data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The three-digit integer $63\\underline{\\hphantom{0}}$ is a multiple of 3. What is the greatest possible difference between two of the possibilities for the units digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 27749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x=\\frac{1-i\\sqrt{3}}{2}$ where $i=\\sqrt{-1}$, then $\\frac{1}{x^2-x}$ is equal to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 27750,
"solution": "To solve for $\\dfrac{1}{x^2-x}$ given $x=\\dfrac{1-i\\sqrt{3}}{2}$, we first need to compute $x^2$ and $x^2 - x$.\n\n1. **Calculate $x^2$:**\n \\[\n x^2 = \\left(\\frac{1-i\\sqrt{3}}{2}\\right)^2 = \\frac{(1-i\\sqrt{3})^2}{4} = \\frac{1 - 2i\\sqrt{3} - 3}{4} =... |
deepscaler | [
{
"role": "user",
"content": "A series of numbers were written: \\(100^{100}, 101^{101}, 102^{102}, \\ldots, 234^{234}\\) (i.e., the numbers of the form \\(n^{n}\\) for natural \\(n\\) from 100 to 234). How many of the numbers listed are perfect squares? (A perfect square is defined as the square of an inte... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 27751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the slope of the line passing through $(-3,5)$ and $(2,-5)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 27752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a WeChat group, members A, B, C, D, and E simultaneously grab 4 red envelopes. Each person can grab at most one red envelope, and all red envelopes are grabbed. Among the 4 red envelopes, there are two worth 2 yuan and two worth 3 yuan (red envelopes with the same amount are... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 27753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ has $AB=8$ and $BC=6$. Point $M$ is the midpoint of diagonal $\\overline{AC}$, and $E$ is on $AB$ with $\\overline{ME} \\perp \\overline{AC}$. What is the area of $\\triangle AME$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{75}{8}"
} | {
"split": "train",
"index": 27754,
"solution": "1. **Calculate the length of diagonal $AC$**:\n Since $ABCD$ is a rectangle, by the Pythagorean Theorem, we have:\n \\[\n AC = \\sqrt{AB^2 + BC^2} = \\sqrt{8^2 + 6^2} = \\sqrt{64 + 36} = \\sqrt{100} = 10.\n \\]\n\n2. **Determine the length of $AM$ and $MC$*... |
deepscaler | [
{
"role": "user",
"content": "If \\( a=\\frac{2}{3}b \\) and \\( b \\neq 0 \\), what is \\( \\frac{9a+8b}{6a} \\) equal to? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 27755,
"solution": "Since \\( a=\\frac{2}{3}b \\), then \\( 3a=2b \\). Since \\( b \\neq 0 \\), then \\( a \\neq 0 \\). Thus, \\( \\frac{9a+8b}{6a}=\\frac{9a+4(2b)}{6a}=\\frac{9a+4(3a)}{6a}=\\frac{21a}{6a}=\\frac{7}{2} \\). Alternatively, \\( \\frac{9a+8b}{6a}=\\frac{3(3a)+8b}{2(3a)}=... |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan (\\theta-\\pi)=2$, then $\\sin ^{2}\\theta+\\sin \\theta\\cos \\theta-2\\cos ^{2}\\theta=$ \\_\\_\\_\\_\\_\\_ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4}{5}"
} | {
"split": "train",
"index": 27756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the share of the Chinese yuan in the currency structure of the National Wealth Fund (NWF) as of 01.07.2021 by one of the following methods:\n\nFirst method:\na) Find the total amount of NWF funds placed in Chinese yuan as of 01.07.2021:\n\\[ CNY_{22} = 1213.76 - 3.36 ... | math | {
"style": "rule",
"ground_truth": "1.2"
} | {
"split": "train",
"index": 27757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system, the length of the chord cut by the ray $θ= \\dfrac {π}{4}$ on the circle $ρ=4\\sin θ$ 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": 27758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the positive numbers $a$ and $b$ satisfy the equation $2+\\log_{2}a=3+\\log_{3}b=\\log_{6}(a+b)$, find the value of $\\frac{1}{a}+\\frac{1}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 27759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the line $18x + 9y = 162$ forming a triangle with the coordinate axes. Calculate the sum of the lengths of the altitudes of this triangle.\nA) 21.21 \nB) 42.43 \nC) 63.64 \nD) 84.85 \nE) 105.06 Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "42.43"
} | {
"split": "train",
"index": 27760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 27761,
"solution": "\nTo solve this problem, we need to determine the probability that the product of two randomly drawn chips, one from each box, is even. Each box contains chips numbered $1$, $2$, and $3$. \n\n#### Step 1: Determine Total Outcomes\nEach box has $3$ chips, and a chip... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( f(x-1)=|x|-|x-2| \\) and \\( f(f(m))=f(2002)-\\frac{7}{2} \\), find the value of the real number \\( m \\). 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": 27762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[\\mathbf{A} = \\renewcommand{\\arraystretch}{1.5} \\begin{pmatrix} \\frac{\\sqrt{3}}{2} & 0 & -\\frac{1}{2} \\\\ 0 & -1 & 0 \\\\ \\frac{1}{2} & 0 & \\frac{\\sqrt{3}}{2} \\end{pmatrix} \\renewcommand{\\arraystretch}{1}.\\]Compute $\\mathbf{A}^{2018}.$ Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} \\frac{1}{2} & 0 & -\\frac{\\sqrt{3}}{2} \\\\ 0 & 1 & 0 \\\\ \\frac{\\sqrt{3}}{2} & 0 & \\frac{1}{2} \\end{pmatrix}"
} | {
"split": "train",
"index": 27763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the m... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 27764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{52}"
} | {
"split": "train",
"index": 27765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On this monthly calendar, the date behind one of the letters is added to the date behind $\\text{C}$. If this sum equals the sum of the dates behind $\\text{A}$ and $\\text{B}$, then the letter is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "P"
} | {
"split": "train",
"index": 27766,
"solution": "1. Let the date behind $C$ be $x$. This means the date behind $C$ is $x$.\n\n2. From the problem statement, the date behind $A$ is $x+1$ and the date behind $B$ is $x+13$. This is derived from the calendar setup where $A$ and $B$ are sequentially after $C$.\n\n3. T... |
deepscaler | [
{
"role": "user",
"content": "Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5$, no collection of $k$ pairs made by... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 27767,
"solution": "Label the left shoes be $L_1,\\dots, L_{10}$ and the right shoes $R_1,\\dots, R_{10}$. Notice that there are $10!$ possible pairings.\n Let a pairing be \"bad\" if it violates the stated condition. We would like a better condition to determine if a given pairing is... |
deepscaler | [
{
"role": "user",
"content": "Let $d$ and $e$ denote the solutions of $3x^2+10x-25=0$. Find $(d-e)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{400}{9}"
} | {
"split": "train",
"index": 27768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measures of a pair of supplementary angles are in the ratio of 7:2. How many degrees are in the measure of their positive difference? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100^\\circ"
} | {
"split": "train",
"index": 27769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Remove five out of twelve digits so that the remaining numbers sum up to 1111.\n\n$$\n\\begin{array}{r}\n111 \\\\\n333 \\\\\n+\\quad 777 \\\\\n999 \\\\\n\\hline 1111\n\\end{array}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1111"
} | {
"split": "train",
"index": 27770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Radii of five concentric circles $\\omega_0,\\omega_1,\\omega_2,\\omega_3,\\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equ... | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 27771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficients of the polynomial \\(P(x)\\) are nonnegative integers, each less than 100. Given that \\(P(10)=331633\\) and \\(P(-10)=273373\\), compute \\(P(1)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 27772,
"solution": "Let \\(P(x)=a_{0}+a_{1}x+a_{2}x^{2}+\\ldots\\). Then \\(\\frac{1}{2}(P(10)+P(-10))=a_{0}+100a_{2}+\\ldots\\) and \\(\\frac{1}{2}(P(10)-P(-10))=10a_{1}+1000a_{3}+\\ldots\\). Since all the coefficients are nonnegative integers, these expressions give us each of the c... |
deepscaler | [
{
"role": "user",
"content": "Two positive integers differ by 6 and their product is 135. What is the larger integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 27773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the data presented in the chart, what was the average daily high temperature in Brixton from September 15th, 2008 to September 21st, 2008, inclusive? The daily high temperatures in degrees Fahrenheit were recorded as 51, 64, 61, 59, 48, 63, and 55. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "57.3"
} | {
"split": "train",
"index": 27774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a polynomial with integer coefficients such that $P(0)+P(90)=2018$. Find the least possible value for $|P(20)+P(70)|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "782"
} | {
"split": "train",
"index": 27775,
"solution": "First, note that $P(x)=x^{2}-3041$ satisfy the condition and gives $|P(70)+P(20)|=|4900+400-6082|=$ 782. To show that 782 is the minimum, we show $2800 \\mid P(90)-P(70)-P(20)+P(0)$ for every $P$, since -782 is the only number in the range $[-782,782]$ that is cong... |
deepscaler | [
{
"role": "user",
"content": "The first few rows of a new sequence are given as follows:\n- Row 1: $3$\n- Row 2: $6, 6, 6, 6$\n- Row 3: $9, 9, 9, 9, 9, 9$\n- Row 4: $12, 12, 12, 12, 12, 12, 12, 12$\n\nWhat is the value of the $40^{\\mathrm{th}}$ number if this arrangement were continued? Let's think step by... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 27776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $1000$ are either a perfect cube or a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 27777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fill the five numbers $2015, 2016, 2017, 2018, 2019$ into the five boxes labeled \" $D, O, G, C, W$ \" such that $D+O+G=C+O+W$. How many different ways can this be done? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 27778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S=\\{1,2, \\ldots, 9\\}$. Compute the number of functions $f: S \\rightarrow S$ such that, for all $s \\in S, f(f(f(s)))=s$ and $f(s)-s$ is not divisible by 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 27779,
"solution": "Since $f(f(f(s)))=s$ for all $s \\in S$, each cycle in the cycle decomposition of $f$ must have length 1 or 3. Also, since $f(s) \\not \\equiv s \\bmod 3$ for all $s \\in S$, each cycle cannot contain two elements $a, b$ such that $a=b \\bmod 3$. Hence each cycle h... |
deepscaler | [
{
"role": "user",
"content": "Points $A, C$, and $B$ lie on a line in that order such that $A C=4$ and $B C=2$. Circles $\\omega_{1}, \\omega_{2}$, and $\\omega_{3}$ have $\\overline{B C}, \\overline{A C}$, and $\\overline{A B}$ as diameters. Circle $\\Gamma$ is externally tangent to $\\omega_{1}$ and $\\om... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 27780,
"solution": "Let the center of $\\omega_{i}$ be $O_{i}$ for $i=1,2,3$ and let $O$ denote the center of $\\Gamma$. Then $O, D$, and $O_{1}$ are collinear, as are $O, E$, and $O_{2}$. Denote by $F$ the point of tangency between $\\Gamma$ and $\\omega_{3}$; then $F, O$, and $O_{3}... |
deepscaler | [
{
"role": "user",
"content": "A box contains 4 white balls and 4 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{35}"
} | {
"split": "train",
"index": 27781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\sqrt{3}b\\cos A - a\\sin B = 0$. $D$ is the midpoint of $AB$, $AC = 2$, and $CD = 2\\sqrt{3}$. Find:\n\n$(Ⅰ)$ The measure of angle $A$;\n\n$(Ⅱ)$ The value of ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{13}"
} | {
"split": "train",
"index": 27782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An environmental agency needs to hire a number of new employees so that 85 of the new employees will be able to monitor water pollution, 73 of the new employees will be able to monitor air pollution, and exactly 27 of the new employees will be able to monitor both. (These 27 ar... | math | {
"style": "rule",
"ground_truth": "131"
} | {
"split": "train",
"index": 27783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand $(x+10)(2y+10)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2xy + 10x + 20y + 100"
} | {
"split": "train",
"index": 27784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $2027$ is prime. Let $T = \\sum \\limits_{k=0}^{72} \\binom{2024}{k}$. What is the remainder when $T$ is divided by $2027$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1369"
} | {
"split": "train",
"index": 27785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The roots of the equation $x^2 + kx + 8 = 0$ differ by 10. Find the greatest possible value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{33}"
} | {
"split": "train",
"index": 27786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has a perimeter of 64 inches and each side has an integer length. How many non-congruent rectangles meet these criteria? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 27787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let set $\\mathcal{A}$ be a 90-element subset of $\\{1,2,3,\\ldots,100\\},$ and let $S$ be the sum of the elements of $\\mathcal{A}.$ Find the number of possible values of $S.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "901"
} | {
"split": "train",
"index": 27788,
"solution": "The smallest $S$ is $1+2+ \\ldots +90 = 91 \\cdot 45 = 4095$. The largest $S$ is $11+12+ \\ldots +100=111\\cdot 45=4995$. All numbers between $4095$ and $4995$ are possible values of S, so the number of possible values of S is $4995-4095+1=901$.\n Alternatively, fo... |
deepscaler | [
{
"role": "user",
"content": "What is the largest value of $x$ that satisfies the equation $\\sqrt{2x}=4x$? 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": "\\frac18"
} | {
"split": "train",
"index": 27789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest \\( n \\) such that the sum of the fourth powers of any \\( n \\) prime numbers greater than 10 is divisible by \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 27790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a kingdom of animals, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 animals of each kind, divided into 100 groups, with each group containing exactly 2 animals of one kind and 1 animal of another kind. A... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 27791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the projection of $\\begin{pmatrix} 1 \\\\ 2 \\\\ 3 \\end{pmatrix}$ onto the plane $3x - y + 4z = 0.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -1/2 \\\\ 5/2 \\\\ 1 \\end{pmatrix}"
} | {
"split": "train",
"index": 27792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrange... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 27793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a + 4b = 33$ and $6a + 3b = 51$, what is the value of $a + b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $6$ -inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inscribed in a right cone with base radius \\(15\\) cm and height \\(30\\) cm. Find the radius \\(r\\) of the sphere, which can be expressed as \\(b\\sqrt{d} - b\\) cm. What is the value of \\(b + d\\)? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"split": "train",
"index": 27796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\\triangle ABC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 27797,
"solution": "\n#### Step 1: Understanding the problem\nWe are given a triangle $ABC$ with vertices $A = (3,0)$, $B = (0,3)$, and $C$ on the line $x + y = 7$. We need to find the area of $\\triangle ABC$.\n\n#### Step 2: Analyzing the line $x + y = 7$\nThe line $x + y = 7$ is pa... |
deepscaler | [
{
"role": "user",
"content": "A rigid board with a mass of \\(m\\) and a length of \\(l = 24\\) meters partially lies on the edge of a horizontal surface, hanging off by two-thirds of its length. To prevent the board from falling, a stone with a mass of \\(2m\\) is placed at its very edge. How far from the ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 27798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\\mathcal{T}$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "413.5"
} | {
"split": "train",
"index": 27799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "550"
} | {
"split": "train",
"index": 27800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the sum of the absolute values of the pairwise differences of five nonnegative numbers is equal to one. Find the smallest possible sum of these numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 27801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find, as a function of $\\, n, \\,$ the sum of the digits of \\[9 \\times 99 \\times 9999 \\times \\cdots \\times \\left( 10^{2^n} - 1 \\right),\\] where each factor has twice as many digits as the previous one. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "9 \\cdot 2^n"
} | {
"split": "train",
"index": 27802,
"solution": "The answer is $9 \\cdot 2^n$ .\nLet us denote the quantity $\\prod_{k=0}^n \\bigl( 10^{2^k}-1 \\bigr)$ as $P_n$ . We wish to find the sum of the digits of $P_n$ .\nWe first note that \\[P_{n-1} < \\prod_{k=0}^{n-1} 10^{2^k} = 10^{2^n-1},\\] so $P_{n-1}$ is a numbe... |
deepscaler | [
{
"role": "user",
"content": "If one vertex of an ellipse and its two foci form the vertices of an equilateral triangle, then find the eccentricity $e$ of the ellipse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 27803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Gang goes to buy milk and finds that it's on special offer that day: each bag costs 2.5 yuan, and there is a \"buy two, get one free\" promotion. Xiao Gang has 30 yuan. What is the maximum number of bags of milk he can buy? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 27804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 3 teachers who have all assigned homework. Determine the number of possible situations where 4 students are doing homework at the same time. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3^{4}"
} | {
"split": "train",
"index": 27805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers in the sequence $101$, $104$, $109$, $116$,$\\ldots$ are of the form $a_n=100+n^2$, where $n=1,2,3,\\ldots$ For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.\n ... | math | {
"style": "rule",
"ground_truth": "401"
} | {
"split": "train",
"index": 27806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{M}$ be a matrix such that\n\\[\\mathbf{M} \\begin{pmatrix} 2 \\\\ -1 \\end{pmatrix} = \\begin{pmatrix} 3 \\\\ 0 \\end{pmatrix} \\quad \\text{and} \\quad \\mathbf{M} \\begin{pmatrix} -3 \\\\ 5 \\end{pmatrix} = \\begin{pmatrix} -1 \\\\ -1 \\end{pmatrix}.\\]Compute $... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 11 \\\\ -1 \\end{pmatrix}"
} | {
"split": "train",
"index": 27807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\\\(f(x)\\\\) is an odd function defined on \\\\(R\\\\) and satisfies \\\\(f(x+1)=f(x-1)\\\\), and when \\\\(x \\in (0,1)\\\\), \\\\(f(x)=2^{x}-2\\\\), then the value of \\\\(f(\\log_{\\frac{1}{2}}24)\\\\) is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 27808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a mathematics competition conducted at a school, the scores $X$ of all participating students approximately follow the normal distribution $N(70, 100)$. It is known that there are 16 students with scores of 90 and above (inclusive of 90). \n(1) What is the approximate total... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 27809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mia is \"helping\" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 27810,
"solution": "1. **Determine the net increase of toys in the box per cycle**: Every 30 seconds, Mia's mom puts 3 toys into the box, and Mia takes 2 toys out. Therefore, the net increase in the number of toys in the box per 30-second cycle is:\n \\[\n 3 - 2 = 1 \\text{ toy}\n... |
deepscaler | [
{
"role": "user",
"content": "Let \\( T = 3 \\times ((1 + i)^{15} - (1 - i)^{15}) \\), where \\( i = \\sqrt{-1} \\). Calculate \\( |T| \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 27811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rectangular box-shaped fort with dimensions 15 feet in length, 12 feet in width, and 6 feet in height, and walls and floor uniformly 1.5 feet thick, calculate the total number of one-foot cubical blocks used to build the fort. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "594"
} | {
"split": "train",
"index": 27812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 27813,
"solution": "1. **Identify the constraints for the professors' seating**: Professors Alpha, Beta, and Gamma must each be seated between two students. This means they cannot occupy the first or last chair in the row, as these positions do not allow a student to be seated on both... |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system xOy, there is a line l₁: x = 2, and a curve C: {x = 2cosϕ, y = 2 + 2sinϕ} (where ϕ is a parameter). With O as the pole and the non-negative half-axis of the x-axis as the polar axis, establish a polar coordinate system. The polar coordinates... | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 27814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\frac{x+3}{x^2+1}$, and $g(x) = x - \\ln(x-p)$.\n(I) Find the equation of the tangent line to the graph of $f(x)$ at the point $\\left(\\frac{1}{3}, f\\left(\\frac{1}{3}\\right)\\right)$;\n(II) Determine the number of zeros of the function $g(x)$, an... | math | {
"style": "rule",
"ground_truth": "6044"
} | {
"split": "train",
"index": 27815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The government decided to privatize civil aviation. For each of the 127 cities in the country, the connecting airline between them is sold to one of the private airlines. Each airline must make all acquired airlines one-way but in such a way as to ensure the possibility of trav... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 27816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four male students and five female students are lined up in a row. Calculate the number of different arrangements with alternating male and female students. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"split": "train",
"index": 27817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two positive integers, \\(A\\) and \\(B\\). The sum of the digits of \\(A\\) is \\(19\\), the sum of the digits of \\(B\\) is \\(20\\), and their addition results in carrying over twice. What is the sum of the digits of \\((\\mathbf{A} + B)\\)? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 27818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a line and a point at a distance of $1 \\mathrm{~cm}$ from the line. What is the volume of a cube where the point is one of the vertices and the line is one of the body diagonals? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{6}}{4}"
} | {
"split": "train",
"index": 27819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a die is rolled consecutively three times, determine the probability that the sequence of numbers facing upwards forms an arithmetic progression. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 27820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of \\(x^3y^5\\) in the expansion of \\(\\left(\\frac{2}{3}x - \\frac{4}{5}y\\right)^8\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{458752}{84375}"
} | {
"split": "train",
"index": 27821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cross, consisting of two identical large squares and two identical small squares, was placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is 810 cm². Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 27822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a 3'' by 3'' square is added at each successive stage, what will be the area of the rectangle at Stage 6, in square inches?\n\n[asy]size(250); real textsize = 10pt;\ndraw(unitsquare);\ndraw(shift(1.5*right)*unitsquare);\ndraw(shift(2.5*right)*unitsquare);\ndraw(shift(4*right... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 27823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n, k_1, \\dots, k_n$ such that $k_1 + \\cdots + k_n = 5n-4$ and \\[ \\frac{1}{k_1} + \\cdots + \\frac{1}{k_n} = 1. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n = 1, k_1 = 1; n = 3, (k_1,k_2,k_3) = (2,3,6); n = 4, (k_1,k_2,k_3,k_4) = (4,4,4,4)"
} | {
"split": "train",
"index": 27824,
"solution": "By the arithmetic-harmonic mean inequality or the Cauchy-Schwarz inequality, \\[ (k_1 + \\cdots + k_n)\\left(\\frac{1}{k_1} + \\cdots + \\frac{1}{k_n} \\right) \\geq n^2. \\] We must thus have $5n-4 \\geq n^2$, so $n \\leq 4$. Without loss of generality, we may sup... |
deepscaler | [
{
"role": "user",
"content": "In the right triangle \\(ABC\\) with \\(\\angle B = 90^\\circ\\), \\(P\\) is a point on the angle bisector of \\(\\angle A\\) inside \\(\\triangle ABC\\). Point \\(M\\) (distinct from \\(A\\) and \\(B\\)) lies on the side \\(AB\\). The lines \\(AP\\), \\(CP\\), and \\(MP\\) int... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 27825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle in the $x y$-plane is such that when projected onto the $x$-axis, $y$-axis, and the line $y=x$, the results are line segments whose endpoints are $(1,0)$ and $(5,0),(0,8)$ and $(0,13)$, and $(5,5)$ and $(7.5,7.5)$, respectively. What is the triangle's area? Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{2}"
} | {
"split": "train",
"index": 27826,
"solution": "Sketch the lines $x=1, x=5, y=8, y=13, y=10-x$, and $y=15-x$. The triangle has to be contained in the hexagonal region contained in all these lines. If all the projections are correct, every other vertex of the hexagon must be a vertex of the triangle, which gives ... |
deepscaler | [
{
"role": "user",
"content": "Three positive integers $a$, $b$, and $c$ satisfy $a\\cdot b\\cdot c=8!$ and $a<b<c$. What is the smallest possible value of $c-a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 27827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In 2019, our county built 4 million square meters of new housing, of which 2.5 million square meters are mid-to-low-priced houses. It is expected that in the coming years, the average annual increase in the area of new housing in our county will be 8% higher than the previous y... | math | {
"style": "rule",
"ground_truth": "2024"
} | {
"split": "train",
"index": 27828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a string of $n$ $7$s, $7777\\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. How many values of $n$ are possible if the inserted $+$ signs create a sum of $7350$ using groups of $7$s, $77$s, $777$s, and possibly $7777$s? Let's think ste... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 27829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number \\( a \\) is randomly chosen from \\( 1, 2, 3, \\cdots, 10 \\), and a number \\( b \\) is randomly chosen from \\( -1, -2, -3, \\cdots, -10 \\). What is the probability that \\( a^{2} + b \\) is divisible by 3? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "37/100"
} | {
"split": "train",
"index": 27830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $n \\ge 1$ call a finite sequence $(a_1, a_2 \\ldots a_n)$ of positive integers progressive if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \\le i \\le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$.... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 27831,
"solution": "If the first term is $x$, then dividing through by $x$, we see that we can find the number of progressive sequences whose sum is $\\frac{360}{x} - 1$, and whose first term is not 1. If $a(k)$ denotes the number of progressive sequences whose sum is $k$ and whose fi... |
deepscaler | [
{
"role": "user",
"content": "A $7 \\times 7$ board is either empty or contains an invisible $2 \\times 2$ ship placed \"by the cells.\" You are allowed to place detectors in some cells of the board and then activate them all at once. An activated detector signals if its cell is occupied by the ship. What i... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 27832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with its center on the line \\( y = b \\) intersects the parabola \\( y = \\frac{3}{4} x^{2} \\) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \\( y = \\frac{3}{4} x + b \\). Find all values of \\( b \\) fo... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{12}"
} | {
"split": "train",
"index": 27833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$. Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 27834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AB + AC$ and $BC = 65$ cm. $P$ is a point on $\\overline{BC}$ such that the perpendicular distances from $P$ to $\\overline{AB}$ and $\\overline{AC}$ are $24$ cm and $36$ cm, respectively. The area of $\\triangle ABC$, in cm $^2$, is Let's thin... | math | {
"style": "rule",
"ground_truth": "2535"
} | {
"split": "train",
"index": 27835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides have integer lengths and $AB=AC$. Circle $\\omega$ has its center at the incenter of $\\triangle ABC$. An excircle of $\\triangle ABC$ is a circle in the exterior of $\\triangle ABC$ that is tangent to one side of the triangle and tangent to the e... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 27836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four distinct integers $a, b, c$, and $d$ are chosen from the set $\\{1,2,3,4,5,6,7,8,9,10\\}$. What is the greatest possible value of $ac+bd-ad-bc$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 27837,
"solution": "We note that $ac+bd-ad-bc=a(c-d)-b(c-d)=(a-b)(c-d)$. Since each of $a, b, c, d$ is taken from the set $\\{1,2,3,4,5,6,7,8,9,10\\}$, then $a-b \\leq 9$ since the greatest possible difference between two numbers in the set is 9 . Similarly, $c-d \\leq 9$. Now, if $a-... |
deepscaler | [
{
"role": "user",
"content": "When two fair dice are thrown, the numbers obtained are $a$ and $b$, respectively. Express the probability that the slope $k$ of the line $bx+ay=1$ is greater than or equal to $-\\dfrac{2}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{6}"
} | {
"split": "train",
"index": 27838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers $1, 2, \\cdots, 1000$, choose $k$ numbers such that any three of the chosen numbers can form the side lengths of a triangle. Find the minimum value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 27839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the conditions $a+acosC=\\sqrt{3}csinA$, $\\left(a+b+c\\right)\\left(a+b-c\\right)=3ab$, $\\left(a-b\\right)\\sin \\left(B+C\\right)+b\\sin B=c\\sin C$. Choose any one of these three conditions and complete the following question, then solve it. In triangle $\\triangle AB... | math | {
"style": "rule",
"ground_truth": "6 + 4\\sqrt{2}"
} | {
"split": "train",
"index": 27840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\theta$ is an angle in the second quadrant and $\\tan(\\theta + \\frac{\\pi}{4}) = \\frac{1}{2}$, find the value of $\\sin\\theta + \\cos\\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{10}}{5}"
} | {
"split": "train",
"index": 27841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $PQRS$ has sides of length 1. Points $M$ and $N$ are on $\\overline{QR}$ and $\\overline{RS},$ respectively, so that $\\triangle PMN$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\\overline{PM}.$ The length of a... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 27842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, $\\overrightarrow{a}\\cdot (\\overrightarrow{b}-\\overrightarrow{a})=0$, find the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 27843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangular prism \\(A-BCD\\), the side edges \\(AB, AC, AD\\) are mutually perpendicular. The areas of triangles \\(\\triangle ABC\\), \\(\\triangle ACD\\), and \\(\\triangle ADB\\) are \\(\\frac{\\sqrt{2}}{2}\\), \\(\\frac{\\sqrt{3}}{2}\\), and \\(\\frac{\\sqrt{6}}{2}\\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}\\pi"
} | {
"split": "train",
"index": 27844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the expanded hexagonal lattice shown below, where each point is one unit from its nearest neighbor. Determine the number of equilateral triangles whose vertices lie on this lattice.\n```asy\nsize(100);\ndot(origin);\ndot(dir(30) + dir(90));\ndot(dir(90));\ndot(dir(90) ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 27845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.\n\n[asy]\nunitsize(0.8cm);\nfor (int i=0; i<7; ++i) {\nfor (int j=0; j<4; ++j) {\ndot((i,j));\n};}\nlabel(\"$A$\",(0,3),W);\nl... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{5}}{3}"
} | {
"split": "train",
"index": 27846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What non-zero, real value of $x$ satisfies $(5x)^4= (10x)^3$? 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{8}{5}"
} | {
"split": "train",
"index": 27847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five people are crowding into a booth against a wall at a noisy restaurant. If at most three can fit on one side, how many seating arrangements accommodate them all? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 27848,
"solution": ""
} |
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