instruction_seed stringlengths 30 7.99k | source stringclasses 1
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Find the absolute value of the sum of the solutions of $2x^2+3x-7=0$. | nvidia |
The equation of the curve traced by the set of all points $(x,y)$ whose distance from the point $(1,1)$ is equal to the distance from the line $x+y=0$ is $y=|x|.$ Find the area enclosed by this curve and the parabola $y=x^2.$ | nvidia |
Alice chose five positive integers and found that their product was even. What is the maximum number of odd integers she could have chosen?
New problem:
In a set of five integers, the sum of the reciprocals of any two integers is always an integer. What is the maximum number of non-multiples of 5 that can be in the s... | nvidia |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | nvidia |
Given that $x$, $y$, and $z$ are positive real numbers such that $(x \cdot y) + z = (x + z) \cdot (y + z)$, find the maximum value of the sum $x + y + z$.
This problem requires a different approach than the original, as it asks for the maximum sum of the three variables rather than the maximum product of the three var... | nvidia |
Alexis lists all the positive integers that are divisible by 25 and have exactly four digits. Determine the ratio of the number of integers ending in 00 to the number of integers ending in 25 or 75. | nvidia |
Let's define $A$ as the set of all positive three-digit integers that have at least two digits that are the same and are not divisible by 7, and $B$ as the set of all positive three-digit integers less than 500 that have at least two digits that are the same. Given that |A|=213 and |B|=112, find |A∪B|. | nvidia |
Find the number of pairs of positive integers $(x, y)$ that satisfy the system of inequalities:
$$\begin{cases}
y > x - 1 \\
y \leq 2x + 1 \\
y - x < 3
\end{cases}$$
and the equation:
$$x + y = 12$$
Determine the values of $x$ and $y$ that satisfy the system. | nvidia |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? | nvidia |
What is the positive integer $M$ for which $$ 12^2 \times 33^2 = 11^2 \times M^2 \? $$ | nvidia |
Let $a, b, c$ be positive real numbers such that $a + b + c = 1.$ Find the set of all possible values of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$ | nvidia |
A regular hexagon is inscribed in a circle with radius $r$. If the length of the diagonal that connects two opposite vertices of the hexagon can be expressed in the form $r\sqrt{a}+r\sqrt{b}$, where $a$ and $b$ are integers, find the value of $a+b$. | nvidia |
Find the maximum value of
\[\frac{x^2 + 2y^2 + 3}{x^2 + y^2 + 1}\]over all real numbers $x$ and $y$ such that $x^2 + y^2 = 4$. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $abc = 2.$ Find the minimum value of
\[(2a + 3b)(b + 3c)(ac + 2).\] | nvidia |
Let $\#$ be a binary operation defined on all positive integers such that for any positive integers $a, b,$ and $c$, the following hold:
$a\,\#\,(b\,\#\,c)=(a+b)\,\#\,c$
and
$a\,\#\,a=a$
Find the solution to the equation $(2\,\#\,6)\,\#\,x=9$ | nvidia |
Find the number of positive integers $n,$ $1 \le n \le 1000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Name the greatest whole number less than $100$ that has an odd number of positive factors. | nvidia |
Let $S$ be the set of all positive integers that have four digits in base $2$. What is the probability that a randomly selected element of $S$, when expressed in base $2$, has exactly two $1$ digits, given that the first digit is $1$? | nvidia |
The centroid (the point where the medians intersect) of a triangle is located at $(2,9)$, while the vertices on the $x$-axis are $(4,0)$ and $(-3,0)$. Determine the sum of the coordinates of the third vertex. | nvidia |
Let $f(x)$ and $g(x)$ be two polynomials such that $f(x)=x^4-x^2+1$ and $g(x)=x^4+x^2+1$. Determine the product of all possible values of $x$ satisfying $\ds\dfrac{f(x)}{g(x)}=-\ds\dfrac{1}{2}$. | nvidia |
A positive number $x$ is such that $\lfloor\sqrt[3]{x}\rfloor=5$.
Find the probability that $x$ is an integer. | nvidia |
Find all real numbers $b$ such that the equation
\[x^4 + bx^3 - 3bx^2 + 2b^2x - b^3 - 2 = 0\]
has exactly two distinct real solutions in $x.$ | nvidia |
Let $x,y,z$ be complex numbers such that $\frac{x+y+z}{2}$, $xy+yz+zx$, and $xyz$ are all real numbers. If $\frac{x+y+z}{2}=-1$, $xy+yz+zx=7$, and $xyz=-5$, find $|x|+|y|+|z|$. | nvidia |
In the Cartesian plane, let $A$ be the point $(0,0)$, $B$ be the point $(3,0)$, and $C$ be a point on the line $y=x$ such that the distance between $C$ and the line segment $\overline{AB}$ is minimized. What is the distance between $C$ and $\overline{AB}$? | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 1$. Find the maximum value of
\[\frac{x^2y^2}{(x + y)^2}.\] | nvidia |
Find the largest positive integer $a$ for which $\frac{1}{x^2 + ax + 2000}$ can be expressed as the sum of two distinct fractions with integer coefficients. | nvidia |
In triangle $ABC$, angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D$. The lengths of the sides of $\triangle ABC$ are integers, and $\sin A=\frac{\sqrt{a^2+b^2}}{ab}$, where $a$ and $b$ are relatively prime positive integers. Determine the value of $a+b$. | nvidia |
Let $\tau (n)$ denote the number of positive integer divisors of $n$ (including $1$ and $n$). Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.
| nvidia |
The cosine of an acute angle $\theta$ is equal to the sum of the tangents of all the angles in the interval $(0, \theta)$ that are integer multiples of $\frac{\pi}{4}$. Find $\theta$ (in radians). | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 16.$ Find the minimum value of $\frac{x}{y} + \frac{y}{x} + 2xy.$ | nvidia |
Let $g : \mathbb{R}^2 \to \mathbb{R}$ be a function such that
\[g(x,y) = g(y,x) \cdot (x-y) + x + y\]for all real numbers $x$ and $y.$
Find the maximum possible value of $g(3,2) - g(2,3).$ | nvidia |
Determine the sum of all real numbers $x$ satisfying
\[(x^2-4x+2)^{x^2-5x+2} = 1.\]
Another problem inspired by this one:
Find the product of all real numbers $y$ satisfying
\[\left(\frac{y^2+3y-2}{y^2-2y-3}\right)^{\frac{y^2+5y-6}{y^2-4y-5}} = -1.\] | nvidia |
Find the sum of all positive integers $n$ such that $n^2$ has a remainder of $4$ when divided by $7$. | nvidia |
Consider a sequence of positive numbers defined by $a_n=\frac{1}{n}$, where $n=1,2,3,\ldots$. Now consider the sequence $b_n$ obtained by taking the sum of every $k$th term of $a_n$ starting from the first term, where $k$ is a positive integer. For which value of $k$ is the sum of the sequence $b_n$ the largest? | nvidia |
The product of two positive integers is 72 and their sum is 18. What is the value of the positive difference of the cubes of the integers? | nvidia |
Find the largest power of 2 that divides the product of all positive integers from 1 to 30. | nvidia |
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
An $\textit{abundant number}$ is a positive integer, the sum of whose distinct proper factors is greater than the number. (The $\textit{proper factors}$ of a number are all of its factors except the number itself.) How many numbers less than 25 are abundant numbers? | nvidia |
Let $x$ be a real number. Consider the following five statements:
$0 < x^2 < 1$
$x^2 > 1$
$-1 < x < 0$
$0 < x < 1$
$0 < x - x^2 < 1$
What is the maximum number of these statements that can be true for any value of $x$? | nvidia |
The sum of the absolute values of three consecutive numbers is greater than 20, and the sum of their squares is less than 100. How many sets of such numbers exist? | nvidia |
A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? | nvidia |
Find all values of $x$ such that $\lfloor x\rfloor + \frac{1}{x} = 5$. | nvidia |
A cube is inscribed inside a sphere. How many of the sphere's points are at the same distance from all the vertices of the cube? | nvidia |
Determine the largest possible value of n for which the polynomial x^n can be expressed as the sum of two or more consecutive polynomials with non-negative integer coefficients. | nvidia |
Find the sum of the real roots of the equation $\cos x = \frac{2x}{1+x^2}$ in the interval $[0, 2\pi]$. | nvidia |
Find the largest number $N$ such that there exists a grid of size $N \times N$ consisting of two types of squares, black and white, where each square shares an edge with at most one square of the same color and at least one square of the opposite color. Additionally, the number of black squares and the number of white ... | nvidia |
What is the product of all positive integers $k$ for which $\mathop{\text{lcm}}[k,12]=48$? | nvidia |
The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
| nvidia |
Find the product of all real values of $r$ for which $\frac{1}{2x}=\frac{r-x}{7}$ has exactly one real solution. | nvidia |
Let $m$ be a positive integer with exactly 10 prime factors, all of which are distinct. Find the number of positive divisors of $m^2$ that are perfect squares. | nvidia |
Find the product of all real values of $r$ for which $\frac{1}{2x}=\frac{r-x}{7}$ has exactly one real solution. | nvidia |
Find the function $f(x)$, with real coefficients, such that $f(1) = 1$ and
\[f(x) + f(y) = f(x+y) + f(xy)\]for all positive real numbers $x$ and $y$.
The function can be represented as a polynomial or other forms of functions. | nvidia |
The sum of the measures of the exterior angles of a regular polygon is 360 degrees. Determine the number of sides of the polygon. | nvidia |
Determine the smallest integer value of $c$ for which the expression $\frac{5x^2+2x+c}{x^2+2x+c}$ is defined for all real numbers except $x=-1$. | nvidia |
Two points on the graph of a function $f(x)$ have coordinates $(1,3)$ and $(2,6)$. Given that the slope of the line passing through these two points is $\frac{7}{3}$ and that $f(3)=5$, find the unique quadratic function that satisfies these conditions. | nvidia |
Find the largest value of $x$ that satisfies the inequality $|x^2-16|\leq5$. | nvidia |
The blue parabola shown is the graph of the equation $y = ax^2 + bx + c$. Find $a+b+c$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
r... | nvidia |
Consider a cubic polynomial with integer coefficients, $p(x)=x^3+a_2x^2+a_1x+a_0$. The polynomial has a rational root $r$, and the remaining roots are the complex conjugates of each other, with the imaginary part equal to $r$. Find all possible values of $r$. | nvidia |
Let $x$ be a real number. Consider the following five statements:
$0 < x^2 < 1$
$x^2 > 1$
$-1 < x < 0$
$0 < x < 1$
$0 < x - x^2 < 1$
What is the maximum number of these statements that can be true for any value of $x$? | nvidia |
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | nvidia |
Find the sum of all the positive terms in the arithmetic sequence 12, 9, 6, $\dots$. | nvidia |
Let $\ds (a_n)$ be a sequence of real numbers such that $a_1=4$ and
$\ds\begin{cases}
a_{n+1}=a_{n-1}+a_n & n\equiv1\pmod{2}\\
a_{n+1}=\ds\frac{a_{n-1}\cdot a_n}{a_{n-1}+a_n} & n\equiv 0\pmod{2}.
\end{cases}$
Find the integer part of the number $\ds a_{13}$. | nvidia |
How many of the positive integers less than or equal to 1000 can be expressed as the sum of the cubes of two nonnegative integers? | nvidia |
Find the smallest value of $d$ such that $3$ is in the range of $g(x)=x^2-2x+d$. | nvidia |
The points $A,B,C,D$ are the vertices of a regular tetrahedron. The point $E$ is such that the midpoint of $AD$ and the midpoint of $BC$ together with $E$ form another regular tetrahedron. How many different planes can be drawn through 3 of the points $\{A,B,C,D,E\}$? | nvidia |
Find the least positive integer $n$ such that $n!$ is congruent to $0 \pmod{17}$, but $n!$ is not congruent to $0 \pmod{19}$. | nvidia |
Find the maximum value of
\[\frac{2x + 3y + 4}{\sqrt{x^2 + y^2 + 2}}\]over all real numbers $x$ and $y.$ | nvidia |
Let $S$ denote a set of positive integers such that no two numbers in $S$ differ by $1$. For each integer $n$, let $f(n)$ denote the largest number of elements in $S$ that are less than or equal to $n$. Determine the positive integer $n$ for which $\frac{f(n)}{n}$ is maximum. | nvidia |
Let $S$ be the set of all positive integers whose digits are $1$ or $2$, with at least one digit being $1$. For each $x \in S$, let $f(x)$ be the sum of the numbers of digits $1$ and $2$ in the decimal representation of $x$. Find the smallest positive integer $x$ such that for every positive integer $n$, $f(x^n) = f(x)... | nvidia |
Determine the value of $x$ such that the expression $\dfrac{2^{2x}}{2^{x+1}}$ simplifies to $\dfrac{1}{2}$. | nvidia |
Find the least positive integer $n$ such that the product of the decimal expansions of $\frac{1}{5}$ and $\frac{1}{7}$ terminates with $n$ zeros.
Answer:
Not explicitly required for this task. | nvidia |
Simplify $\displaystyle\frac{1+\sqrt{2}}{2-\sqrt{3}}$, where $\sqrt{2}$ and $\sqrt{3}$ are positive real numbers. | nvidia |
Consider the polynomial equation $x^6 - 2x^3 + 2 = 0$. Determine the smallest positive integer $m$ such that the sum of the $m^{\text{th}}$ powers of the roots of the given equation equals 0. | nvidia |
The sum of the squares of two positive integers $a$ and $b$ is equal to the product of their sum and difference. Determine all possible values of $a$ and $b$.
Let $a^2+b^2=(a+b)(a-b)$. Rearranging, we have $a^2+b^2=a^2-ab+ab-b^2$.
Therefore, $ab=0$.
Since $a$ and $b$ are positive integers, this is impossible. Hence,... | nvidia |
Let $g(x)$ be a function such that $g(1) = 2$ and
\[g(x+y) = g\left(\frac{x^2 + y^2}{x+y}\right) + \frac{(x-y)^2}{x+y}\]for all positive real numbers $x$ and $y.$ Find $g(x).$ | nvidia |
Find the number of distinct possible values of $a$ such that for any complex number $z$, the equation $(z-a)(z-2a)(z-3a)=(z-4a)(z-5a)(z-6a)$ holds. | nvidia |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | nvidia |
Let $S$ be the set of all positive integers $n$ such that $n^2 + n$ is a perfect square. Prove that $\gcd(n, m) = 1$ for any two distinct elements $n$ and $m$ in $S$. | nvidia |
The function $g$ is quadratic and satisfies $g(d+1)-g(d) = 4d+3$ for all real numbers $d$. Find $g(4)-g(2)$. | nvidia |
The function $$f(\theta)=\begin{vmatrix}
1 & 1 & \cos\theta \\
1 & 1 & \sin\theta \\
1 & 1 & \sin2\theta
\end{vmatrix}$$ takes real values for all real angles $\theta$. Find the minimum value of $f(\theta)$. | nvidia |
For positive integers $m$ and $n$, determine the greatest possible value of $\cos(\ln m)-\sin(\ln n)$. | nvidia |
Let $p(x)$ be a polynomial with real coefficients such that $p(x) \geq 0$ for all real $x$ and
\[\lim_{x \to -\infty} p(x) = \lim_{x \to \infty} p(x) = 1.\]Find the minimum value of
\[p(x) + \frac{1}{p(x)} + p(x + 1) + \frac{1}{p(x + 1)}\]for all real $x$. | nvidia |
Find the value of $a$ that satisfies the equation
$$a \cdot 5^b + 2 = a \cdot 2^c + 3$$
where $b$ and $c$ are integers such that $b + c = 6$. | nvidia |
The sum of the squares of two consecutive integers is 74. Find the difference between their cubes. | nvidia |
The product of the roots of the quadratic polynomial $x^2+ax+b$ is equal to $16$, and the sum of the roots is equal to $-4$. Determine the value of the product of all possible values of $a$. | nvidia |
A rectangle is inscribed in a semicircle of radius 4, with one of its vertices at the center of the semicircle and the other three vertices on the semicircle. If the acute angle between the two radii to the vertices on the semicircle is $\theta$, find the length of the side of the rectangle closest to the diameter.
Mu... | nvidia |
What is the smallest positive integer $n$ such that $n^2 \equiv 361 \pmod{17}$? | nvidia |
The product of the first $n$ positive integers is equal to the sum of the squares of the first $m$ positive integers. Find $m$ in terms of $n$. | nvidia |
Let $p$ be a prime and let $n$ be a positive integer. For which values of $n$ does there exist a primitive root modulo $p^n$ that is not a primitive root modulo $p$? | nvidia |
The graph of the polynomial
$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$
has five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?
$\textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$ | nvidia |
A polynomial $P(x)$ of degree $n$ satisfies $P(P(x)) = x^n$ for all $x$. Find the sum of the coefficients of $P(x)$. | nvidia |
What is the least positive integer with exactly $10$ positive factors? | nvidia |
Find the sum of all positive odd factors of $720$ that are not perfect squares. | nvidia |
A circle is inscribed in a right triangle with legs 3 and 4. Find the ratio of the area of the triangle to the area of the circle. | nvidia |
Find the sum of cubes of all roots of the equation
$$x^3-7x^2+3x-1=0$$
if it is known that the sum of the roots is equal to 7. | nvidia |
Find the number of non-congruent trapezoids with integer side lengths, where the lengths of the parallel sides are in geometric progression, and the sum of the lengths of the non-parallel sides is equal to the length of the longer parallel side. | nvidia |
What is the smallest positive integer that can be added to 0.9 to obtain a number that can be expressed as the sum of two different terminating decimals? | nvidia |
Let $g(x)$ be a function such that
$$g(x) =
\begin{cases}
|x|+2 &\text{if }x<0, \\
x^2+1&\text{if }x\ge 0.
\end{cases}
$$
Determine the value of $a$ such that $g(a) = g(-a)$. | nvidia |
Let g(x)=3cos(2πx)+1. Determine the number of values of x on the interval −2≤x≤2 that satisfy the equation g(g(x))=−g(x). | nvidia |
Find the value of $x$ such that $\left|\frac{x}{2}\right|+\left|\frac{2}{x}\right|=3$. | nvidia |
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