data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Calculate $3.6 \\times 0.25 + 1.5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"split": "train",
"index": 3700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α \\in (0, \\frac{π}{3})$ satisfies the equation $\\sqrt{6} \\sin α + \\sqrt{2} \\cos α = \\sqrt{3}$, find the values of:\n1. $\\cos (α + \\frac{π}{6})$\n2. $\\cos (2α + \\frac{π}{12})$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{30} + \\sqrt{2}}{8}"
} | {
"split": "train",
"index": 3701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle can be formed having side lengths 4, 5, and 8. It is impossible, however, to construct a triangle with side lengths 4, 5, and 9. Ron has eight sticks, each having an integer length. He observes that he cannot form a triangle using any three of these sticks as side le... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 3702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x \\) and \\( y \\) be real numbers with \\( x > y \\) such that \\( x^{2} y^{2} + x^{2} + y^{2} + 2xy = 40 \\) and \\( xy + x + y = 8 \\). Find the value of \\( x \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 + \\sqrt{7}"
} | {
"split": "train",
"index": 3703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers \\( n \\) between 10 and 1000 have the property that the sum of the digits of \\( n \\) is 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 3704,
"solution": "We note that the sum of the digits of 1000 is not 3. Every other positive integer in the given range has two or three digits. For the sum of the digits of an integer to be 3, no digit can be greater than 3. If a two-digit integer has sum of digits equal to 3, then i... |
deepscaler | [
{
"role": "user",
"content": "Given lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$, find the value of $a$ such that the lines are parallel. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 3705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum distance from a point on the curve y=e^{x} to the line y=x. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 3706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Selina takes a sheet of paper and cuts it into 10 pieces. She then takes one of these pieces and cuts it into 10 smaller pieces. She then takes another piece and cuts it into 10 smaller pieces and finally cuts one of the smaller pieces into 10 tiny pieces. How many pieces of pa... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 3707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{7}"
} | {
"split": "train",
"index": 3708,
"solution": "\nTo solve this problem, we need to determine the probability that three randomly chosen vertices of a cube will form a plane that intersects the interior of the cube. We will use a combinatorial approach to count the favorable and total outcomes.\n\n#### Step 1: To... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{v} = \\begin{pmatrix} 4 \\\\ -5 \\end{pmatrix}$ and $\\mathbf{w} = \\begin{pmatrix} 13 \\\\ 2 \\end{pmatrix}$. Find the area of the parallelogram formed by vectors $\\mathbf{v}$ and $2\\mathbf{w}$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "146"
} | {
"split": "train",
"index": 3709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle \\(ABCD\\), \\(\\overline{AB}=30\\) and \\(\\overline{BC}=15\\). Let \\(E\\) be a point on \\(\\overline{CD}\\) such that \\(\\angle CBE=45^\\circ\\) and \\(\\triangle ABE\\) is isosceles. Find \\(\\overline{AE}.\\) Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 3710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \\leq a \\leq 100$ and $b \\geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is ... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 3711,
"solution": "By Vietas, the sum of the roots is $-a$ and the product is $b$. Therefore, both roots are nonpositive. For each value of $a$ from $1$ to $100$, the number of $b$ values is the number of ways to sum two numbers between $0$ and $a-1$ inclusive to $a$. This is just $1 ... |
deepscaler | [
{
"role": "user",
"content": "The probability of snow for each of the next four days is $\\frac{3}{4}$. However, if it snows any day before the last day, the probability of snow on the last day increases to $\\frac{4}{5}$. What is the probability that it will snow at least once during these four days? Expre... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1023}{1280}"
} | {
"split": "train",
"index": 3712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "$19"
} | {
"split": "train",
"index": 3713,
"solution": "1. **Identify the total number of days and total cost**: Al takes the pills for two weeks, which is 14 days. The total cost for these two weeks is $\\$546$.\n\n2. **Calculate daily expenditure**: \n \\[\n \\text{Daily cost} = \\frac{\\text{Total cost}}{\\text{Nu... |
deepscaler | [
{
"role": "user",
"content": "To shift the graph of the function $y=\\sin \\left(2x- \\frac{\\pi}{4}\\right)$ to the graph of the function $y = \\sin(2x)$, determine the horizontal shift required. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"split": "train",
"index": 3714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the arithmetic sequence $\\{a_n\\}$, $S_n$ denotes the sum of its first $n$ terms. Given that $a_4 + a_8 = 4$, find the value of $S_{11} + a_6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 3715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle has a radius of 15 units. Suppose a chord in this circle bisects a radius perpendicular to the chord and the distance from the center of the circle to the chord is 9 units. What is the area of the smaller segment cut off by the chord?\nA) $100 \\pi$\nB) $117.29$\nC) $1... | math | {
"style": "rule",
"ground_truth": "117.29"
} | {
"split": "train",
"index": 3716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the next 3 days, a meteorological station forecasts the weather with an accuracy rate of 0.8. The probability that the forecast is accurate for at least two consecutive days is ___. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.768"
} | {
"split": "train",
"index": 3717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 5$, $BC = 12$, and $AC = 13$. Let $BM$ be the median from vertex $B$ to side $AC$. If $BM = m \\sqrt{2}$, then find $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{2}"
} | {
"split": "train",
"index": 3718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $.3\\overline{25}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{161}{495}"
} | {
"split": "train",
"index": 3719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many sequences of 6 digits $x_1, x_2, \\ldots, x_6$ can we form, given the condition that no two adjacent $x_i$ have the same parity? Leading zeroes are allowed. (Parity means 'odd' or 'even'; so, for example, $x_2$ and $x_3$ cannot both be odd or both be even.) Let's thin... | math | {
"style": "rule",
"ground_truth": "31,250"
} | {
"split": "train",
"index": 3720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both these circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answe... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 3721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A local community club consists of four leaders and a number of regular members. Each year, the current leaders leave the club, and every regular member is responsible for recruiting three new members. At the end of the year, four new leaders are elected from outside the club. ... | math | {
"style": "rule",
"ground_truth": "4100"
} | {
"split": "train",
"index": 3722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, ... | math | {
"style": "rule",
"ground_truth": "5979"
} | {
"split": "train",
"index": 3723,
"solution": "1. **Identify the pattern of numbers said by Tadd**: Tadd's turns involve saying an increasing number of consecutive numbers. The first few blocks of numbers Tadd says are:\n - 1st block: [1]\n - 2nd block: [7-10] (4 numbers)\n - 3rd block: [22-28] (7 numbers)... |
deepscaler | [
{
"role": "user",
"content": "If the probability of producing a Grade B product is $0.03$, and the probability of producing a Grade C product is $0.02$, calculate the probability of randomly inspecting a product and finding it to be a qualified product. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "0.95"
} | {
"split": "train",
"index": 3724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Find the constant term in the expansion of ${\\left( \\frac{1}{x}- \\sqrt{\\frac{x}{2}}\\right)}^{9}$;\n\n(2) Given that ${x}^{10}={a}_{0}+{a}_{1}\\left( x+2 \\right)+{a}_{2}{\\left( x+2 \\right)}^{2}+... +{a}_{10}{\\left( x+2 \\right)}^{10}$, find the value of ${{a}_{1}+{a... | math | {
"style": "rule",
"ground_truth": "-1023"
} | {
"split": "train",
"index": 3725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The matrices\n\\[\\begin{pmatrix} a & 1 & b \\\\ 2 & 2 & 3 \\\\ c & 5 & d \\end{pmatrix} \\quad \\text{and} \\quad \\begin{pmatrix} -5 & e & -11 \\\\ f & -13 & g \\\\ 2 & h & 4 \\end{pmatrix}\\]are inverses. Find $a + b + c + d + e + f + g + h.$ Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 3726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gauss is a famous German mathematician, known as the \"Prince of Mathematics\". There are 110 achievements named after his name \"Gauss\". For $x\\in R$, let $[x]$ represent the largest integer not greater than $x$, and let $\\{x\\}=x-[x]$ represent the non-negative fractional ... | math | {
"style": "rule",
"ground_truth": "3027+ \\sqrt{3}"
} | {
"split": "train",
"index": 3727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\left|\\frac12 - \\frac38i\\right|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac58"
} | {
"split": "train",
"index": 3728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle of radius $2$, there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"split": "train",
"index": 3729,
"solution": "1. **Identify the Geometry of the Problem:**\n Let $AB$ be a line segment of length $2$, tangent to a circle $O$ at its midpoint $P$. The radius of circle $O$, $OP$, is $2$. Since $P$ is the midpoint of $AB$, we have $AP = PB = 1$.\n\n2. **Use the Right Triangle P... |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 3730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of $x^4$ in the expansion of $(1-2x^2)^5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 3731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\lceil{\\sqrt{x}}\\rceil=15$, how many possible integer values of $x$ are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 3732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the decimal representation of $\frac{1}{7}$, the 100th digit to the right of the decimal is? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 3733,
"solution": "The digits to the right of the decimal place in the decimal representation of $\frac{1}{7}$ occur in blocks of 6, repeating the block of digits 142857. Since $16 \times 6=96$, then the 96th digit to the right of the decimal place is the last in one of these blocks; ... |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, the graph of the quadratic function $y=a\\left(x-h\\right)^{2}+k\\left(a \\lt 0\\right)$ passes through the points $A\\left(-3,m\\right)$ and $B\\left(-1,n\\right)$.<br/>$(1)$ When $m=n$, find the length of the line segment $AB$... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"split": "train",
"index": 3734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The restaurant has two types of tables: square tables that can seat 4 people, and round tables that can seat 9 people. If the number of diners exactly fills several tables, the restaurant manager calls this number a \"wealth number.\" Among the numbers from 1 to 100, how many \... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 3735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graphs of $y=|x|$ and $y=-x^2-3x-2$ are drawn. For every $x$, a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 3736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that real numbers \\( a \\) and \\( b \\) are such that the equation \\( a x^{3} - x^{2} + b x - 1 = 0 \\) has three positive real roots, find the minimum value of \\( P = \\frac{5 a^{2} - 3 a b + 2}{a^{2}(b - a)} \\) for all such \\( a \\) and \\( b \\). Let's think step... | math | {
"style": "rule",
"ground_truth": "12 \\sqrt{3}"
} | {
"split": "train",
"index": 3737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an acute-angled triangle, the sides $a$ and $b$ are the roots of the equation $x^{2}-2 \\sqrt {3}x+2=0$. The angles $A$ and $B$ satisfy the equation $2\\sin (A+B)- \\sqrt {3}=0$. Find the value of the side $c$ and the area of $\\triangle ABC$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\dfrac { \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 3738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(x,3)$ and $\\overrightarrow{b}=(-1,y-1)$, and $\\overrightarrow{a}+2\\overrightarrow{b}=(0,1)$, find the value of $|\\overrightarrow{a}+\\overrightarrow{b}|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 3739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of $\\frac{1}{x}$ in the expansion of $((1-x^{2})^{4}(\\frac{x+1}{x})^{5})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-29"
} | {
"split": "train",
"index": 3740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a $3 \\times 3$ grid (each cell is a $1 \\times 1$ square), two identical pieces are placed, with at most one piece per cell. There are ___ distinct ways to arrange the pieces. (If two arrangements can be made to coincide by rotation, they are considered the same arrangement... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\\log_{10} b$, what is the median of the set $\\{0, 1, a, b,\n1/b\\}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "b"
} | {
"split": "train",
"index": 3742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), point \\(N\\) lies on side \\(AB\\) such that \\(AN = 3NB\\); the median \\(AM\\) intersects \\(CN\\) at point \\(O\\). Find \\(AB\\) if \\(AM = CN = 7\\) cm and \\(\\angle NOM = 60^\\circ\\). Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{7}"
} | {
"split": "train",
"index": 3743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two unit squares are selected at random without replacement from an $n \\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\\frac{1}{2015}$. Let's... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 3744,
"solution": "There are 3 cases in this problem. Number one, the center squares. Two, the edges, and three, the corners. The probability of getting one center square and an adjacent square is $\\frac{(n-2)(n-2)}{n^2}$ multiplied by $\\frac{4}{n^2 -1}$. Add that to the probability... |
deepscaler | [
{
"role": "user",
"content": "Let the real numbers \\(a_{1}, a_{2}, \\cdots, a_{100}\\) satisfy the following conditions: (i) \\(a_{1} \\geqslant a_{2} \\geqslant \\cdots \\geqslant a_{100} \\geqslant 0\\); (ii) \\(a_{1}+a_{2} \\leqslant 100\\); (iii) \\(a_{3}+a_{4} + \\cdots + a_{100} \\leqslant 100\\). Fi... | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 3745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$ with the general term formula $a_n = -n^2 + 12n - 32$, determine the maximum value of $S_n - S_m$ for any $m, n \\in \\mathbb{N^*}$ and $m < n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the principal (smallest positive) period of the function\n\n$$\ny=(\\arcsin (\\sin (\\arccos (\\cos 3 x))))^{-5}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 3747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "According to statistical data, the daily output of a factory does not exceed 200,000 pieces, and the daily defect rate $p$ is approximately related to the daily output $x$ (in 10,000 pieces) by the following relationship:\n$$\np= \\begin{cases} \n\\frac{x^{2}+60}{540} & (0<x\\l... | math | {
"style": "rule",
"ground_truth": "\\frac{100}{9}"
} | {
"split": "train",
"index": 3748,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest integer \\( n > 1 \\) such that \\(\\frac{1^2 + 2^2 + 3^2 + \\ldots + n^2}{n}\\) is a square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"split": "train",
"index": 3749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe t... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 3750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure below, how many ways are there to select 5 bricks, one in each row, such that any two bricks in adjacent rows are adjacent? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 3751,
"solution": "The number of valid selections is equal to the number of paths which start at a top brick and end at a bottom brick. We compute these by writing 1 in each of the top bricks and letting lower bricks be the sum of the one or two bricks above them. Thus, the number ins... |
deepscaler | [
{
"role": "user",
"content": "On $5\\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile. Let's th... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 3752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1. When the smallest number is removed, the average is increased by 1. When both the largest and the smallest numbers are removed, the average of the remaining four num... | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 3753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of real number $a$ , such that for every $a$ , equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:\n(a) $x_0$ is an even integer;\n(b) $|x_0|<1000$ . Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "999"
} | {
"split": "train",
"index": 3754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a football tournament, 15 teams participated, each playing exactly once against every other team. A win awarded 3 points, a draw 1 point, and a loss 0 points.\n\nAfter the tournament ended, it was found that some 6 teams each scored at least $N$ points. What is the maximum p... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 3755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, with $D$ on $AC$ and $F$ on $BC$, given $AB \\perp AC$, $AF \\perp BC$, and $BD = DF = FC = 1$. If also $D$ is the midpoint of $AC$, find the length of $AC$.\nA) 1\nB) $\\sqrt{2}$\nC) $\\sqrt{3}$\nD) 2\nE) $\\sqrt[3]{4}$ Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 3756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the probability that a 4 × 4 square grid becomes a single uniform color (all white or all black) after rotation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{32768}"
} | {
"split": "train",
"index": 3757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, $a_{n+1} + (-1)^n a_n = 2n - 1$. Calculate the sum of the first 12 terms of $\\{a_n\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 3758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \\leq k \\leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in the given... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"split": "train",
"index": 3759,
"solution": "Label each of the bottom squares as $x_0, x_1 \\ldots x_9, x_{10}$.\n Through induction, we can find that the top square is equal to ${10\\choose0}x_0 + {10\\choose1}x_1 + {10\\choose2}x_2 + \\ldots {10\\choose10}x_{10}$. (This also makes sense based on a combinator... |
deepscaler | [
{
"role": "user",
"content": "Find the number of real zeros of $x^{3}-x^{2}-x+2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 3760,
"solution": "Let $f(x)=x^{3}-x^{2}-x+2$, so $f^{\\prime}(x)=3 x^{2}-2 x-1$. The slope is zero when $3 x^{2}-2 x-1=0$, where $x=-\\frac{1}{3}$ and $x=1$. Now $f\\left(\\frac{1}{3}\\right)>0$ and $f(1)>0$, so there are no zeros between $x=-\\frac{1}{3}$ and $x=1$. Since \\lim _{x ... |
deepscaler | [
{
"role": "user",
"content": "What is $\\left(20 - \\left(2010 - 201\\right)\\right) + \\left(2010 - \\left(201 - 20\\right)\\right)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 3761,
"solution": "We start by simplifying the expression inside the parentheses:\n\n1. Calculate $2010 - 201$:\n \\[\n 2010 - 201 = 1809\n \\]\n\n2. Substitute this result back into the first part of the expression:\n \\[\n 20 - (2010 - 201) = 20 - 1809 = -1789\n \\]\n\n3... |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $ABC$ with sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively, and it is given that $(3b-c)\\cos A = a\\cos C$.\n(1) Find the value of $\\cos A$;\n(2) If the area of $\\triangle ABC$ is $S=2\\sqrt{2}$, find the minimum value of the... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}+2\\sqrt{2}"
} | {
"split": "train",
"index": 3762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$, $y$, and $z$ are positive numbers satisfying \\[\nx+\\frac{1}{y}=4,\\ \\ \\ y+\\frac{1}{z}=1,\\text{ and }z+\\frac{1}{x}=\\frac{7}{3},\n\\]find the value of $xyz$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 3763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $AB = 13\\text{ cm},$ $DC = 20\\text{ cm},$ and $AD = 5\\text{ cm}.$ What is the length of $AC,$ to the nearest tenth of a centimeter?\n\n[asy]\ndraw((0,0)--(5,12)--(21,12)--(5,0)--cycle,black+linewidth(1));\ndraw((5,12)--(5,0),black+linewidth(1));\ndraw((0,0)--... | math | {
"style": "rule",
"ground_truth": "24.2"
} | {
"split": "train",
"index": 3764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y = -2x^2 + 4x + m$.\n1. For what value of $m$ does the parabola intersect the x-axis at exactly one point?\n2. If two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the parabola have $x$-coordinates satisfying $x_1 > x_2 > 2$, compare the values of $y_1$ and $y_... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 3765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $AB = 9,$ $BC = 10,$ and $AC = 11.$ If $D$ and $E$ are chosen on $\\overline{AB}$ and $\\overline{AC}$ so that $AD = 4$ and $AE = 7,$ then find the area of triangle $ADE.$\n\n[asy]\nunitsize (1 cm);\n\npair A, B, C, D, E;\n\nA = (2,3);\nB = (0,0);\nC = (6,0)... | math | {
"style": "rule",
"ground_truth": "\\frac{280 \\sqrt{2}}{33}"
} | {
"split": "train",
"index": 3766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Beaumont High School, there are 20 players on the basketball team. All 20 players are taking at least one of biology or chemistry. (Biology and chemistry are two different science courses at the school.) If there are 8 players taking biology and 4 players are taking both ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 3767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of the following infinite series:\n\n$$\n1-\\frac{1}{a}+\\frac{1}{a^{2}}-\\frac{1}{a^{3}}+\\frac{1}{a^{4}}-\\frac{1}{a^{5}}+\\ldots\n$$\n\nif \\( a \\) is a positive number greater than 1. What will the sum be as the value of \\( a \\) approaches 1? Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 3768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\) be a fixed integer with \\( n \\geq 2 \\). Determine the minimal constant \\( c \\) such that the inequality\n\n\\[ \\sum_{1 \\leq i < j \\leq n} x_i x_j (x_i^2 + x_j^2) \\leq c \\left( \\sum_{i=1}^{n} x_i \\right)^4 \\]\n\nholds for all non-negative real numbers ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 3769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the distance between the center of the circle with equation $x^2+y^2=-4x+6y-12$ and the point $(1,7)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a,$ $b,$ $x,$ and $y$ are real numbers such that $ax+by=3,$ $ax^2+by^2=7,$ $ax^3+by^3=16,$ and $ax^4+by^4=42,$ find $ax^5+by^5.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 3771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sqrt{3}\\sin x\\cos x + \\cos^2 x + a$.\n\n(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$;\n\n(2) If the sum of the maximum and minimum values of $f(x)$ in the interval $[-\\frac{\\pi}{6}, \\frac{\\pi}{3}]$... | math | {
"style": "rule",
"ground_truth": "a = -\\frac{1}{4}"
} | {
"split": "train",
"index": 3772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is chosen randomly from within a circular region with radius $r$. A related concentric circle with radius $\\sqrt{r}$ contains points that are closer to the center than to the boundary. Calculate the probability that a randomly chosen point lies closer to the center tha... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 3773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a right triangle with $m(\\widehat{A})=90^\\circ$ . Let $APQR$ be a square with area $9$ such that $P\\in [AC]$ , $Q\\in [BC]$ , $R\\in [AB]$ . Let $KLMN$ be a square with area $8$ such that $N,K\\in [BC]$ , $M\\in [AB]$ , and $L\\in [AC]$ . What i... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a real coefficient fourth-degree polynomial with a leading coefficient of 1 that has four imaginary roots, where the product of two of the roots is \\(32+\\mathrm{i}\\) and the sum of the other two roots is \\(7+\\mathrm{i}\\), determine the coefficient of the quadratic t... | math | {
"style": "rule",
"ground_truth": "114"
} | {
"split": "train",
"index": 3775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \\sqrt{a^{2} + b^{2}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "289"
} | {
"split": "train",
"index": 3776,
"solution": "1. **Express the given equation in terms of $a$ and $b$:** \n Let $z = a + bi$. Then, the given equation $z + |z| = 2 + 8i$ can be rewritten as:\n \\[\n a + bi + \\sqrt{a^2 + b^2} = 2 + 8i\n \\]\n Here, $a$ and $b$ are real numbers, and $\\sqrt{a^2 + b^2}... |
deepscaler | [
{
"role": "user",
"content": "Find the positive real number $x$ such that $\\lfloor x \\rfloor \\cdot x = 54$. Express $x$ as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.714285714285714"
} | {
"split": "train",
"index": 3777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a tetrahedron V-ABC with edge length 10, point O is the center of the base ABC. Segment MN has a length of 2, with one endpoint M on segment VO and the other endpoint N inside face ABC. If point T is the midpoint of segment MN, then the area of the trajectory formed by point... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 3778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain collection of numbered indexed cards includes one card with a 1 written on it, two cards with a 2, and so forth up to $n$ cards showing an $n,$ for some positive integer $n$. Determine $n,$ if the average value of a card in this collection is 2017. Let's think step b... | math | {
"style": "rule",
"ground_truth": "3025"
} | {
"split": "train",
"index": 3779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle with radius $4$, find the area of the region formed by all line segments of length $4$ that are tangent to this circle at their midpoints.\nA) $2\\pi$\nB) $4\\pi$\nC) $8\\pi$\nD) $16\\pi$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 3780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of the quadrilateral prism \\( A B C D A_{1} B_{1} C_{1} D_{1} \\) is a rhombus \\( A B C D \\) with \\( B D = 12 \\) and \\( \\angle B A C = 60^{\\circ} \\). A sphere passes through the vertices \\( D, A, B, B_{1}, C_{1}, D_{1} \\).\n\na) Find the area of the circle o... | math | {
"style": "rule",
"ground_truth": "192\\sqrt{3}"
} | {
"split": "train",
"index": 3781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a convex trapezoid such that $\\angle A B C=\\angle B C D=90^{\\circ}, A B=3, B C=6$, and $C D=12$. Among all points $X$ inside the trapezoid satisfying $\\angle X B C=\\angle X D A$, compute the minimum possible value of $C X$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\sqrt{113}-\\sqrt{65}"
} | {
"split": "train",
"index": 3782,
"solution": "Let $P=A D \\cap B C$. Then, we see that the locus if $X$ is the arc $\\widehat{B D}$ of $\\odot(P B D)$. Let $O$ be its center, and $R$ be the radius of $\\odot(P B D)$, then the answer is $C O-R$. Let $T$ be the second intersection of $\\odot(P B D)$ and $C D$. We... |
deepscaler | [
{
"role": "user",
"content": "At a national competition, 31 participants are accommodated in a hotel where each participant gets his/her own room. The rooms are numbered from 1 to 31. When all participants have arrived, except those assigned to rooms 15, 16, and 17, what is the median room number of the rem... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 3783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A(3,5)$ and $B(7,10)$ are the endpoints of a diameter of a circle graphed in a coordinate plane. How many square units are in the area of the circle? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41\\pi}{4}"
} | {
"split": "train",
"index": 3784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi + 2"
} | {
"split": "train",
"index": 3785,
"solution": "To solve the problem, we first analyze the given equation $x^2 + y^2 = |x| + |y|$ and consider the symmetry in the four quadrants.\n\n#### Step 1: Analyze the equation in different cases\nWe consider the equation in four cases based on the signs of $x$ and $y$:\n- *... |
deepscaler | [
{
"role": "user",
"content": "The noon temperatures for ten consecutive days were $78^{\\circ}$, $80^{\\circ}$, $82^{\\circ}$, $85^{\\circ}$, $88^{\\circ}$, $90^{\\circ}$, $92^{\\circ}$, $95^{\\circ}$, $97^{\\circ}$, and $95^{\\circ}$ Fahrenheit. The increase in temperature over the weekend days (days 6 to ... | math | {
"style": "rule",
"ground_truth": "88.2"
} | {
"split": "train",
"index": 3786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $WXYZ$ be a rhombus with diagonals $WY = 20$ and $XZ = 24$. Let $M$ be a point on $\\overline{WX}$, such that $WM = MX$. Let $R$ and $S$ be the feet of the perpendiculars from $M$ to $\\overline{WY}$ and $\\overline{XZ}$, respectively. Find the minimum possible value of $RS... | math | {
"style": "rule",
"ground_truth": "\\sqrt{244}"
} | {
"split": "train",
"index": 3787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a polar coordinate system with the pole at point $O$, the curve $C\\_1$: $ρ=6\\sin θ$ intersects with the curve $C\\_2$: $ρ\\sin (θ+ \\frac {π}{4})= \\sqrt {2}$. Determine the maximum distance from a point on curve $C\\_1$ to curve $C\\_2$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "3+\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 3788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\cos (x+\\frac{π}{3})$, determine the horizontal shift of the graph of the function $y=\\sin x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 3789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $u$ and $v$ be real numbers such that \\[(u + u^2 + u^3 + \\cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \\cdots + v^{10}) + 10v^{11} = 8.\\] Determine, with proof, which of the two numbers, $u$ or $v$ , is larger. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "v"
} | {
"split": "train",
"index": 3790,
"solution": "The answer is $v$ .\nWe define real functions $U$ and $V$ as follows: \\begin{align*} U(x) &= (x+x^2 + \\dotsb + x^8) + 10x^9 = \\frac{x^{10}-x}{x-1} + 9x^9 \\\\ V(x) &= (x+x^2 + \\dotsb + x^{10}) + 10x^{11} = \\frac{x^{12}-x}{x-1} + 9x^{11} . \\end{align*} We wish ... |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "152"
} | {
"split": "train",
"index": 3791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are isosceles right triangles. If the area of square $ABCD$ is 25, find the area of $KLMN$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 3792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$ . (Note: $n$ is written in the usual base ten notation.) Let's think ... | math | {
"style": "rule",
"ground_truth": "9999"
} | {
"split": "train",
"index": 3793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $(x-a+3)$ if the value of $x$ is $(a+7)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four people in a room. For every two people, there is a $50 \\%$ chance that they are friends. Two people are connected if they are friends, or a third person is friends with both of them, or they have different friends who are friends of each other. What is the proba... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{32}"
} | {
"split": "train",
"index": 3795,
"solution": "We label the four people in the room $A, B, C$, and $D$. We represent each person by a point. There are six possible pairs of friends: $AB, AC, AD, BC, BD$, and $CD$. We represent a friendship by joining the corresponding pair of points and a non-friendship by not j... |
deepscaler | [
{
"role": "user",
"content": "Find all integers $n \\ge 3$ such that among any $n$ positive real numbers $a_1$ , $a_2$ , $\\dots$ , $a_n$ with \\[\\max(a_1, a_2, \\dots, a_n) \\le n \\cdot \\min(a_1, a_2, \\dots, a_n),\\] there exist three that are the side lengths of an acute triangle. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\{n \\ge 13\\}"
} | {
"split": "train",
"index": 3796,
"solution": "Without loss of generality, assume that the set $\\{a\\}$ is ordered from least to greatest so that the bounding condition becomes $a_n \\le n \\cdot a_1.$ Now set $b_i \\equiv \\frac{a_i}{a_1},$ and since a triangle with sidelengths from $\\{a\\}$ will be similar t... |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$. There exists a point $M$ in the first quadrant of ellipse $C$ such that $|MF_{1}|=|F_{1}F_{2}|$. Line $F_{1}M$ intersects the $y$-axis at point $A$, and $F_{2}A$ bisects $... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"split": "train",
"index": 3797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the center of a circular field, there is a geologists' house. Eight straight roads radiate from it, dividing the field into 8 equal sectors. Two geologists set off on a journey from their house, each traveling at a speed of 4 km/h along a road chosen at random. Determine the... | math | {
"style": "rule",
"ground_truth": "0.375"
} | {
"split": "train",
"index": 3798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let square $WXYZ$ have sides of length $8$. An equilateral triangle is drawn such that no point of the triangle lies outside $WXYZ$. Determine the maximum possible area of such a triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16\\sqrt{3}"
} | {
"split": "train",
"index": 3799,
"solution": ""
} |
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