equation and apply logarithmic properties:**\n \\[\n \\log p + \\log q = \\log(p+q)\n \\]\n Using the property that $\\log a + \\log b = \\log(ab)$, we can rewrite the left side:\n \\[\n \\log(pq) = \\log(p+q)\n \\]\n\n2. **S... |
deepscaler | [
{
"role": "user",
"content": "Two lines with slopes $-\\frac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 21841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $9 \\cdot (-5) - (7 \\cdot -2) + (8 \\cdot -6)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-79"
} | {
"split": "train",
"index": 21842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n$, let, $\\tau(n)$ be the number of positive integer divisors of $n$. How many integers $1 \\leq n \\leq 50$ are there such that $\\tau(\\tau(n))$ is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 21843,
"solution": "Note that $\\tau(n)$ is odd if and only if $n$ is a perfect square. Thus, it suffices to find the number of integers $n$ in the given range such that $\\tau(n)=k^{2}$ for some positive integer $k$. If $k=1$, then we obtain $n=1$ as our only solution. If $k=2$, we s... |
deepscaler | [
{
"role": "user",
"content": "\nA high-tech Japanese company has presented a unique robot capable of producing construction blocks that can be sold for 90 monetary units each. Due to a shortage of special chips, it is impossible to replicate or even repair this robot if it goes out of order in the near futu... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 21844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \\ge x^n+y^n+z^n$$ holds. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "n \\ge 0"
} | {
"split": "train",
"index": 21845,
"solution": "\nWe are given the conditions \\( \\max(a, b, c, x, y, z) = a \\), \\( a + b + c = x + y + z \\), and \\( abc = xyz \\). We need to find all integer \\( n \\) such that the inequality\n\\[\na^n + b^n + c^n \\ge x^n + y^n + z^n\n\\]\nholds for any positive real numb... |
deepscaler | [
{
"role": "user",
"content": "Annie and Bonnie are running laps around a $400$-meter oval track. They started together, but Annie has pulled ahead, because she runs $25\\%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 21846,
"solution": "\n#### Step-by-step Analysis:\n\n1. **Understanding the Problem:**\n Annie and Bonnie start running together on a 400-meter track. Annie runs 25% faster than Bonnie. We need to determine how many laps Annie will have run when she first passes Bonnie.\n\n2. **Sett... |
deepscaler | [
{
"role": "user",
"content": "Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 21847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the parallelogram \\(ABCD\\), the longer side \\(AD\\) is 5. The angle bisectors of angles \\(A\\) and \\(B\\) intersect at point \\(M\\).\n\nFind the area of the parallelogram, given that \\(BM = 2\\) and \\(\\cos \\angle BAM = \\frac{4}{5}\\). Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 21848,
"solution": ""
} |
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