data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find the $3 \\times 3$ matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\mathbf{v} = -4 \\mathbf{v}\\]for all three-dimensional vectors $\\mathbf{v}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -4 & 0 & 0 \\\\ 0 & -4 & 0 \\\\ 0 & 0 & -4 \\end{pmatrix}"
} | {
"split": "train",
"index": 34649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The TV station is broadcasting 5 different advertisements, among which there are 3 different commercial advertisements and 2 different Olympic promotional advertisements. The last advertisement must be an Olympic promotional advertisement, and the two Olympic promotional advert... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 34650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cheburashka spent his money to buy as many mirrors from Galya's store as Gena bought from Shapoklyak's store. If Gena were buying from Galya, he would have 27 mirrors, and if Cheburashka were buying from Shapoklyak, he would have 3 mirrors. How many mirrors would Gena and Chebu... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 34651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_{n}\\}$ is an arithmetic sequence, $a_{9}+a_{12} \\lt 0$, $a_{10}\\cdot a_{11} \\lt 0$, and the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$, denoted as $S_{n}$, has a maximum value, determine the maximum value of $n$ when $S_{n} \\gt 0... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 34652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\\sin \\frac{\\theta}{2} \\cdot (1 + \\cos \\theta)\\]for $0 < \\theta < \\pi.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4 \\sqrt{3}}{9}"
} | {
"split": "train",
"index": 34653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $(a+b-c)(a+b+c)=ab$. \n$(1)$ Determine the measure of angle $C$. \n$(2)$ If $c=2a\\cos B$ and $b=2$, find the area of $\\triangle ABC$. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 34654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive real numbers $x \\neq 1$ and $y \\neq 1$ satisfy $\\log_2{x} = \\log_y{16}$ and $xy = 64$. What is $(\\log_2{\\tfrac{x}{y}})^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 34655,
"solution": "1. **Convert the logarithmic equation:** Given $\\log_2{x} = \\log_y{16}$, we can use the change of base formula to rewrite this as:\n \\[\n \\log_2{x} = \\frac{\\log_2{16}}{\\log_2{y}} = \\frac{4}{\\log_2{y}}\n \\]\n\n2. **Express $x$ in terms of $y$:** We a... |
deepscaler | [
{
"role": "user",
"content": "Given a set of data $x_{1}$, $x_{2}$, …, $x_{20}$, if the expectation of this set of data is $3$ and the variance is $3$, determine the expectation and variance of $2x_{1}+3$, $2x_{2}+3$, …, $2x_{20}+3$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a certain crosswalk, the pedestrian signal alternates between red and green lights, with the red light lasting for $40s$. If a pedestrian arrives at the crosswalk and encounters a red light, the probability that they need to wait at least $15s$ for the green light to appear ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{8}"
} | {
"split": "train",
"index": 34657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 34658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a\nrunning tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either playe... | math | {
"style": "rule",
"ground_truth": "5/11"
} | {
"split": "train",
"index": 34659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anya wants to buy ice cream that costs 19 rubles. She has two 10-ruble coins, two 5-ruble coins, and one 2-ruble coin in her pocket. Anya randomly picks three coins from her pocket without looking. Find the probability that the selected coins will be enough to pay for the ice c... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"split": "train",
"index": 34660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A spinner has four sections labeled 1, 2, 3, and 4, each section being equally likely to be selected. If you spin the spinner three times to form a three-digit number, with the first outcome as the hundreds digit, the second as the tens digit, and the third as the unit digit, w... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 34661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all the subsets of $\\{1,2,3, \\ldots, 2018,2019\\}$ having exactly 100 elements. For each subset, take the greatest element. Find the average of all these greatest elements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 34662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\tan x = 2,$ then find $\\tan \\left( x + \\frac{\\pi}{4} \\right).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 34663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob are playing a game where Alice declares, \"My number is 36.\" Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is stric... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 34664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sides opposite to the internal angles A, B, and C of triangle ABC are a, b, and c respectively, if -c cosB is the arithmetic mean of $\\sqrt {2}$a cosB and $\\sqrt {2}$b cosA, find the maximum value of sin2A•tan²C. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"split": "train",
"index": 34665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parabola $y=x^{2}-7$, find the length of the line segment $|AB|$ where $A$ and $B$ are two distinct points on it that are symmetric about the line $x+y=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{2}"
} | {
"split": "train",
"index": 34666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a 6 by 6 grid of points, what fraction of the larger square's area is inside the shaded square if the shaded square is rotated 45 degrees with vertices at points (2,2), (3,3), (2,4), and (1,3)? Express your answer as a common fraction. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{18}"
} | {
"split": "train",
"index": 34667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, it is known that $P$ is a moving point on the graph of the function $f(x)=\\ln x$ ($x > 0$). The tangent line $l$ at point $P$ intersects the $x$-axis at point $E$. A perpendicular line to $l$ through point $P$ intersects the $x$-axis a... | math | {
"style": "rule",
"ground_truth": "\\dfrac {1}{2}(e+ \\dfrac {1}{e})"
} | {
"split": "train",
"index": 34668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 0.1s are there in 1.9? How many 0.01s are there in 0.8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 34669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) such that both \\( n^2 \\) and \\( (n+1)^2 \\) contain the digit 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 34670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a rectangle $AEFJ$ inside a regular decagon $ABCDEFGHIJ$. What is the ratio of the area of the rectangle to the area of the decagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2:5"
} | {
"split": "train",
"index": 34671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If a particular frustum has an altitude of $24$ centimeters, the area of its lower base is $225\\pi$ sq cm and the area of its upper base is $25\\pi$ sq cm, what is the altitud... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^2}{25} + \\frac{y^2}{9} = 1$, and the line $l: 4x - 5y + 40 = 0$. Is there a point on the ellipse for which the distance to line $l$ is minimal? If so, what is the minimal distance? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{\\sqrt{41}}"
} | {
"split": "train",
"index": 34673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four points $B,$ $A,$ $E,$ and $L$ are on a straight line, as shown. The point $G$ is off the line so that $\\angle BAG = 120^\\circ$ and $\\angle GEL = 80^\\circ.$ If the reflex angle at $G$ is $x^\\circ,$ then what does $x$ equal?\n\n[asy]\ndraw((0,0)--(30,0),black+linewidth(... | math | {
"style": "rule",
"ground_truth": "340"
} | {
"split": "train",
"index": 34674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$, $\\sin A = \\frac{8}{17}$ and $\\sin B = 1$. Find $\\sin C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{17}"
} | {
"split": "train",
"index": 34675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all prime numbers $p,q,r$ , such that $\\frac{p}{q}-\\frac{4}{r+1}=1$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(7, 3, 2), (3, 2, 7), (5, 3, 5)"
} | {
"split": "train",
"index": 34676,
"solution": "The given equation can be rearranged into the below form:\n$4q = (p-q)(r+1)$ \n \n$Case 1: 4|(p-q)$ \nthen we have\n$q = ((p-q)/4)(r+1)$ $=> (p-q)/4 = 1$ and $q = r + 1$ $=> r = 2, q = 3$ and $p = 7$ \n \n$Case 2: 4|(r+1)$ \nthen we have\n$q = (p-q)((r+1)/4)$ $=... |
deepscaler | [
{
"role": "user",
"content": "A unicorn is tethered by a $20$-foot silver rope to the base of a magician's cylindrical tower whose radius is $8$ feet. The rope is attached to the tower at ground level and to the unicorn at a height of $4$ feet. The unicorn has pulled the rope taut, the end of the rope is $4... | math | {
"style": "rule",
"ground_truth": "813"
} | {
"split": "train",
"index": 34677,
"solution": "Note that by Power of a Point, the point the unicorn is at has power $4 \\cdot 20 = 80$ which implies that the tangent from that point to the tower is of length $\\sqrt{80}=4\\sqrt{5},$ however this is length of the rope projected into 2-D. If we let $\\theta$ be t... |
deepscaler | [
{
"role": "user",
"content": "Compute, in terms of $n$, $\\sum_{k=0}^{n}\\binom{n-k}{k} 2^{k}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\cdot 2^{n}+(-1)^{n}}{3}"
} | {
"split": "train",
"index": 34678,
"solution": "Let $T_{n}=\\sum_{k=0}^{n}\\binom{n-k}{k} 2^{k}$. From Pascal's recursion for binomial coefficients, we can find $T_{n}=2 T_{n-2}+T_{n-1}$, with $T_{0}=1$ and $T_{1}=1$. The characteristic polynomial of this recursion is $x^{2}-x-2=0$, which has roots 2 and -1. Thu... |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR,$ $S$ is on $\\overline{PQ}$ such that $PS:SQ = 4:1,$ and $T$ is on $\\overline{QR}$ such that $QT:TR = 4:1.$ If lines $ST$ and $PR$ intersect at $U,$ then find $\\frac{ST}{TU}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 34679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function $g(x) = \\frac{x^2}{2} + 2x - 1$. Determine the sum of all distinct numbers $x$ such that $g(g(g(x))) = 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 34680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many ordered triples $(a, b, c)$ of positive integers are the equations $abc+9=ab+bc+ca$ and $a+b+c=10$ satisfied? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 34681,
"solution": "Subtracting the first equation from the second, we obtain $1-a-b-c+ab+bc+ca-abc=(1-a)(1-b)(1-c)=0$. Since $a, b$, and $c$ are positive integers, at least one must equal 1. Note that $a=b=c=1$ is not a valid triple, so it suffices to consider the cases where exactly... |
deepscaler | [
{
"role": "user",
"content": "Point \\( M \\) lies on the edge \\( AB \\) of cube \\( ABCD A_1 B_1 C_1 D_1 \\). Rectangle \\( MNLK \\) is inscribed in square \\( ABCD \\) in such a way that one of its vertices is at point \\( M \\), and the other three vertices are located on different sides of the base squ... | math | {
"style": "rule",
"ground_truth": "1:4"
} | {
"split": "train",
"index": 34682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the parallelogram generated by $\\begin{pmatrix} 3 \\\\ 1 \\\\ -2 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ -3 \\\\ 4 \\end{pmatrix}.$\n\n[asy]\nunitsize(0.4 cm);\n\npair A, B, C, D;\n\nA = (0,0);\nB = (7,2);\nC = (1,3);\nD = B + C;\n\ndraw(A--B,Arrow(6));\n... | math | {
"style": "rule",
"ground_truth": "10 \\sqrt{3}"
} | {
"split": "train",
"index": 34683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The center of the circle inscribed in a trapezoid is at distances of 5 and 12 from the ends of one of the non-parallel sides. Find the length of this side. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 34684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a function $h(x),$ for positive integer values of $x,$ by \\[h(x) = \\left\\{\\begin{aligned} \\log_2 x & \\quad \\text{ if } \\log_2 x \\text{ is an integer} \\\\ 1 + h(x + 1) & \\quad \\text{ otherwise}. \\end{aligned} \\right.\\]Compute $h(100).$ Let's think step by s... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 34685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kelvin the frog lives in a pond with an infinite number of lily pads, numbered $0,1,2,3$, and so forth. Kelvin starts on lily pad 0 and jumps from pad to pad in the following manner: when on lily pad $i$, he will jump to lily pad $(i+k)$ with probability $\\frac{1}{2^{k}}$ for ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 34686,
"solution": "Suppose we combine all of the lily pads with numbers greater than 2019 into one lily pad labeled $\\infty$. Also, let Kelvin stop once he reaches one of these lily pads. Now at every leap, Kelvin has an equal chance of landing on 2019 as landing on $\\infty$. Furth... |
deepscaler | [
{
"role": "user",
"content": "\nGiven the sets\n$$\n\\begin{array}{l}\nA=\\left\\{(x, y) \\mid x=m, y=-3m+2, m \\in \\mathbf{Z}_{+}\\right\\}, \\\\\nB=\\left\\{(x, y) \\mid x=n, y=a\\left(a^{2}-n+1\\right), n \\in \\mathbf{Z}_{+}\\right\\},\n\\end{array}\n$$\n\nfind the total number of integers $a$ such tha... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 34687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Below is the graph of $y = a \\sin bx$ for some constants $a < 0$ and $b > 0.$ Find $a.$\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal g(real x)\n{\n\treturn (-2*sin(x/3));\n}\n\ndraw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3*pi,3*pi,-3,3,pi/2,1);\nlaye... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 34688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cybil and Ronda are sisters. The 10 letters from their names are placed on identical cards so that each of 10 cards contains one letter. Without replacement, two cards are selected at random from the 10 cards. What is the probability that one letter is from each sister's name? ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 34689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( c \\) be the constant term in the expansion of \\( \\left(2 x+\\frac{b}{\\sqrt{x}}\\right)^{3} \\). Find the value of \\( c \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 34690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\ln x+ax (a\\in R)$.\n(1) When $a=- \\frac{1}{3}$, find the extreme values of the function $f(x)$ in the interval $[e,e^{2}]$;\n(2) When $a=1$, the function $g(x)=f(x)- \\frac{2}{t}x^{2}$ has only one zero point. Find the value of the positive number $... | math | {
"style": "rule",
"ground_truth": "t=2"
} | {
"split": "train",
"index": 34691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are real numbers, and $x^{2}+2xy-y^{2}=7$, find the minimum value of $x^{2}+y^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 34692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $m$ are there from 1 to 1996, such that $\\frac{m^{2}+7}{m+4}$ is not a reduced fraction? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 34693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petrov writes down odd numbers: \\(1, 3, 5, \\ldots, 2013\\), and Vasechkin writes down even numbers: \\(2, 4, \\ldots, 2012\\). Each of them calculates the sum of all the digits of all their numbers and tells it to the star student Masha. Masha subtracts Vasechkin's result fro... | math | {
"style": "rule",
"ground_truth": "1007"
} | {
"split": "train",
"index": 34694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangles $\\triangle DEF$ and $\\triangle D'E'F'$ lie in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(15,0)$, $D'(15,25)$, $E'(25,25)$, $F'(15,10)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\\triangle DEF$ to $... | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 34695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $111_4+323_4+132_4$. Express your answer in base $4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1232_4"
} | {
"split": "train",
"index": 34696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers $k$ for which there exists a nonzero, 2-dimensional vector $\\mathbf{v}$ such that\n\\[\\begin{pmatrix} 3 & 4 \\\\ 6 & 3 \\end{pmatrix} \\mathbf{v} = k \\mathbf{v}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{6}"
} | {
"split": "train",
"index": 34697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of all positive integral values of $m$ such that $m^2 - 40m + 399 = q$ for some prime number $q$. Note that there is at least one such $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 34698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is $k\\%$ acid. From jar C, $\\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the soluti... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 34699,
"solution": "One may first combine all three jars in to a single container. That container will have $10$ liters of liquid, and it should be $50\\%$ acidic. Thus there must be $5$ liters of acid.\n Jar A contained $45\\% \\cdot 4L$, or $1.8L$ of acid, and jar B $48\\% \\cdot 5L... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $x$ such that $x + 8901$ results in a palindrome? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 34700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every real number $x$, let $\\lfloor x\\rfloor$ denote the greatest integer not exceeding $x$, and let $f(x) = \\lfloor x\\rfloor(2014^{x-\\lfloor x\\rfloor}-1)$. The set of all numbers $x$ such that $1\\leq x<2014$ and $f(x)\\leq 1$ is a union of disjoint intervals. What i... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 34701,
"solution": "1. **Define the function and interval**: Given the function \\( f(x) = \\lfloor x \\rfloor (2014^{x - \\lfloor x \\rfloor} - 1) \\), we need to find the set of \\( x \\) such that \\( 1 \\leq x < 2014 \\) and \\( f(x) \\leq 1 \\).\n\n2. **Simplify the function**: L... |
deepscaler | [
{
"role": "user",
"content": "\nA traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each person told the traveler whether the neighbor to their right was truthful or deceitful. Based on these statements, the traveler was able to d... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 34702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 34703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\sin (\\pi x+\\varphi)-2\\cos (\\pi x+\\varphi)$ $(0 < \\varphi < \\pi)$, its graph is symmetric about the line $x=1$. Find $\\sin 2\\varphi$. 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": 34704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a_{1}, a_{2}, \\cdots, a_{n}\\) be an increasing sequence of positive integers. For a positive integer \\(m\\), define\n\\[b_{m}=\\min \\left\\{n \\mid a_{n} \\geq m\\right\\} (m=1,2, \\cdots),\\]\nthat is, \\(b_{m}\\) is the smallest index \\(n\\) such that \\(a_{n} \\g... | math | {
"style": "rule",
"ground_truth": "42399"
} | {
"split": "train",
"index": 34705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nAn economist-cryptographer received a cryptogram from the ruler that contained a secret decree to introduce a per-unit tax in a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that it was not possible to collect a hi... | math | {
"style": "rule",
"ground_truth": "8640"
} | {
"split": "train",
"index": 34706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of numbers $x_{1},x_{2},x_{3},\\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50} = m/n,$ where $m$ and $n$ are relatively prime posi... | math | {
"style": "rule",
"ground_truth": "173"
} | {
"split": "train",
"index": 34707,
"solution": "Let the sum of all of the terms in the sequence be $\\mathbb{S}$. Then for each integer $k$, $x_k = \\mathbb{S}-x_k-k \\Longrightarrow \\mathbb{S} - 2x_k = k$. Summing this up for all $k$ from $1, 2, \\ldots, 100$,\n\\begin{align*}100\\mathbb{S}-2(x_1 + x_2 + \\cdo... |
deepscaler | [
{
"role": "user",
"content": "Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(2,2),$ $(4,6),$ $(6,12),$ and $(8,12),$ there is one point where the graphs of $y=h(2x)$ and $y=2j(x)$ must intersect. What is the sum of the coordinates of that point? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 34708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side of a triangle are 2, 2, and $\\sqrt{6} - \\sqrt{2}.$ Enter the angles of the triangle in degrees, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75^\\circ, 75^\\circ"
} | {
"split": "train",
"index": 34709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( M \\) and \\( N \\) are the midpoints of sides \\( BC \\) and \\( AD \\) of quadrilateral \\( ABCD \\). It is known that \\(\\angle B = 150^\\circ\\), \\(\\angle C = 90^\\circ\\), and \\(AB = CD\\). Find the angle between the lines \\(MN\\) and \\(BC\\). Let's think ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 34710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gamma and Delta both participated in a two-day science quiz. Each attempted questions totaling 600 points after the second day. On the first day, Gamma scored 210 points out of 350 points attempted, and on the second day scored 150 points out of 250 points attempted. Delta, who... | math | {
"style": "rule",
"ground_truth": "\\frac{359}{600}"
} | {
"split": "train",
"index": 34711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many lattice points (points with integer coordinates) are inside (but not on the boundary) the region formed by the right branch of the hyperbola $x^{2} - y^{2} = 1$ and the line $x = 100$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9800"
} | {
"split": "train",
"index": 34712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $3206_7$ to a base 10 integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1133"
} | {
"split": "train",
"index": 34713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On each cell of a $200 \\times 200$ grid, we place a car, which faces in one of the four cardinal directions. In a move, one chooses a car that does not have a car immediately in front of it, and slides it one cell forward. If a move would cause a car to exit the grid, the car ... | math | {
"style": "rule",
"ground_truth": "6014950"
} | {
"split": "train",
"index": 34714,
"solution": "Let $n=100$. The answer is $\\frac{1}{2} n\\left(12 n^{2}+3 n-1\\right)=6014950$. A construction for an $8 \\times 8$ grid instead (so $n=4$ ): ![](https://cdn.mathpix.com/cropped/2024_08_14_43b5d29562bf3caac770g-11.jpg?height=583&width=573&top_left_y=237&top_left_... |
deepscaler | [
{
"role": "user",
"content": "Given $a=1$, $b=2$, $C=\\frac{2π}{3}$ in triangle $\\triangle ABC$, calculate the value of $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{9}"
} | {
"split": "train",
"index": 34715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A city uses a lottery system for assigning car permits, with 300,000 people participating in the lottery and 30,000 permits available each month.\n1. If those who win the lottery each month exit the lottery, and those who do not win continue in the following month's lottery, wi... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 34716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ with distinct terms, given that $a_3a_5=3a_7$, and $S_3=9$. \n$(1)$ Find the general formula for the sequence $\\{a_n\\}$. \n$(2)$ Let $T_n$ be the sum of the first $n$ terms of the sequence $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 34717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the different prime factors of $150280?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 34718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers from 1 to 8 are placed at the vertices of a cube so that the sum of the numbers at any three vertices, which lie on the same face, is at least 10. What is the minimum possible sum of the numbers on one face? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 34719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(ABCD\\) is a square, points \\(E\\) and \\(F\\) lie on the side \\(BC\\) and \\(CD\\) respectively, such that \\(BE = CF = \\frac{1}{3} AB\\). \\(G\\) is the intersection of \\(BF\\) and \\(DE\\). If\n\\[\n\\frac{\\text{Area of } ABGD}{\\text{Area of } ABCD} = \\f... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 34720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We suppose that $AB=1$, and that the oblique segments form an angle of $45^{\\circ}$ with respect to $(AB)$. There are $n$ vertices above $(AB)$. \n\nWhat is the length of the broken line? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 34721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In equilateral triangle $ABC$ with side length 2, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_{1}$ and $A_{2}$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_{1}$ and $B... | math | {
"style": "rule",
"ground_truth": "66-36\\sqrt{3}"
} | {
"split": "train",
"index": 34722,
"solution": "Since everything is equilateral it's easy to find the side length of the wanted triangle. By symmetry, it's just $AA_{1}+2A_{1}B_{2}=3AA_{1}-AB$. Using the definition of a parabola, $AA_{1}=\\frac{\\sqrt{3}}{2}A_{1}B$ so some calculation gives a side length of $2(1... |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$. Find the sum of the first $60$ terms of $\\{a_n\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1830"
} | {
"split": "train",
"index": 34723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a city with 10 parallel streets and 10 streets crossing them at right angles, what is the minimum number of turns that a closed bus route passing through all intersections can have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 34724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\Omega=\\left\\{(x, y, z) \\in \\mathbb{Z}^{3}: y+1 \\geq x \\geq y \\geq z \\geq 0\\right\\}\\). A frog moves along the points of \\(\\Omega\\) by jumps of length 1. For every positive integer \\(n\\), determine the number of paths the frog can take to reach \\((n, n, ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\binom{3 n}{n}}{2 n+1}"
} | {
"split": "train",
"index": 34725,
"solution": "Let \\(\\Psi=\\left\\{(u, v) \\in \\mathbb{Z}^{3}: v \\geq 0, u \\geq 2 v\\right\\}\\). Notice that the map \\(\\pi: \\Omega \\rightarrow \\Psi\\), \\(\\pi(x, y, z)=(x+y, z)\\) is a bijection between the two sets; moreover \\(\\pi\\) projects all allowed paths of t... |
deepscaler | [
{
"role": "user",
"content": "For each integer $n$ greater than 1, let $F(n)$ be the number of solutions of the equation $\\sin x = \\sin nx$ on the interval $[0, \\pi]$. What is $\\sum_{n=2}^{2007} F(n)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2{,}016{,}532"
} | {
"split": "train",
"index": 34726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation \\( x^{2} + mx + 1 + 2i = 0 \\) has real roots. Find the minimum value of the modulus of the complex number \\( m \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2 + 2\\sqrt{5}}"
} | {
"split": "train",
"index": 34727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight people are sitting around a circular table for a meeting, and the recorder is sitting between the leader and the deputy leader. Calculate the total number of different seating arrangements possible, considering arrangements that can be made identical through rotation as t... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 34728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a terminal with fifteen gates arranged in a straight line with exactly $90$ feet between adjacent gates. A passenger's departure gate is assigned at random. Later, the gate is changed to another randomly chosen gate. Calculate the probability that the passenger walks $... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 34729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a spelling bee $50\\%$ of the students were eliminated after the first round. Only $\\frac{1}{3}$ of the remaining students were still in the contest after the second round. If 24 students were still in the contest after the second round, how many students began the contest?... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 34730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 34731,
"solution": "1. **Understanding the Problem**: Mary divides a circle into 12 sectors with central angles that are integers and form an arithmetic sequence. The sum of these angles must be $360^\\circ$ since they complete a circle.\n\n2. **Setting Up the Equations**: Let $a_1$ b... |
deepscaler | [
{
"role": "user",
"content": "Given the expression $2-(-3)-4\\times(-5)-6-(-7)-8\\times(-9)+10$, evaluate this expression. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 34732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a bag, there are a total of 12 balls that are identical in shape and mass, including red, black, yellow, and green balls. When drawing a ball at random, the probability of getting a red ball is $\\dfrac{1}{3}$, the probability of getting a black or yellow ball is $\\dfrac{5}... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{12}"
} | {
"split": "train",
"index": 34733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$, where $a_3$ and $a_7$ are the two roots of the quadratic equation $x^2+7x+9=0$, calculate the value of $a_5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 34734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a table consisting of $n$ rows and $m$ columns, numbers are written such that the sum of the elements in each row is 1248, and the sum of the elements in each column is 2184. Find the numbers $n$ and $m$ for which the expression $2n - 3m$ takes the smallest possible natural ... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 34735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vertices of a regular hexagon are labeled $\\cos (\\theta), \\cos (2 \\theta), \\ldots, \\cos (6 \\theta)$. For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds fo... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 34736,
"solution": "The existence of the Chebyshev polynomials, which express $\\cos (n \\theta)$ as a polynomial in $\\cos (\\theta)$, imply that Bob draws a blue line between $\\cos (\\theta)$ and each other vertex, and also between $\\cos (2 \\theta)$ and $\\cos (4 \\theta)$, betwe... |
deepscaler | [
{
"role": "user",
"content": "Let $a_{0}, a_{1}, a_{2}, \\ldots$ denote the sequence of real numbers such that $a_{0}=2$ and $a_{n+1}=\\frac{a_{n}}{1+a_{n}}$ for $n \\geq 0$. Compute $a_{2012}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{4025}"
} | {
"split": "train",
"index": 34737,
"solution": "Calculating out the first few terms, note that they follow the pattern $a_{n}=\\frac{2}{2 n+1}$. Plugging this back into the recursion shows that it indeed works."
} |
deepscaler | [
{
"role": "user",
"content": "The sides of a triangle have lengths of $13$, $84$, and $85$. Find the length of the shortest altitude. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12.8470588235"
} | {
"split": "train",
"index": 34738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $O$ be the center of the square $ABCD$. If $3$ points are chosen from $O$, $A$, $B$, $C$, and $D$ at random, calculate the probability that the $3$ points are collinear. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 34739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When plotted in the standard rectangular coordinate system, trapezoid $PQRS$ has vertices $P(2, -4)$, $Q(2, 3)$, $R(7, 10)$, and $S(7, 2)$. What is the area of trapezoid $PQRS$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37.5"
} | {
"split": "train",
"index": 34740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone is inverted and filled with water to 2/3 of its height. What percent of the cone's volume and surface area (not including the base) are filled with water and exposed to air, respectively? Express your answer as a decimal to the nearest ten-thousandth. Let'... | math | {
"style": "rule",
"ground_truth": "55.5556\\%"
} | {
"split": "train",
"index": 34741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic $4x^2 - 40x + 100$ can be written in the form $(ax+b)^2 + c$, where $a$, $b$, and $c$ are constants. What is $2b-3c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 34742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( P \\) be the midpoint of the height \\( VH \\) of a regular square pyramid \\( V-ABCD \\). If the distance from point \\( P \\) to a lateral face is 3 and the distance to the base is 5, find the volume of the regular square pyramid. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 34743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a Martian civilization, all logarithms whose bases are not specified as assumed to be base $b$, for some fixed $b\\ge2$. A Martian student writes down \\[3\\log(\\sqrt{x}\\log x)=56\\] \\[\\log_{\\log x}(x)=54\\] and finds that this system of equations has a single real numb... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 34744,
"solution": "Let $y = \\log _{b} x$ Then we have \\[3\\log _{b} (y\\sqrt{x}) = 56\\] \\[\\log _{y} x = 54\\] which gives \\[y^{54} = x\\] Plugging this in gives \\[3\\log _{b} (y \\cdot y^{27}) = 3\\log _{b} y^{28} = 56\\] which gives \\[\\log _{b} y = \\dfrac{2}{3}\\] so \\[b^... |
deepscaler | [
{
"role": "user",
"content": "Given \\( a \\in \\mathbb{R} \\), the complex number \\( z = \\frac{(a-i)(1+i)}{i} \\), if \\( \\bar{z} = z \\), then find the value of \\( a \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 34745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 34746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Out of three hundred eleventh-grade students, 77% received excellent and good grades on the first exam, 71% on the second exam, and 61% on the third exam. What is the minimum number of participants who received excellent and good grades on all three exams? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 34747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g$ be a function defined for all real numbers that satisfies $g(3+x) = g(3-x)$ and $g(8+x) = g(8-x)$ for all $x$. If $g(0) = 0$, determine the least number of roots $g(x) = 0$ must have in the interval $-1000 \\leq x \\leq 1000$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "402"
} | {
"split": "train",
"index": 34748,
"solution": ""
} |
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