data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A semicircle with a radius of 1 is drawn inside a semicircle with a radius of 2. A circle is drawn such that it touches both semicircles and their common diameter. What is the radius of this circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"split": "train",
"index": 39649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a function $f(x)$ with domain $I$, if there exists an interval $\\left[m,n\\right]\\subseteq I$ such that $f(x)$ is a monotonic function on the interval $\\left[m,n\\right]$, and the range of the function $y=f(x)$ for $x\\in \\left[m,n\\right]$ is $\\left[m,n\\right]$, then... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 39650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The matrices\n\\[\\begin{pmatrix} 3 & -8 \\\\ a & 11 \\end{pmatrix} \\quad \\text{and} \\quad \\begin{pmatrix} 11 & b \\\\ 4 & 3 \\end{pmatrix}\\]are inverses. Enter the ordered pair $(a,b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-4,8)"
} | {
"split": "train",
"index": 39651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the following diagram, \\(ABCD\\) is a square, and \\(E\\) is the center of the square \\(ABCD\\). \\(P\\) is a point on a semi-circle with diameter \\(AB\\). \\(Q\\) is a point on a semi-circle with diameter \\(AD\\). Moreover, \\(Q, A,\\) and \\(P\\) are collinear (that is... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 39652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), the perpendicular bisectors of sides \\(AB\\) and \\(AC\\) are drawn, intersecting lines \\(AC\\) and \\(AB\\) at points \\(N\\) and \\(M\\) respectively. The length of segment \\(NM\\) is equal to the length of side \\(BC\\) of the triangle. The angle at... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 39653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a school bus leaves school with 48 students on board, and one-half of the students get off the bus at each of the first three stops, how many students remain on the bus after the third stop? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 39654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 39655,
"solution": "1. **Identify the triangle and variables**: Let's denote the length of each congruent side of the isosceles triangle as $a$. Let $\\theta$ be the measure of each of the two congruent (acute) angles at the base, and $\\phi$ be the measure of the vertex angle, which ... |
deepscaler | [
{
"role": "user",
"content": "A sequence consists of the digits $122333444455555 \\ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the 4501st and 4052nd digits of this sequence. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 39656,
"solution": "Note that $n$ contributes $n \\cdot d(n)$ digits, where $d(n)$ is the number of digits of $n$. Then because $1+\\cdots+99=4950$, we know that the digits of interest appear amongst copies of two digit numbers. Now for $10 \\leq n \\leq 99$, the number of digits in t... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), \\(AC = 18 \\, \\text{cm}\\) and \\(BC = 21 \\, \\text{cm}\\). Point \\(K\\) is the midpoint of side \\(BC\\), and point \\(M\\) is the midpoint of side \\(AB\\). Point \\(N\\) lies on side \\(AC\\) such that \\(AN = 6 \\, \\text{cm}\\). Additionally, \\(... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $333 + 33 + 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "369"
} | {
"split": "train",
"index": 39658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the digits left and right of the decimal point in the decimal form of the number \\[ (\\sqrt{2} + \\sqrt{3})^{1980}. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.9"
} | {
"split": "train",
"index": 39659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N \\ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$ . Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other i... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers \\( x \\) and \\( y \\) satisfy \\( 4x^2 - 5xy + 4y^2 = 5 \\). Let \\( S = x^2 + y^2 \\). Find the value of \\( \\frac{1}{S_{\\max}} + \\frac{1}{S_{\\min}} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8/5"
} | {
"split": "train",
"index": 39661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all 6-digit multiples of 22 of the form $5d5,\\!22e$ where $d$ and $e$ are digits. What is the maximum value of $d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 39662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the relationship between \\(\\arcsin \\cos \\arcsin x\\) and \\(\\arccos \\sin \\arccos x\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 39663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Thirty-nine students from seven classes came up with 60 problems, with students of the same class coming up with the same number of problems (not equal to zero), and students from different classes coming up with a different number of problems. How many students came up with on... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 39664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers \\( n \\) that satisfy\n\\[ 20 < n^2 < 200. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What number is placed in the shaded circle if each of the numbers $1,5,6,7,13,14,17,22,26$ is placed in a different circle, the numbers 13 and 17 are placed as shown, and Jen calculates the average of the numbers in the first three circles, the average of the numbers in the mid... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39666,
"solution": "The averages of groups of three numbers are equal if the sums of the numbers in each group are equal, because in each case the average equals the sum of the three numbers divided by 3. Therefore, the averages of three groups of three numbers are equal if the sum of... |
deepscaler | [
{
"role": "user",
"content": "A club has increased its membership to 12 members and needs to elect a president, vice president, secretary, and treasurer. Additionally, they want to appoint two different advisory board members. Each member can hold only one position. In how many ways can these positions be f... | math | {
"style": "rule",
"ground_truth": "665,280"
} | {
"split": "train",
"index": 39667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x \\gt 0$, $y \\gt 0$, $x+2y=1$, calculate the minimum value of $\\frac{{(x+1)(y+1)}}{{xy}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8+4\\sqrt{3}"
} | {
"split": "train",
"index": 39668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As $p$ ranges over the primes greater than $5$, how many different remainders can $p^2$ leave upon division by $120$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 39669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n, m \\) be positive integers. Define the sets \\( A = \\{1, 2, \\ldots, n\\} \\) and \\( B_{n}^{m} = \\{ (a_1, a_2, \\ldots, a_m) \\mid a_i \\in A, i = 1, 2, \\ldots, m \\} \\) that satisfy:\n1. \\( |a_i - a_{i+1}| \\neq n-1 \\) for \\( i = 1, 2, \\ldots, m-1 \\);\n2. ... | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 39670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cuboid has dimensions of 2 units by 2 units by 2 units. It has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3', P_4'.$ Vertices $P_2, P_3,$ and $P_4$ are adjacent to $P_1$, and vertices $P_i' (i = 1,2,3,4)$ are opposite to $P_i$. A regular octahedron has one vertex in each of ... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 39671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $O$ is the center of the regular octagon $ABCDEFGH$, and $Y$ is the midpoint of side $\\overline{CD}$. Calculate the fraction of the area of the octagon that is shaded, where the shaded region includes triangles $\\triangle DEO$, $\\triangle EFO$, $\\triangle FGO$, and ha... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 39672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The geometric series $a+ar+ar^2+\\cdots$ has a sum of $12$, and the terms involving odd powers of $r$ have a sum of $5.$ What is $r$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"split": "train",
"index": 39673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[f(f(x) + y) = f(x + y) + xf(y) - xy - x + 1\\]for all real numbers $x$ and $y.$\n\nLet $n$ be the number of possible values of $f(1),$ and let $s$ be the sum of all possible values of $f(1).$ Find $n \\times s.... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 39674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(m, n) = 3m + n + (m + n)^2 \\). Find \\( \\sum_{m=0}^{\\infty} \\sum_{n=0}^{\\infty} 2^{-f(m, n)} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"split": "train",
"index": 39675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $P$ is chosen in the interior of $\\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\\triangle ABC$, the resulting smaller triangles $t_{1}$, $t_{2}$, and $t_{3}$ in the figure, have areas $4$, $9$, and $49$, respectively. Find the are... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 39676,
"solution": "By the transversals that go through $P$, all four triangles are similar to each other by the $AA$ postulate. Also, note that the length of any one side of the larger triangle is equal to the sum of the sides of each of the corresponding sides on the smaller triangl... |
deepscaler | [
{
"role": "user",
"content": "If $2137^{753}$ is multiplied out, the units' digit in the final product is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39677,
"solution": "1. **Identify the units digit of the base number**: The units digit of $2137$ is $7$. Therefore, the units digit of $2137^{753}$ will be the same as the units digit of $7^{753}$.\n\n2. **Determine the pattern of units digits for powers of $7$**:\n - $7^1$ has a u... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $n$ less than $1000$ have the property that the number of positive integers less than $n$ which are coprime to $n$ is exactly $\\tfrac n3$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 39678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\( \\frac{2^{3}-1}{2^{3}+1} \\cdot \\frac{3^{3}-1}{3^{3}+1} \\cdot \\frac{4^{3}-1}{4^{3}+1} \\cdot \\frac{5^{3}-1}{5^{3}+1} \\cdot \\frac{6^{3}-1}{6^{3}+1} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43/63"
} | {
"split": "train",
"index": 39679,
"solution": "Use the factorizations \\( n^{3}-1=(n-1)\\left(n^{2}+n+1\\right) \\) and \\( n^{3}+1=(n+1)\\left(n^{2}-n+1\\right) \\) to write \\( \\frac{1 \\cdot 7}{3 \\cdot 3} \\cdot \\frac{2 \\cdot 13}{4 \\cdot 7} \\cdot \\frac{3 \\cdot 21}{5 \\cdot 13} \\cdot \\frac{4 \\cdot ... |
deepscaler | [
{
"role": "user",
"content": "Given that quadrilateral \\(ABCD\\) is a right trapezoid with the upper base \\(AD = 8\\) cm, the lower base \\(BC = 10\\) cm, and the right leg \\(CD = 6\\) cm. Point \\(E\\) is the midpoint of \\(AD\\), and point \\(F\\) is on \\(BC\\) such that \\(BF = \\frac{2}{3} BC\\). Po... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 39680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of queens that can be placed on an $8 \\times 8$ chessboard so that each queen can attack at least one other queen? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the list $2, x, y, 5$, the sum of any two adjacent numbers is constant. What is the value of $x-y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 39682,
"solution": "Since the sum of any two adjacent numbers is constant, then $2 + x = x + y$. This means that $y = 2$ and makes the list $2, x, 2, 5$. This means that the sum of any two adjacent numbers is $2 + 5 = 7$, and so $x = 5$. Therefore, $x - y = 5 - 2 = 3$."
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $A B C$, let the parabola with focus $A$ and directrix $B C$ intersect sides $A B$ and $A C$ at $A_{1}$ and $A_{2}$, respectively. Similarly, let the parabola with focus $B$ and directrix $C A$ intersect sides $B C$ and $B A$ at $B_{1}$ and $B_{2}$, respectively. Fi... | math | {
"style": "rule",
"ground_truth": "\\frac{6728}{3375}"
} | {
"split": "train",
"index": 39683,
"solution": "By the definition of a parabola, we get $A A_{1}=A_{1} B \\sin B$ and similarly for the other points. So $\\frac{A B_{2}}{A B}=\\frac{A C_{1}}{A C}$, giving $B_{2} C_{1} \\| B C$, and similarly for the other sides. So $D E F$ (WLOG, in that order) is similar to $A ... |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is on the $y$-axis with $y$-coordinate greater than 0 and less than 100. A circle is drawn through $P, Q(4,4)$ and $O(0,0)$. How many possible positions for $P$ are there so that the radius of this circle is an integer? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 39684,
"solution": "Suppose that $P$ has coordinates $P(0,2a)$ for some real number $a$. Since $P$ has $y$-coordinate greater than 0 and less than 100, then $0 < 2a < 100$ or $0 < a < 50$. We determine an expression for the radius of the circle in terms of $a$ and then determine how m... |
deepscaler | [
{
"role": "user",
"content": "Let $O$ and $H$ denote the circumcenter and orthocenter of triangle $ABC,$ respectively. If $AO = AH,$ then enter all possible values of $\\angle A$ (in degrees), separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60^\\circ, 120^\\circ"
} | {
"split": "train",
"index": 39685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1,$ $x_2,$ $x_3,$ $x_4,$ $x_5$ be the roots of the polynomial $f(x) = x^5 + x^2 + 1,$ and let $g(x) = x^2 - 2.$ Find\n\\[g(x_1) g(x_2) g(x_3) g(x_4) g(x_5).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-23"
} | {
"split": "train",
"index": 39686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C$ are given by: $\\begin{cases} x=2\\cos\\theta \\\\ y=\\sin\\theta \\end{cases}$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 39687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn the product\n\n\\[ \n24^{a} \\cdot 25^{b} \\cdot 26^{c} \\cdot 27^{d} \\cdot 28^{e} \\cdot 29^{f} \\cdot 30^{g} \n\\]\n\nseven numbers \\(1, 2, 3, 5, 8, 10, 11\\) were assigned to the exponents \\(a, b, c, d, e, f, g\\) in some order. Find the maximum number of zeros that ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 39688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\\\(\\cos \\left(\\alpha+ \\frac{\\pi}{6}\\right)-\\sin \\alpha= \\frac{2 \\sqrt{3}}{3} \\\\), determine the value of \\\\(\\sin \\left(\\alpha- \\frac{7\\pi}{6}\\right) \\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 39689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the equation $2kx+2m=6-2x+nk$ has a solution independent of $k$, the value of $4m+2n$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\sqrt{3+2\\sqrt{2}}-\\sqrt{3-2\\sqrt{2}}$ is equal to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 39691,
"solution": "We are given the expression $\\sqrt{3+2\\sqrt{2}}-\\sqrt{3-2\\sqrt{2}}$ and need to find its value among the choices provided.\n\n1. **Square the expression** to simplify:\n \\[\n \\left(\\sqrt{3+2\\sqrt{2}} - \\sqrt{3-2\\sqrt{2}}\\right)^2 = (\\sqrt{3+2\\sqrt{... |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $XY = 15$, $XZ = 17$, and $YZ = 24$. The medians $XM$, $YN$, and $ZL$ of triangle $XYZ$ intersect at the centroid $G$. Let $Q$ be the foot of the altitude from $G$ to $YZ$. Find $GQ$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"split": "train",
"index": 39692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has length $AC = 40$ and width $AE = 24$. Point $B$ is positioned one-third of the way along $AC$ from $A$ to $C$, and point $F$ is halfway along $AE$. Find the total area enclosed by rectangle $ACDE$ and the semicircle with diameter $AC$ minus the area of quadrilat... | math | {
"style": "rule",
"ground_truth": "800 + 200\\pi"
} | {
"split": "train",
"index": 39693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with the midpoints of sides \\(BC\\), \\(AC\\), and \\(AB\\) denoted as \\(D\\), \\(E\\), and \\(F\\) respectively, it is known that the medians \\(AD\\) and \\(BE\\) are perpendicular to each other, with lengths \\(\\overline{AD} = 18\\) and \\(\\ove... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 39694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $WXYZ$ is a trapezoid such that $\\overline{WX}\\parallel \\overline{ZY}$ and $\\overline{WY}\\perp\\overline{ZY}$. If $YZ = 15$, $\\tan Z = \\frac{4}{3}$, and $\\tan X = \\frac{3}{2}$, what is the length of $XY$? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{20\\sqrt{13}}{3}"
} | {
"split": "train",
"index": 39695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $((-3)^2)^4(-3)^8 \\cdot 2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "86093442"
} | {
"split": "train",
"index": 39696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equations: \n(1) $(x-3)^2+2x(x-3)=0$ \n(2) $x^2-4x+1=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2-\\sqrt{3}"
} | {
"split": "train",
"index": 39697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\\begin{cases}x=2\\cos \\alpha \\\\ y=\\sin \\alpha \\\\ \\end{cases}$ (where $\\alpha$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive $... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 39698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's consider two fictional states, Sunland and Moonland, which have different license plate formats. Sunland license plates have the format LLDDLLL (where 'L' stands for a letter and 'D' for a digit), while Moonland license plates have the format LLLDDD. Assuming all 10 digit... | math | {
"style": "rule",
"ground_truth": "1170561600"
} | {
"split": "train",
"index": 39699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square with a side length of 12 cm, the midpoints of its adjacent sides are connected to each other and to the opposite side of the square. Find the radius of the circle inscribed in the resulting triangle. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5} - \\sqrt{2}"
} | {
"split": "train",
"index": 39700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x|x|.$ Find $f^{-1}(4) + f^{-1}(-100).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 39701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is inside $\\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\\triangle ABC$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 39702,
"solution": "Because we're given three concurrent cevians and their lengths, it seems very tempting to apply Mass points. We immediately see that $w_E = 3$, $w_B = 1$, and $w_A = w_D = 2$. Now, we recall that the masses on the three sides of the triangle must be balanced out, s... |
deepscaler | [
{
"role": "user",
"content": "Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\\angle ABD$ exceeds $\\angle AHG$ by $12^\\circ$. Find the degree me... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 39703,
"solution": "Let $O$ be the center of the circle with $ABCDE$ on it.\n Let $x$ be the degree measurement of $\\overarc{ED}=\\overarc{DC}=\\overarc{CB}=\\overarc{BA}$ in circle $O$\n and $y$ be the degree measurement of $\\overarc{EF}=\\overarc{FG}=\\overarc{GH}=\\overarc{HI}=\\... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a subset of $\\{1, 2, . . . , 500\\}$ such that no two distinct elements of S have a\nproduct that is a perfect square. Find, with proof, the maximum possible number of elements\nin $S$ . Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "306"
} | {
"split": "train",
"index": 39704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a complex number $z$ satisfying the equation $|z-1|=|z+2i|$ (where $i$ is the imaginary unit), find the minimum value of $|z-1-i|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{5}}{10}"
} | {
"split": "train",
"index": 39705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos (\\omega x+\\varphi)$ ($\\omega > 0$, $|\\varphi| \\leqslant \\frac {\\pi}{2}$), when $x=- \\frac {\\pi}{4}$, the function $f(x)$ can achieve its minimum value, and when $x= \\frac {\\pi}{4}$, the function $y=f(x)$ can achieve its maximum value. M... | math | {
"style": "rule",
"ground_truth": "- \\frac {\\pi}{2}"
} | {
"split": "train",
"index": 39706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles triangle $ABC$, with $AB=AC$, is fixed in the plane. A point $P$ is randomly placed within the triangle. The length of each equal side $AB$ and $AC$ is 6 units and the base $BC$ is 8 units. What is the probability that the area of triangle $PBC$ is more than one-th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 39707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $a \\neq b$, $c = \\sqrt{3}$, and $\\sqrt{3} \\cos^2 A - \\sqrt{3} \\cos^2 B = \\sin A \\cos A - \\sin B \\cos B$.\n\n(I) Find the measure of angle $C$;\n\n... | math | {
"style": "rule",
"ground_truth": "\\frac{24\\sqrt{3} + 18}{25}"
} | {
"split": "train",
"index": 39708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, let $a_{n}$ be the smallest nonnegative integer such that there is only one positive integer at most $n$ that is relatively prime to all of $n, n+1, \\ldots, n+a_{n}$. If $n<100$, compute the largest possible value of $n-a_{n}$. Let's think step b... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39709,
"solution": "Note that 1 is relatively prime to all positive integers. Therefore, the definition of $a_{n}$ can equivalently be stated as: \"$a_{n}$ is the smallest nonnegative integer such that for all integers $x, 2 \\leq x \\leq n$, $x$ shares a prime factor with at least on... |
deepscaler | [
{
"role": "user",
"content": "A scale drawing of a park shows that one inch represents 800 feet. A line segment in the drawing that is 4.75 inches long represents how many feet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3800"
} | {
"split": "train",
"index": 39710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $c=a\\cos B+b\\sin A$. \n(I) Find angle $A$. \n(II) If $a=2$, find the maximum area of $\\triangle ABC$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}+1"
} | {
"split": "train",
"index": 39711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parabola, given by the equation $y^{2}=2px (p > 0)$, has a focus that lies on the line $l$. This line intersects the parabola at two points, $A$ and $B$. A circle with the chord $AB$ as its diameter has the equation $(x-3)^{2}+(y-2)^{2}=16$. Find the value of $p$. Let's think... | math | {
"style": "rule",
"ground_truth": "p = 2"
} | {
"split": "train",
"index": 39712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the digits of a 5 -digit number is 180 . How many such numbers exist? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 39713,
"solution": "Let the digits be $a, b, c, d, e$. Then $a b c d e=180=2^{2} \\cdot 3^{2} \\cdot 5$. We observe that there are 6 ways to factor 180 into digits $a, b, c, d, e$ (ignoring differences in ordering): $180=$ $1 \\cdot 1 \\cdot 4 \\cdot 5 \\cdot 9=1 \\cdot 1 \\cdot 5 \\c... |
deepscaler | [
{
"role": "user",
"content": "289. A remarkable number. Find a number such that its fractional part, its integer part, and the number itself form a geometric progression. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 39714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C$: $\\frac{x^2}{a^2} - y^2 = 1$ $(a > 0)$ and the line $l$: $x + y = 1$, which intersect at two distinct points $A$ and $B$.\n\n1. Find the range of values for $a$.\n2. Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\\overrightarrow{P... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{13}"
} | {
"split": "train",
"index": 39715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For which smallest \\( n \\) do there exist \\( n \\) numbers in the interval \\( (-1, 1) \\) such that their sum is 0, and the sum of their squares is 36? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 39716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the real number sequence: -1, $a_1$, $a_2$, $a_3$, -81 forms a geometric sequence, determine the eccentricity of the conic section $x^2+ \\frac{y^2}{a_2}=1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 39717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\\text{th}}$ integer in the list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31254"
} | {
"split": "train",
"index": 39718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6, BC=5=DA,$and $CD=4.$ Draw circles of radius 3 centered at $A$ and $B,$ and circles of radius 2 centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radiu... | math | {
"style": "rule",
"ground_truth": "134"
} | {
"split": "train",
"index": 39719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $(8x - 5)^2$ given that the number $x$ satisfies the equation $7x^2 + 6 = 5x + 11$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2865 - 120\\sqrt{165}}{49}"
} | {
"split": "train",
"index": 39720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work wit... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 39721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x)$ that always satisfies the following conditions on its domain $\\mathbb{R}$: \n① $f(x) = f(-x)$, ② $f(2+x) = f(2-x)$, when $x \\in [0, 4)$, $f(x) = -x^2 + 4x$. \n(1) Find $f(8)$. \n(2) Find the number of zeros of $f(x)$ in $[0, 2015]$. Let's think step... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 39722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest three-digit multiple of 9 whose digits' sum is 18? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "990"
} | {
"split": "train",
"index": 39723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set $\\{8, -7, 2, -4, 20\\}$. Find the smallest sum that can be achieved by adding three different numbers from this set. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 39724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of the x term in the expansion of $(x^2-x-2)^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 39725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the \"Black White Pair\" game, a common game among children often used to determine who goes first, participants (three or more) reveal their hands simultaneously, using the palm (white) or the back of the hand (black) to decide the winner. If one person shows a gesture diff... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 39726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain sequence of numbers, the first number is $1$, and, for all $n\\ge 2$, the product of the first $n$ numbers in the sequence is $n^2$. The sum of the third and the fifth numbers in the sequence is Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{61}{16}"
} | {
"split": "train",
"index": 39727,
"solution": "1. **Identify the pattern in the sequence**: Given that the product of the first $n$ numbers in the sequence is $n^2$, we can express this as:\n \\[\n a_1 \\cdot a_2 \\cdot \\ldots \\cdot a_n = n^2\n \\]\n where $a_i$ represents the $i$-th number in the seq... |
deepscaler | [
{
"role": "user",
"content": "The roots of the equation $2\\sqrt{x} + 2x^{-\\frac{1}{2}} = 5$ can be found by solving: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4x^2-17x+4 = 0"
} | {
"split": "train",
"index": 39728,
"solution": "1. **Substitute and Simplify**: Let $t = \\sqrt{x}$. Then, $t^2 = x$ and $x^{-\\frac{1}{2}} = \\frac{1}{t}$. Substitute these into the given equation:\n \\[\n 2\\sqrt{x} + 2x^{-\\frac{1}{2}} = 5 \\implies 2t + \\frac{2}{t} = 5.\n \\]\n\n2. **Form a Quadratic ... |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$, $\\sin A = \\frac{3}{5}$ 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{4}{5}"
} | {
"split": "train",
"index": 39729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f:\\mathbb{R}\\to\\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x - 1 \\text{ or } f(x) = -x - 1."
} | {
"split": "train",
"index": 39730,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that for all real numbers \\( x \\) and \\( y \\), the following functional equation holds:\n\n\\[\nf(x)f(y) + f(x+y) = xy.\n\\]\n\n**Step 1: Plug-in Specific Values**\n\nFirst,... |
deepscaler | [
{
"role": "user",
"content": "A square has vertices at \\((-2a, -2a), (2a, -2a), (-2a, 2a), (2a, 2a)\\). The line \\( y = -\\frac{x}{2} \\) cuts this square into two congruent quadrilaterals. Calculate the perimeter of one of these quadrilaterals divided by \\( 2a \\). Express your answer in simplified radi... | math | {
"style": "rule",
"ground_truth": "4 + \\sqrt{5}"
} | {
"split": "train",
"index": 39731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^{3} + ax^{2} - 2x + 1$ has an extremum at $x=1$. \n$(1)$ Find the value of $a$; \n$(2)$ Determine the monotonic intervals and extremum of $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 39732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[(15 - x)(8 - x)(15 + x)(8 + x).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6480.25"
} | {
"split": "train",
"index": 39733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of a semicircle with an area of ______ square meters is 15.42 meters. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14.13"
} | {
"split": "train",
"index": 39734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A one-meter gas pipe has rusted through in two places. Determine the probability that all three resulting parts can be used as connectors to gas stoves, given that according to regulations, the stove must not be closer than 25 cm to the main gas pipe. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "1/16"
} | {
"split": "train",
"index": 39735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The denominator of a geometric progression \\( b_{n} \\) is \\( q \\), and for some natural \\( n \\geq 2 \\),\n\n$$\n\\log_{4} b_{2}+\\log_{4} b_{3}+\\ldots+\\log_{4} b_{n}=4 \\cdot \\log_{4} b_{1}\n$$\n\nFind the smallest possible value of \\( \\log_{q} b_{1}^{2} \\), given t... | math | {
"style": "rule",
"ground_truth": "-30"
} | {
"split": "train",
"index": 39736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 39737,
"solution": "1. **Identify the prime factors of $12!$ and their exponents:** \n We start by finding the exponents of the prime factors of $12!$. For any prime $p$, the exponent of $p$ in $n!$ is given by:\n \\[\n \\sum_{k=1}^{\\infty} \\left\\lfloor \\frac{n}{p^k} \\righ... |
deepscaler | [
{
"role": "user",
"content": "Within the range of 0° to 360°, find the angle(s) with the same terminal side as -120°. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 39738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Voldemort bought a book for $\\$5$. It was one-tenth of its original price. What was the original price in dollars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$50"
} | {
"split": "train",
"index": 39739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kevin Kangaroo begins his journey on a number line at 0 with a goal of reaching a rock located at 1. However, he hops only $\\frac{1}{4}$ of the distance toward the rock with each leap. After each leap, he tires, reducing his subsequent hop to $\\frac{1}{4}$ of the remaining di... | math | {
"style": "rule",
"ground_truth": "\\frac{14197}{16384}"
} | {
"split": "train",
"index": 39740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn a math competition with problems $A$, $B$, and $C$, there are 39 participants, each of whom answered at least one question correctly. Among those who answered problem $A$ correctly, the number of participants who answered only problem $A$ is 5 more than those who also answ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 39741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of triangle $PQR$ are in the ratio $3:4:5$. Segment $QS$ is the angle bisector drawn to the longest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the shorter subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Express ... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{7}"
} | {
"split": "train",
"index": 39742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have 10 distinguishable socks in my drawer: 4 white, 4 brown, and 2 blue. In how many ways can I choose a pair of socks, provided that I get two socks of the same color? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 39743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is very boring to look at a black-and-white clock face, so Clive painted the number 12 red exactly at noon and decided to paint the current hour red every 57 hours.\n\na) How many numbers on the clock face will be painted?\n\nb) How many numbers will be red if Clive paints t... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10,001 students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of \\( k \\) societies. Suppo... | math | {
"style": "rule",
"ground_truth": "5000"
} | {
"split": "train",
"index": 39745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many rectangles can be formed where each of the four vertices are points on a 4x4 grid with points spaced evenly along the grid lines? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 39746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of \n\\[3003 + \\frac{1}{3} \\left( 3002 + \\frac{1}{3} \\left( 3001 + \\dots + \\frac{1}{3} \\left( 4 + \\frac{1}{3} \\cdot 3 \\right) \\right) \\dotsb \\right).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9006"
} | {
"split": "train",
"index": 39747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[\\bold{A} = \\begin{pmatrix} 0 & 1 & 2 \\\\ 1 & 0 & 1 \\\\ 2 & 1 & 0 \\end{pmatrix}.\\]There exist constants $p$, $q$, and $r$ such that\n\\[\\bold{A}^3 + p \\bold{A}^2 + q \\bold{A} + r \\bold{I} = \\bold{0},\\]where $\\bold{I}$ and $\\bold{0}$ are the $3 \\times 3$ id... | math | {
"style": "rule",
"ground_truth": "(0,-6,-4)"
} | {
"split": "train",
"index": 39748,
"solution": ""
} |
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