level stringclasses 6
values | type stringclasses 7
values | data_source stringclasses 1
value | prompt stringlengths 86 4.38k | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|---|
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points $(x_1,y_1),$ $(x_2,y_2),$ $(x_3,y_3),$ and $(x_4,y_4).$ Find
\[x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 3600,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$
\[\sqrt{\frac{a}{b + c + d}} + \sqrt{\frac{b}{a + c + d}} + \sqrt{\frac{c}{a + b + d}} + \sqrt{\frac{d}{a + b + c}} > m.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 3601,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | An ellipse has foci $(2, 2)$ and $(2, 6)$, and it passes through the point $(14, -3).$ Given this, we can write the equation of the ellipse in standard form as \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find the ordered quadruple $(a, b, h, k)$.
(E... | math | {
"ground_truth": " (8\\sqrt3, 14, 2, 4)",
"style": "rule"
} | {
"index": 3602,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | How many of the first $1000$ positive integers can be expressed in the form
\[\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$? Let's think step by step and output the final answe... | math | {
"ground_truth": "600",
"style": "rule"
} | {
"index": 3603,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | For a positive integer $n,$ let
\[a_n = \sum_{k = 0}^n \frac{1}{\binom{n}{k}} \quad \text{and} \quad b_n = \sum_{k = 0}^n \frac{k}{\binom{n}{k}}.\]Simplify $\frac{a_n}{b_n}.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{2}{n}",
"style": "rule"
} | {
"index": 3604,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | 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 output the final answer within \boxed{}. | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 3605,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The numbers $a_1,$ $a_2,$ $a_3,$ $b_1,$ $b_2,$ $b_3,$ $c_1,$ $c_2,$ $c_3$ are equal to the numbers $1,$ $2,$ $3,$ $\dots,$ $9$ in some order. Find the smallest possible value of
\[a_1 a_2 a_3 + b_1 b_2 b_3 + c_1 c_2 c_3.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "214",
"style": "rule"
} | {
"index": 3606,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the maximum value of
\[f(x) = 3x - x^3\]for $0 \le x \le \sqrt{3}.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 3607,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the equation of the directrix of the parabola $x = -\frac{1}{6} y^2.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "x = \\frac{3}{2}",
"style": "rule"
} | {
"index": 3608,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | An ellipse has its foci at $(-1, -1)$ and $(-1, -3).$ Given that it passes through the point $(4, -2),$ its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$ Let's think step by step and output the final answe... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 3609,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the quadratic function $f(x) = x^2 + ax + b$ such that
\[\frac{f(f(x) + x)}{f(x)} = x^2 + 1776x + 2010.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "x^2 + 1774x + 235",
"style": "rule"
} | {
"index": 3610,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | What is the value of the sum
\[
\sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,
\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{49}{4}",
"style": "rule"
} | {
"index": 3611,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find a quadratic with rational coefficients and quadratic term $x^2$ that has $\sqrt{3}-2$ as a root. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "x^2+4x+1",
"style": "rule"
} | {
"index": 3612,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | A parabola has vertex $V = (0,0)$ and focus $F = (0,1).$ Let $P$ be a point in the first quadrant, lying on the parabola, so that $PF = 101.$ Find $P.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "(20,100)",
"style": "rule"
} | {
"index": 3613,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Is
\[f(x) = \log (x + \sqrt{1 + x^2})\]an even function, odd function, or neither?
Enter "odd", "even", or "neither". Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\text{odd}",
"style": "rule"
} | {
"index": 3614,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | A positive real number $x$ is such that \[
\sqrt[3]{1-x^3} + \sqrt[3]{1+x^3} = 1.
\]Find $x^6.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{28}{27}",
"style": "rule"
} | {
"index": 3615,
"split": "train"
} |
Level 1 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Express the following sum as a simple fraction in lowest terms.
$$\frac{1}{1\times2} + \frac{1}{2\times3} + \frac{1}{3\times4} + \frac{1}{4\times5} + \frac{1}{5\times6}$$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": 3616,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i^2 = -1.$ Find $b.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "51",
"style": "rule"
} | {
"index": 3617,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Given that $x<1$ and \[(\log_{10} x)^2 - \log_{10}(x^2) = 48,\]compute the value of \[(\log_{10}x)^3 - \log_{10}(x^3).\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-198",
"style": "rule"
} | {
"index": 3618,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $a,$ $b,$ and $c$ be the roots of $x^3 - 7x^2 + 5x + 2 = 0.$ Find
\[\frac{a}{bc + 1} + \frac{b}{ac + 1} + \frac{c}{ab + 1}.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{15}{2}",
"style": "rule"
} | {
"index": 3619,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find all real values of $x$ which satisfy
\[\frac{1}{x + 1} + \frac{6}{x + 5} \ge 1.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "(-5,-2] \\cup (-1,3]",
"style": "rule"
} | {
"index": 3620,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "200",
"style": "rule"
} | {
"index": 3621,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | For integers $a$ and $T,$ $T \neq 0,$ a parabola whose general equation is $y = ax^2 + bx + c$ passes through the points $A = (0,0),$ $B = (2T,0),$ and $C = (2T + 1,28).$ Let $N$ be the sum of the coordinates of the vertex point. Determine the largest value of $N.$ Let's think step by step and output the final answer... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 3622,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12},$ find all possible values for $x.$
(Enter your answer as a comma-separated list.) Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "4, 6",
"style": "rule"
} | {
"index": 3623,
"split": "train"
} |
Level 1 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find all roots of the polynomial $x^3+x^2-4x-4$. Enter your answer as a list of numbers separated by commas. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-1,2,-2",
"style": "rule"
} | {
"index": 3624,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{2}{b}.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 3625,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Compute
\[\frac{\lfloor \sqrt[4]{1} \rfloor \cdot \lfloor \sqrt[4]{3} \rfloor \cdot \lfloor \sqrt[4]{5} \rfloor \dotsm \lfloor \sqrt[4]{2015} \rfloor}{\lfloor \sqrt[4]{2} \rfloor \cdot \lfloor \sqrt[4]{4} \rfloor \cdot \lfloor \sqrt[4]{6} \rfloor \dotsm \lfloor \sqrt[4]{2016} \rfloor}.\] Let's think step by step and ou... | math | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"index": 3626,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find all real numbers $p$ so that
\[x^4 + 2px^3 + x^2 + 2px + 1 = 0\]has at least two distinct negative real roots. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\left( \\frac{3}{4}, \\infty \\right)",
"style": "rule"
} | {
"index": 3627,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the minimum of the function
\[\frac{xy}{x^2 + y^2}\]in the domain $\frac{2}{5} \le x \le \frac{1}{2}$ and $\frac{1}{3} \le y \le \frac{3}{8}.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{6}{13}",
"style": "rule"
} | {
"index": 3628,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Three of the four endpoints of the axes of an ellipse are, in some order, \[(-2, 4), \; (3, -2), \; (8, 4).\]Find the distance between the foci of the ellipse. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "2 \\sqrt{11}",
"style": "rule"
} | {
"index": 3629,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the remainder when $x^{2015} + 1$ is divided by $x^8 - x^6 + x^4 - x^2 + 1.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-x^5 + 1",
"style": "rule"
} | {
"index": 3630,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did not att... | math | {
"ground_truth": "\\frac{349}{500}",
"style": "rule"
} | {
"index": 3631,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $f(x) = ax^6 + bx^4 - cx^2 + 3.$ If $f(91) = 1$, find $f(91) + f(-91)$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 3632,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $f(x) = x^2 + 6x + c$ for all real numbers $x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{11 - \\sqrt{13}}{2}",
"style": "rule"
} | {
"index": 3633,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the minimum value of
\[x^2 + 2xy + 3y^2 - 6x - 2y,\]over all real numbers $x$ and $y.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-11",
"style": "rule"
} | {
"index": 3634,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $f(x) = x^2-2x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c)))) = 3$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 3635,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $ f(x) = x^3 + x + 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) = - 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "899",
"style": "rule"
} | {
"index": 3636,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the maximum value of
\[\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_1,\]over all real numbers $\theta_1,$ $\theta_2,$ $\theta_3,$ $\theta_4,$ and $\theta_5.$ Let's think step by step and output the final answer wit... | math | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": 3637,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | If
\begin{align*}
a + b + c &= 1, \\
a^2 + b^2 + c^2 &= 2, \\
a^3 + b^3 + c^3 &= 3,
\end{align*}find $a^4 + b^4 + c^4.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{25}{6}",
"style": "rule"
} | {
"index": 3638,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The terms of the sequence $(a_i)$ defined by $a_{n + 2} = \frac {a_n + 2009} {1 + a_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "90",
"style": "rule"
} | {
"index": 3639,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $z$ be a complex number with $|z| = \sqrt{2}.$ Find the maximum value of
\[|(z - 1)^2 (z + 1)|.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "4 \\sqrt{2}",
"style": "rule"
} | {
"index": 3640,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | For $-25 \le x \le 25,$ find the maximum value of $\sqrt{25 + x} + \sqrt{25 - x}.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 3641,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $f$ be a function satisfying $f(xy) = f(x)/y$ for all positive real numbers $x$ and $y$. If $f(500) = 3$, what is the value of $f(600)$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": 3642,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Compute $$\sum_{n=1}^{\infty} \frac{3n-1}{2^n}.$$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 3643,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $x,$ $y,$ and $z$ be nonnegative numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of
\[2xy \sqrt{6} + 8yz.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\sqrt{22}",
"style": "rule"
} | {
"index": 3644,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$.
Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "86",
"style": "rule"
} | {
"index": 3645,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the focus of the parabola $y = -3x^2 - 6x.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\left( -1, \\frac{35}{12} \\right)",
"style": "rule"
} | {
"index": 3646,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 3647,
"split": "train"
} |
Level 1 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.5 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
... | math | {
"ground_truth": "\\text{E}",
"style": "rule"
} | {
"index": 3648,
"split": "train"
} |
Level 1 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Compute $\sqrt{(31)(30)(29)(28)+1}.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "869",
"style": "rule"
} | {
"index": 3649,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The polynomial
$$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 3650,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The function $f$ takes nonnegative integers to real numbers, such that $f(1) = 1,$ and
\[f(m + n) + f(m - n) = \frac{f(2m) + f(2n)}{2}\]for all nonnnegative integers $m \ge n.$ Find the sum of all possible values of $f(10).$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 3651,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Real numbers $x,$ $y,$ and $z$ satisfy the following equality:
\[4(x + y + z) = x^2 + y^2 + z^2.\]Let $M$ be the maximum value of $xy + xz + yz,$ and let $m$ be the minimum value of $xy + xz + yz.$ Find $M + 10m.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "28",
"style": "rule"
} | {
"index": 3652,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the range of the function \[f(x) = \frac{x}{x^2-x+1},\]where $x$ can be any real number. (Give your answer in interval notation.) Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "[-\\tfrac13, 1]",
"style": "rule"
} | {
"index": 3653,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "671",
"style": "rule"
} | {
"index": 3654,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the ordered pair $(a,b)$ of positive integers, with $a < b,$ for which
\[\sqrt{1 + \sqrt{21 + 12 \sqrt{3}}} = \sqrt{a} + \sqrt{b}.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "(1,3)",
"style": "rule"
} | {
"index": 3655,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | If $x$ is real, compute the maximum integer value of
\[\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "41",
"style": "rule"
} | {
"index": 3656,
"split": "train"
} |
Level 1 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of these values of $a$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-16",
"style": "rule"
} | {
"index": 3657,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Calculate $\frac{1}{4} \cdot \frac{2}{5} \cdot \frac{3}{6} \cdot \frac{4}{7} \cdots \frac{49}{52} \cdot \frac{50}{53}$. Express your answer as a common fraction. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{1}{23426}",
"style": "rule"
} | {
"index": 3658,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that \[
n^2 = x^2+y^2+z^2+2xy+2yz+2zx+3x+3y+3z-6
\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 3659,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The function $f$ satisfies \[
f(x) + f(2x+y) + 5xy = f(3x - y) + 2x^2 + 1
\]for all real numbers $x,y$. Determine the value of $f(10)$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-49",
"style": "rule"
} | {
"index": 3660,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $1000x^3+27= (ax^2 + bx +c )(d x^2 +ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "11,\\!090",
"style": "rule"
} | {
"index": 3661,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | For positive integers $n$, define $S_n$ to be the minimum value of the sum
\[\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},\]where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is $17$. Find the unique positive integer $n$ for which $S_n$ is also an integer. Let's think step by step and output the final answer within \... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 3662,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $x,$ $y,$ $z$ be real numbers, all greater than 3, so that
\[\frac{(x + 2)^2}{y + z - 2} + \frac{(y + 4)^2}{z + x - 4} + \frac{(z + 6)^2}{x + y - 6} = 36.\]Enter the ordered triple $(x,y,z).$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "(10,8,6)",
"style": "rule"
} | {
"index": 3663,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The set of points $(x,y)$ such that $|x - 3| \le y \le 4 - |x - 1|$ defines a region in the $xy$-plane. Compute the area of this region. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 3664,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The planet Xavier follows an elliptical orbit with its sun at one focus. At its nearest point (perigee), it is 2 astronomical units (AU) from the sun, while at its furthest point (apogee) it is 12 AU away. When Xavier is midway along its orbit, as shown, how far is it from the sun, in AU?
[asy]
unitsize(1 cm);
path... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 3665,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "98",
"style": "rule"
} | {
"index": 3666,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $x,$ $y,$ $z$ be real numbers such that
\begin{align*}
x + y + z &= 4, \\
x^2 + y^2 + z^2 &= 6.
\end{align*}Let $m$ and $M$ be the smallest and largest possible values of $x,$ respectively. Find $m + M.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | {
"index": 3667,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the distance between the foci of the ellipse
\[\frac{x^2}{20} + \frac{y^2}{4} = 7.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "8 \\sqrt{7}",
"style": "rule"
} | {
"index": 3668,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | One focus of the ellipse $\frac{x^2}{2} + y^2 = 1$ is at $F = (1,0).$ There exists a point $P = (p,0),$ where $p > 0,$ such that for any chord $\overline{AB}$ that passes through $F,$ angles $\angle APF$ and $\angle BPF$ are equal. Find $p.$
[asy]
unitsize(2 cm);
pair A, B, F, P;
path ell = xscale(sqrt(2))*Circle((... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 3669,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "17",
"style": "rule"
} | {
"index": 3670,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $x,$ $y,$ $z$ be real numbers such that $4x^2 + y^2 + 16z^2 = 1.$ Find the maximum value of
\[7x + 2y + 8z.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": 3671,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | For what values of $x$ is $\frac{\log{(3-x)}}{\sqrt{x-1}}$ defined? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "(1,3)",
"style": "rule"
} | {
"index": 3672,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $a,$ $b,$ $c$ be positive real numbers. Find the smallest possible value of
\[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 3673,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that
\[P(1) = P(3) = P(5) = P(7) = a\]and
\[P(2) = P(4) = P(6) = P(8) = -a.\]What is the smallest possible value of $a$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": " 315",
"style": "rule"
} | {
"index": 3674,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $a,$ $b,$ $c$ be complex numbers such that
\begin{align*}
ab + 4b &= -16, \\
bc + 4c &= -16, \\
ca + 4a &= -16.
\end{align*}Enter all possible values of $abc,$ separated by commas. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "64",
"style": "rule"
} | {
"index": 3675,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The polynomial equation \[x^3 + bx + c = 0,\]where $b$ and $c$ are rational numbers, has $5-\sqrt{2}$ as a root. It also has an integer root. What is it? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": 3676,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The function $y=\frac{x^3+8x^2+21x+18}{x+2}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 3677,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 3678,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find all solutions to the equation\[ \sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "81, 256",
"style": "rule"
} | {
"index": 3679,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the number of ordered quadruples $(a,b,c,d)$ of nonnegative real numbers such that
\begin{align*}
a^2 + b^2 + c^2 + d^2 &= 4, \\
(a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 16.
\end{align*} Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 3680,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $x,$ $y,$ $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{1}{x + y} + \frac{1}{x + z} + \frac{1}{y + z}.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": 3681,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$ The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\left( \\frac{15}{4}, 0 \\right)",
"style": "rule"
} | {
"index": 3682,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | If
\[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 3 \quad \text{and} \quad \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 0,\]find $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 3683,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The partial fraction decomposition of
\[\frac{x^2 - 19}{x^3 - 2x^2 - 5x + 6}\]is
\[\frac{A}{x - 1} + \frac{B}{x + 2} + \frac{C}{x - 3}.\]Find the product $ABC.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 3684,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Suppose that all four of the numbers \[2 - \sqrt{5}, \;4+\sqrt{10}, \;14 - 2\sqrt{7}, \;-\sqrt{2}\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 3685,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $x$ and $y$ be real numbers such that $x + y = 3.$ Find the maximum value of
\[x^4 y + x^3 y + x^2 y + xy + xy^2 + xy^3 + xy^4.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\frac{400}{11}",
"style": "rule"
} | {
"index": 3686,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Determine the value of the expression
\[\log_2 (27 + \log_2 (27 + \log_2 (27 + \cdots))),\]assuming it is positive. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 3687,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the largest positive integer $n$ such that
\[\sin^n x + \cos^n x \ge \frac{1}{n}\]for all real numbers $x.$ Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 3688,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $a$ and $b$ be real numbers. Consider the following five statements:
$\frac{1}{a} < \frac{1}{b}$
$a^2 > b^2$
$a < b$
$a < 0$
$b < 0$
What is the maximum number of these statements that can be true for any values of $a$ and $b$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 3689,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "26",
"style": "rule"
} | {
"index": 3690,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find the roots of $z^2 - z = 5 - 5i.$
Enter the roots, separated by commas. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "3 - i, -2 + i",
"style": "rule"
} | {
"index": 3691,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Suppose $f(x) = 6x - 9$ and $g(x) = \frac{x}{3} + 2$. Find $f(g(x)) - g(f(x))$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 3692,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Which type of conic section is described by the equation \[\sqrt{x^2 + (y-1)^2} + \sqrt{(x-5)^2 + (y+3)^2} = 10?\]Enter "C" for circle, "P" for parabola, "E" for ellipse, "H" for hyperbola, and "N" for none of the above. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "\\text{(E)}",
"style": "rule"
} | {
"index": 3693,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Find all $t$ such that $x-t$ is a factor of $6x^2+13x-5.$
Enter your answer as a list separated by commas. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-\\frac{5}{2}",
"style": "rule"
} | {
"index": 3694,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 3695,
"split": "train"
} |
Level 3 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | When the graph of $y = 2x^2 - x + 7$ is shifted four units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$. Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "28",
"style": "rule"
} | {
"index": 3696,
"split": "train"
} |
Level 5 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | Let $a$ and $b$ be the roots of $k(x^2 - x) + x + 5 = 0.$ Let $k_1$ and $k_2$ be the values of $k$ for which $a$ and $b$ satisfy
\[\frac{a}{b} + \frac{b}{a} = \frac{4}{5}.\]Find
\[\frac{k_1}{k_2} + \frac{k_2}{k_1}.\] Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "254",
"style": "rule"
} | {
"index": 3697,
"split": "train"
} |
Level 2 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | The function $f$ is linear and satisfies $f(d+1)-f(d) = 3$ for all real numbers $d$. What is $f(3)-f(5)$? Let's think step by step and output the final answer within \boxed{}. | math | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": 3698,
"split": "train"
} |
Level 4 | Intermediate Algebra | DigitalLearningGmbH/MATH-lighteval | A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$? Let's think step by step and output the final... | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": 3699,
"split": "train"
} |
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