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A right triangle with legs of length $a$ and $b$ is inscribed in a circle. If the area of the circle is $16\pi$ square units, find the maximum possible value of the area of the right triangle, in square units. | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $a+b+c=1$. Find the maximum value of $\sqrt{a+b} + \sqrt{b+c} + \sqrt{c+a}$. | nvidia |
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. | nvidia |
Find the smallest positive integer $m$ such that $\frac{m}{m+77}$ is equal to a repeating decimal with a single-digit repeating block. | nvidia |
Find the largest positive integer n for which the sum of the products of the non-zero digits of the integers from 1 to 10^n inclusive is divisible by 13. | nvidia |
A triangle has vertices at $A(0,0)$, $B(1,0)$, and $C(0,1)$.
A point $P$ is chosen inside the triangle such that the product of its distances from the sides of the triangle is maximized.
Find the coordinates of point $P$. | nvidia |
In the equation
$$5x^2-kx+1=0$$
determine $k$ such that the difference of the roots be equal to unity.
New problem:
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Determine all integer values of $n$ such that the cube of the sum of the squares of two consecutive positive integers is less than 6000. | nvidia |
Find the smallest positive integer $n$ such that $n!$ is divisible by the cube of its largest prime factor. | nvidia |
Let $m$ and $n$ be real numbers such that $2m + 3n \le 6$ and $m + 5n \le 9.$ Find the largest possible value of $m + 2n$. | nvidia |
The sides of a cube are covered with red, white, and blue stickers, such that each face has a different color. How many different ways can the cube be oriented such that the red face is on top and the white face is on the front? | nvidia |
Find the smallest positive integer $k$ such that for any integer $n$, the polynomial $f(x) = (x+n)(x+2n)(x+3n)(x+4n)$ has a factor of $x^k$. | nvidia |
Let $f(x)$ be a function such that its graph passes through the points $(1, 3), (2, 4),$ and $(3, 5)$.
Find the sum of all possible values of $f(f(f(2)))$. | nvidia |
Find the largest possible value of the expression $$\frac{(a+b)(b+c)(c+a)}{abc},$$where $a > b > c > 0$ are real numbers. | nvidia |
Find the sum of all positive integers less than or equal to 100 that are divisible by both 3 and 5. | nvidia |
Find the number of distinct positive integers $n$ such that $n^2 + 2n + 1$ is a perfect square. | nvidia |
A line passes through the points $(x_1,y_1)$ and $(x_2,y_2)$. Given that the midpoint of the segment formed by these two points is $(1,2)$, and the angle between the line and the positive $x$-axis is $\frac{\pi}{6}$, find the coordinates of the point $(x_2,y_2)$ if $(x_1,y_1)=(3,-1)$. | nvidia |
Find the number of positive integers less than or equal to 1000 that are not divisible by 7 or 11. | nvidia |
The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.
| nvidia |
Consider a positive integer $n$ such that $12$ divides $n^2$ and $n^2$ divides $432$. What is the number of possible values of $n$? | nvidia |
A smaller hexagon is inscribed inside a larger regular hexagon in such a way that the sides of the smaller hexagon are tangent to the sides of the larger hexagon at their midpoints. What percent of the area of the larger hexagon is the area of the smaller hexagon? | nvidia |
Given a positive integer $n$, we define a set $A_n$ as follows:
$$A_n=\left\{\frac{m(n+1)}{2}| m \in \mathbb{Z}, n|m\right\}$$
We call an integer $a \in A_n$ "strict" if there exists an integer $b$ such that $a+b=n$ and $a|b$.
Find the smallest value of $n>5$ such that there exist at least 3 strict elements in $A_n$... | nvidia |
Consider a circle with radius $r$. Seven points are randomly chosen inside or on this circle. Let $b$ be the smallest possible number such that it is always possible to select one pair of points from these seven such that the distance between them is equal to or less than $b$. Find the value of $b$ in terms of $r$. | nvidia |
What is the smallest number of different sizes of rectangles that can be used to tile a 9x9 square such that each rectangle shares at least one side with another rectangle of the same size? | nvidia |
Given that the graph of the function $f(x) = |x-1| + |x+1|$ consists of three line segments, determine the area of the triangle enclosed by this graph and the x-axis. | nvidia |
Find the smallest positive integer $m$ such that $m!$ is divisible by $10^6$. | nvidia |
Suppose that $a$ and $b$ are positive integers such that $(a+bi)^2 = 3+4i$. What is $a+bi$?
New Problem:
Let $a$ and $b$ be positive integers such that $(a+bi)^3 = 2+11i$. What is $a+bi$? | nvidia |
Find all the solutions to the equation
$$a^{\lfloor x \rfloor} + b^{\lfloor x \rfloor} = a^{x} + b^{x}$$
where $a$ and $b$ are positive reals and $\lfloor x \rfloor$ denotes the largest integer that is less than or equal to $x$. For each $a$ and $b$, determine the number of solutions for $x$. | nvidia |
Find all values of $x$ such that $x^3-2x^2-5x+6>0$. Express your answer in interval notation. | nvidia |
Let $x$ be a real number. Define
$$\binom{x}{0}=1, \qquad \binom{x}{n}=\frac{x(x-1)\dots (x-n+1)}{n!}, \qquad n=1,2,\dots.$$
Prove that for any positive integer $n,$
$$\sum_{k=0}^{n} (-1)^{n-k}\binom{n}{k}\binom{x+k}{n}=\begin{cases} 0& \text{if} & x=0,1,2,\dots,n-1\\ 1&\text{if}& x=n. \end{cases}$$ | nvidia |
Find the maximum value of
\[\frac{x + 2y + 3}{\sqrt{x^2 + y^2 + 1}}\]over all real numbers $x$ and $y.$ | nvidia |
Find the largest integer $n$ such that the increasing sequence $\left\lfloor \frac{2^{k_1}+2^{k_2}+\ldots+2^{k_n}}{n}\right\rfloor$, where $k_i\geq 1$ are distinct integers for all $i$, contains all positive integers. | nvidia |
Determine the smallest positive integer $n$ such that the product of all prime numbers less than or equal to $n$ is greater than 1000. | 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, and the sum of the absolute values of the coefficients of these linear factors is a perfect square. | nvidia |
Find the sum of all $x$ such that the expression $\frac{x^2+2x+1}{x^2+2x-3}$ is equal to zero.
Note: This problem requires a different approach because instead of finding the product of the values that make the expression undefined (i.e., the values that make the denominator zero), we are asked to find the sum of the ... | nvidia |
Determine the range of values for the real number $a$ such that the inequality
\[\frac{ax+1}{x^2+2x+1} > \frac{3x+1}{x^2+x+1}\]holds for all real numbers $x$ satisfying $x < 0$. | nvidia |
Let $x, y, z$, and $w$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}x^3+y^3&=&z^3+w^3&=&216,\\ xz&=&yw&=&36.\end{array}\]
If $P=xyzw$, compute the value of $\lfloor \log_{10}P \rfloor$. | nvidia |
[asy] draw(circle((0,8sqrt(3)),4sqrt(3)),black+linewidth(.75)); draw(circle((0,4sqrt(3)),2sqrt(3)),black+linewidth(.75)); draw((-8/3,16sqrt(3)/3)--(-4/3,8sqrt(3)/3)--(0,0)--(4/3,8sqrt(3)/3)--(8/3,16sqrt(3)/3),dot); MP("B",(-8/3,16*sqrt(3)/3),W);MP("B'",(8/3,16*sqrt(3)/3),E); MP("A",(-4/3,8*sqrt(3)/3),W);MP("A'",(4/3,8*... | nvidia |
Find a positive integer that is divisible by 18 and whose cube root is a number between 8 and 8.1. | nvidia |
Let $a_1 , a_2 , \dots$ be a sequence for which $a_1=2$ , $a_2=3$, and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$? | nvidia |
Determine the number of diagonals that can be drawn for a nonagon. | nvidia |
Let $x$ be a positive number such that $\sqrt[3]{x} \cdot \sqrt[4]{x} \cdot \sqrt[5]{x} = 24$. Find $x$. | nvidia |
In the simplified expansion of the expression \[(x+y+z+w)^{10}+(x-y-z-w)^{10},\]what is the ratio of the number of terms with all positive coefficients to the number of terms with all negative coefficients? | nvidia |
Find the product of all integer divisors of $105$ that also divide $14$. (Recall that the divisors of an integer may be positive or negative.) | nvidia |
What is the smallest positive integer $n$ such that $531n \equiv 1067n \pmod{24}?$ | nvidia |
Let $a_1,a_2,\cdots ,a_{2017}$ be a sequence of real numbers such that for every $n\ge 3$, $a_n=a_{n-1}-2a_{n-2}$. What can we say about the value of the expression $$P=\prod_{k=1}^{2017}(1+a_k)?$$ | nvidia |
Find the product of all real values of $x$ that satisfy the equation $\frac{x^2}{x-4} + 3 = x$. | nvidia |
Find the number of positive integers smaller than 1000 that are divisible by exactly three different prime numbers. | nvidia |
Find the largest number in the sequence 1, 4, 7, 10,... that is a perfect square. | nvidia |
The product of two positive integers is $3x^2+5x+2$, where $x$ is a positive integer. If one of the integers is a divisor of the other, what is the smallest possible value of the larger integer? | nvidia |
Find the sum of all real values of $x$ that satisfy the system of equations \begin{align*}
x^2+xy&=1,\\
x^2+y^2&=2.
\end{align*} | nvidia |
How many four-digit numbers can be formed by using the digits 1, 2, 3, 4, 5, and 6 without repetition such that the sum of the two middle digits is equal to the sum of the two outer digits? | nvidia |
The graphs of two functions $f(x)$ and $g(x)$ intersect at the points $(1,1)$, $(3,4)$, and $(5,7)$. The graphs of $y=f(x-1)$ and $y=g(x+1)$ also intersect at one point. Find the product of the coordinates of that point. | nvidia |
Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real. | nvidia |
Consider a set of real numbers $\{x_1, x_2, \dots, x_n\}$. Define the function $f(x_i)$ as the smallest integer that is greater than or equal to $x_i$, for each $i$. Given that $f(x_1) + f(x_2) + \dots + f(x_n) = k$, where $k$ is an integer, find the largest possible value of $n$ if $|x_i| < 1$ for all $i$. | nvidia |
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. | nvidia |
Let $f$ be a function such that
$$f(x) = x + \frac{1}{2} f(x-1),\quad x\ge 2.$$
Find the smallest value of $x$ such that $f(x)\ge 3000$. | nvidia |
The value of $\log_{10}{26483}$ is between the consecutive integers $x$ and $y$. Find $x+y$. | nvidia |
The function $g(x)$ is defined on the real numbers such that $g(g(x)) = x^2$ and $g(x^2 + 1) = x^2 - 1$ for all real numbers $x.$ Find $g(2).$ | nvidia |
Given that the lengths of the sides of a triangle with positive area are $2 + \sqrt{3}$, $3 + \sqrt{2}$, and $x + \sqrt{y}$, where $x$ and $y$ are positive integers, find the number of possible values for $x$ and $y$. | nvidia |
Find the number of sequences $a_1, a_2, \dots$ of positive integers such that for any two positive integers $m$ and $n$ the following equation holds:
$$a_{m+n} + a_{mn} = a_m a_n + \gcd(m,n),$$
where $\gcd(m,n)$ is the greatest common divisor of $m$ and $n$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | nvidia |
Find the smallest positive integer that has four different prime factors, with the additional condition that the sum of these prime factors is a perfect square. | nvidia |
In the equation $ax^2+bx+c=0$, the sum of the roots is $-\frac{b}{a}$ and the product of the roots is $\frac{c}{a}$. If the sum of the roots is equal to the product of the roots, determine the value of $a$ such that the equation has real and distinct roots. | nvidia |
The number $1000!$ can be expressed as the product of powers of prime numbers. Determine the largest power of $7$ that divides $1000!$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 6.\]Find the minimum value of $x^3 y^2 z.$ | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
In the equation $x^3 - kx^2 + 1 = 0$, determine $k$ such that the sum of the squares of the roots is equal to $-1$. | nvidia |
The function $f(x)$ satisfies
\[f(x+y)=f(x)f(y)-2x^2y\]
for all real numbers $x$ and $y.$ If $f(1)=3$, find $f(3)$. | nvidia |
Find the smallest positive integer $x$ that has exactly four distinct positive divisors and is relatively prime to 120. | nvidia |
Find the number of positive integers less than 100 that are multiples of both 3 and 5, but not of 7. | nvidia |
Let's consider a sequence of numbers where each term is the cube of the previous term. What is the sum of all positive integers less than 100 that are cubes of cubes of perfect cubes? | nvidia |
Find the smallest positive integer $k$ such that the three fractions $\frac{4}{7}$, $\frac{2}{5}$, and $\frac{3}{k}$ have a common denominator. | nvidia |
Find the number of real roots of the equation
$$2^x + 3^x + 5^x = x^2$$
such that $x$ is an integer. | nvidia |
Let $a$ be a real number randomly selected from the interval $[-12,12]$. Then, $p$ and $q$ are two relatively prime positive integers such that $p/q$ is the probability that the equation $x^4+16a^2=(2a^2-6a)x^2$ has $\textit{at least}$ two distinct real solutions. Find the value of $p+q$. | nvidia |
The function $g$ has the property that for each real number $x$ in its domain, $-x$ is also in its domain and \[
g(x) + g(-x) = x^2.
\]What is the largest set of real numbers that can be in the domain of $g$? | nvidia |
Find the smallest positive integer $m$ such that the number of divisors of $m^2 - m$ is exactly twice the number of divisors of $m$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 1$. Find the set of all possible values of $\frac{x}{y} + \frac{y}{x}$. | nvidia |
Let $f(x)$ and $g(x)$ be functions such that $f(g(x))=e^x$ and $g(f(x))=x^2+1$ for all real numbers $x$. If $f(2)=1$, find $f(g(1))$. | nvidia |
Find the sum of the reciprocals of the positive divisors of 50. | nvidia |
Let the foot of the perpendicular from the origin to the plane $ax+by+cz+d=0$ be $(x_0,y_0,z_0)$. Prove that the equation of the plane can be written as $xx_0+yy_0+zz_0-(x_0^2+y_0^2+z_0^2)=0$. Use this fact to find the equation of the plane, given that $(x_0,y_0,z_0)=(12,-4,3)$. | nvidia |
Let $S$ be a set of positive integers such that no two distinct elements in $S$ are congruent modulo $13$, and no two distinct elements in $S$ are congruent modulo $17$. Find the largest possible cardinality of $S$. | nvidia |
A positive real number $x$ is such that \[
\sqrt[3]{1-x^3} + \sqrt[3]{1+x^3} = 1.
\]Find $x^6.$ | nvidia |
Find the number of prime numbers that can be written in the form $2^n+1$ where $n$ is an integer from 1 to 10. | nvidia |
In the Cartesian plane, a square with side length 2 is centered at the origin. A point is chosen uniformly at random from the interior of this square. What is the probability that the point's distance from the origin is greater than its distance from the point (1,1)? | nvidia |
Find the sum of all angles $x \in [0^\circ, 360^\circ]$ that satisfy
\[\tan^3 x + \cot^3 x = \frac{1}{\sin x \cos x}.\] | nvidia |
Find the value of $x$ such that $x+\frac{1}{x}=5$. | nvidia |
Let $b$ and $c$ be real numbers. If the polynomial $x^2+bx+c$ has exactly one real root and $b=c+1$, find the value of the product of all possible values of $c$. | nvidia |
Consider a set of functions $f_n(x)$ defined recursively as follows:
$f_n(0) = 0$
$f_n(x) = n \cdot f_{n-1}(x-1) + 1$ for $x > 0$
Determine the smallest positive integer $n$ such that the domain of $f_n(x)$ contains at least 1000 distinct values. | nvidia |
There are ten points selected in a plane such that every subset of three points forms a triangle (i.e., no three points are collinear). Consider all sets of five points out of these ten such that all its four-element subsets are convex quadrilaterals. What is the maximum possible total number of such sets of five point... | nvidia |
There are 2 complex numbers $x+yi$, $z+wi$. If $x=2$, $z=-x$, and the sum of the numbers is $3i$, find $y+w$. | nvidia |
A bookshelf has 5 shelves, and each shelf can hold a certain number of books. The number of books that can be placed on each shelf is a consecutive integer, and the total number of books that can be placed on the bookshelf is 30. Determine the number of books that can be placed on the middle shelf. | nvidia |
Let $\mathbf{B}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{B}^4 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{B}^3$ can be. If you think the answer is infinite, then enter "infinite". | nvidia |
The length of the segment between the points $(2a, a-4)$ and $(4, -1)$ is $2\sqrt{10}$ units. What is the product of all possible values for $a$?
The points $A(2,5)$, $B(7,11)$, and $C(13,19)$ form a triangle. Find the value of $a$ such that the point $(3a,a+1)$ lies on the circumcircle of triangle $ABC$. | nvidia |
Three fair coins are tossed, and the numbers $x,y,z$ of heads obtained are recorded. Find the probability that $x+y+z$ is a prime number. | nvidia |
Find the sum of all positive integers $n$ such that $n^2-2n-3<5$. | nvidia |
Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 8$, $BC=AD=11$, and $CA=DB=14$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | nvidia |
In a right triangle with an angle of $30^{\circ}$, the length of the shorter leg is given as 6 inches. Determine the length of the longer leg, given that the sum of the areas of the two triangles formed by dropping an altitude from the vertex of the $30^{\circ}$ angle to the hypotenuse is equal to 12 square inches. | nvidia |
Find the value of the digit represented by $x$ so that the number $32x45$ is divisible by 6 and the sum of its digits is a perfect square. | nvidia |
What is the sum of all positive integers less than 100 that are squares of perfect squares? | nvidia |
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