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Find the value of $x$ such that $\sqrt{x+2} = 8$. | nvidia |
An irregular pentagon has a vertex angle of 120 degrees and its side opposite to that vertex forms a 60-degree angle with the exterior of the pentagon. Determine the sum of the degree measures of the other three interior angles. | nvidia |
The sum of the squares of three consecutive positive integers is equal to 365. What is the middle integer? | nvidia |
The pattern of Pascal's triangle is illustrated in the diagram shown. What is the fourth element in Row 15 of Pascal's triangle? $$
\begin{array}{ccccccccccccc}\vspace{0.1in}
\textrm{Row 0}: & \qquad & & & & & 1 & & & & & & \\ \vspace{0.1in}
\textrm{Row 1}: & \qquad & & & & 1 & & 1 & & & & &\\ \vspace{0.1in}
\textrm{R... | nvidia |
The sum of the squares of two consecutive integers is 185. What are the integers? | nvidia |
Let $g(n)$ denote the number of positive divisors of the integer $n$. Find the smallest positive integer $k$ such that for all integers $n \geq k$, the inequality $g(n) < \frac{n}{\log_2(n)}$ holds. | nvidia |
Find the sum of all distinct positive integer divisors of the number $2^3\cdot3^2\cdot5\cdot7$, excluding 1 and the number itself. | nvidia |
Compute $1-2+3-4+ \dots -98+99$ . | nvidia |
Find the number of positive integers less than or equal to 1000 that have at least 4 distinct positive divisors. | nvidia |
Find the number of positive whole number divisors of 432 that are not multiples of 3. | nvidia |
Given a triangle with vertices at $(1,1)$, $(2,3)$, and $(-1,2)$, find the equation of the line that passes through the midpoint of the side joining $(1,1)$ and $(2,3)$ and the centroid of the triangle. | nvidia |
Find the smallest positive integer $n$ such that for every coloring of the lattice points in the plane with $n$ colors, there exist two points $(a,b)$ and $(c,d)$ such that $(a,b)$ is colored the same as $(c,d)$ and $a-b=c-d$. | nvidia |
On the Cartesian plane, a circle centered at the point $(x,y)$ passes through the points $(-9,-14)$, $(23,32)$, and $(46,16)$. The coordinates $x$ and $y$ satisfy the equation $2x-3y=-100$. Determine the radius of the circle. | nvidia |
How many four-digit numbers can be formed using only the digits 1, 2, 3, and 4, with the condition that the number of 1's and 3's must be equal? | nvidia |
Let $a$, $b$ be positive real numbers. Find the minimum value of $|a-2|+|b-3|+|a+b-5|$. | nvidia |
Find the sum of the series
$$\sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)(n+3)}$$ | nvidia |
Let $c$ and $d$ be positive real numbers such that each of the equations $x^2 + cx + 3d = 0$ and $x^2 + 3dx + c = 0$ has real roots. Find the smallest possible value of $c + d.$ | nvidia |
What is the positive difference between the sum of the squares of two consecutive positive integers and the product of their squares? | nvidia |
A square is divided into smaller congruent squares such that every horizontal and vertical line drawn through the center of the square intersects at least one of the smaller squares. If the total number of smaller squares is 99, determine the side length of the original square. | nvidia |
Given the equation
$$\frac{x^2-2mx+m^2}{(x-1)^2}=n,$$
determine the value of $m$ such that for any real value of $x$, $n$ is never negative. | nvidia |
A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $19\,836$ is divisible by $9$ but $19\,825$ is not.
If $D\,767\,E89$ is divisible by $9,$ where $D$ and $E$ each represent a single digit, what is the sum of all possible values of the sum $D+E?$ | nvidia |
A binomial $x^3+ax^2+bx+c$ is such that its square has only three terms.
Determine $b$ if $a$ and $c$ are known. | nvidia |
Three real numbers $x, y, z$ are such that $x+y+z=6, x^2+y^2+z^2=18, x^3+y^3+z^3=54$. Compute $(xy+yz+zx)^2$. | nvidia |
Problem:
Let $a,$ $b,$ $c,$ $d$ be positive real numbers such that the roots of $x^2 - ax + b = 0$ are $c$ and $d,$ and the roots of $x^2 - cx + d = 0$ are $a$ and $b.$ Find the maximum possible value of $abcd.$ | nvidia |
What is the least positive integer $n$ such that $4125$ is a factor of $n!$? | nvidia |
Let $a$, $b$, $c$, and $d$ be distinct positive integers such that $a^2+b^2$, $b^2+c^2$, $c^2+d^2$, and $d^2+a^2$ are all equal. Find the smallest possible value of $a+b+c+d$. | nvidia |
Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be... | nvidia |
Find the square root of $\dfrac{8!}{70}$.
Write another problem inspired by this one:
Let $x!$ be the product of the integers from 1 to $x$. Prove that the sum of the roots of the equation
$$\frac{x!}{x^2} + \frac{(x+1)!}{(x+1)^2} + \frac{(x+2)!}{(x+2)^2} = \frac{(x+3)!}{(x+3)^2}$$
is equal to $x+1$. | nvidia |
In a triangle, two of the side lengths are $7a$ and $8a$, where $a$ is a positive real number. The angle between these sides is $\theta$. If the area of the triangle is 30 square units, find the value of $a$. | nvidia |
For all positive integers $n$, let $g(n)=\log_{2015} n^3$. Find $g(7)+g(9)+g(15)$. | nvidia |
If $x^2+ax+b$ is a perfect square trinomial, determine $a$ and $b$ such that the product of its roots is $-14$. | nvidia |
Determine the sum of the cubes of the two largest prime numbers smaller than 10. | nvidia |
Find the least positive integer that is divisible by both 36 and 48, and whose square root is not an integer. | nvidia |
Given that
$$\frac{1}{(x^2+1)(x^2+4)(x^2+9)}=\frac{Ax+B}{x^2+1}+\frac{Cx+D}{x^2+4}+\frac{Ex+F}{x^2+9}$$
determine the value of $A+C+E$. | nvidia |
On the complex plane, a particle starts at the point $a + bi$ and ends at the point $c + di,$ along the line connecting the two points. Along the way, the particle intersects the circle centered at the origin with radius 2 at two points. If the angle between the line segment connecting these two points and the positive... | nvidia |
Find the largest positive integer $a$ for which $x^2-ax+2009$ has no real roots. | nvidia |
Find the integer $n$, $0 \le n \le 11$, such that \[n \equiv 10389 \pmod{12}.\] | nvidia |
Find the sum of all integer values $k$ for which $\binom{30}{15}+\binom{30}{k}=\binom{31}{16}$. | nvidia |
The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers? | nvidia |
Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians. | nvidia |
A $\textit{perfect square}$ is a number that can be expressed as the product of an integer with itself. The number 65 can be expressed as the sum of two perfect squares in multiple ways. What is the smallest positive difference between two such perfect squares? | nvidia |
Let's denote $$g(x)=
\begin{cases}
kx^2-x+1, \quad if x<\frac{\pi}{2}\\
sin(x), \qquad \qquad if x\ge\frac{\pi}{2}.
\end{cases}
$$
Knowing that $\lim_{x\rightarrow\frac{\pi}{2}^-} g(x) = \lim_{x\rightarrow\frac{\pi}{2}^+} g(x)$, find k. | nvidia |
Find the length of the segment on the number line whose endpoints satisfy $|x-\sqrt[3]{27}|=2$. | nvidia |
Find the number of positive integers $n \le 1000$ such that $15n$ is a perfect square. | nvidia |
Determine the value of $x$ such that the first digit to the right of the decimal point of the fraction $\frac{3}{7+x}$ is 5. | nvidia |
What is the maximum possible number of triangles that can be determined by 8 points in a plane, such that no three points are collinear? | nvidia |
There are three real numbers $x$ that are not in the domain of $$f(x) = \frac{1}{1+\frac{1}{1+\frac 1x}}.$$ What is the sum of those three numbers? | nvidia |
Let $a$ and $b$ be positive integers such that $\gcd(a,b) = 2$. Determine the smallest possible value of $\gcd(a^2,3b^2)$. | nvidia |
Determine the smallest positive integer $n$ such that $n!$ is divisible by $11^3$. | nvidia |
A magic square is a $3\times 3$ grid filled with distinct positive integers in the range from $1$ to $9$ such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. If the sum of the numbers in the magic square is $36$, how many possible values of the top left... | nvidia |
Given that the parabola $y=ax^2+bx+c$ passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$, determine the value of $a+b+c$. | nvidia |
Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors and $n$ is the largest integer less than $100$ with exactly three positive divisors. What is $m+n$? | nvidia |
How many positive integers that are less than 200 have a sum of digits equal to 10? | nvidia |
Consider the system of equations
$$\frac{x}{a} = \frac{y}{b} = \frac{x+y}{c} = \frac{xy}{d},$$
where $a, b, c, d$ are positive integers. Find the smallest possible value of $a+b+c+d$ such that the system has a non-trivial integer solution. | nvidia |
Determine the largest possible value of the parameter $a$ such that all real solutions to the equation
\[\cos^2 x - \left(a + \frac{1}{2}\right) \cos x + \frac{a}{2} = 0\]are in the interval $[0, 2\pi).$ | nvidia |
Find the sum of all the perfect squares between 1 and 100 that are also multiples of 3. | nvidia |
Determine the number of pairs of distinct positive integers $(m, n),$ $1 \le m < n \le 1000,$ such that $m^2 + n^2$ can be expressed as the sum of squares of two other distinct positive integers. | nvidia |
Find the integer $k$ that satisfies $0\le k<29$ and $$-500\equiv k\pmod{29}~?$$ | nvidia |
If $x$ and $y$ are the roots of the equation $z^2+2z-3=0$, find the value of $x^4+y^4$. | nvidia |
Given a positive integer $n$, what is the smallest integer that always divides into the difference between the sum of the factors of $n$ and $n$ itself? | nvidia |
A sequence $c_1, c_2, \ldots$ is defined as follows: $c_1 = 2$, and for all positive integers $n$ greater than 1, $c_n$ is the product of all positive integers $k$ such that $k \le n$ and $\gcd(k, n) = 1$. For example, $c_6 = 5$ because the only positive integers $k \le 6$ that are relatively prime to 6 are 1 and 5. Ca... | nvidia |
In triangle $PQR$, $PQ=6$, $QR=7$, and $RP=8$. Points $G$ and $H$ are on ray $PQ$ with $PQ<PG<PH$. The point $K \neq R$ is a point of intersection of the incircles of $\triangle PGR$ and $\triangle QHR$ satisfying $GK=3$ and $HK=9$. Then $HQ$ can be expressed as $\tfrac{e+f\sqrt{g}}{h}$, where $e$, $f$, $g$, and $h$ ar... | nvidia |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | nvidia |
A point in space has rectangular coordinates $(x,y,z)$, where $x$ and $y$ are positive and $z<0$. Its spherical coordinates are $(\rho, \theta, \phi)$, where $\rho$ is a prime number and $0<\theta<\frac{\pi}{2}$. The angle $\phi$ is such that $\cos \phi = -\frac{1}{3}$.
Determine the value of $\rho$. | nvidia |
What is the smallest positive integer that can be expressed as a sum of two distinct perfect squares in exactly three different ways? | nvidia |
Let f be a function such that for all positive integers x and y it holds that $f(xy)=f(x)+f(y)$. It is known that $f(2) = 2$ and $f(3)=3$. What is the value of $f(12)$? | nvidia |
Given a sphere centered at the origin with radius 1, find the minimum possible area of a triangle inscribed in the sphere, given that two of its vertices lie on a fixed diameter of the sphere. | nvidia |
Let $f(x)$ be a function such that $f(x+y)=e^xf(y)$ for all real numbers $x$ and $y$. If $f(0)=1$, find $f(2)$. | nvidia |
Find the minimum value of $\cos x + \cos y + \cos z$, where $x$, $y$, and $z$ are positive real numbers that satisfy $\sin x + \sin y + \sin z = 2$. | nvidia |
Consider a cylindrical vessel of height $6$ cm and radius $r$ cm. Find the value of $r$ such that the volume of water that can be held by the vessel when it is placed on a horizontal surface and filled to the brim is exactly twice the volume of the water that can be held by the vessel when it is placed on its side (i.e... | nvidia |
Let $m$ and $n$ be positive integers such that $\ds\f{m}{n}=0.\overline{123}$. Determine $m + n + 2024$. | nvidia |
Consider the set of all real functions that satisfy the following condition: if the value of a function at any point is multiplied by the argument, the result will differ from the value of the function at any other point by a constant. Is there a function among them that takes the same value at two different points? | nvidia |
Let $x$ be a number such that $\sqrt[3]{1+\sqrt{x}}+\sqrt[3]{1-\sqrt{x}}=\sqrt[3]{5}$.
Find $x$. | nvidia |
Find the smallest possible surface area of a rectangular box with a fixed volume of 240 cubic inches, assuming that the dimensions of the box are natural numbers. | nvidia |
Find the sum of all positive integers less than 100 that are cubes of prime numbers. | nvidia |
How many four-digit numbers greater than 2999 can be formed such that the product of the middle two digits exceeds 5? | nvidia |
Determine the sum of all real values of $a$ for which the equation
$$\frac{2}{\sqrt{x}}=\frac{a\sqrt{x}-1}{5}$$
has exactly one positive solution. | nvidia |
Consider a set $T$ of complex numbers of the form $x + yi$, where $x$ and $y$ are real numbers, such that $x^2 + y^2 = 1$. Find the smallest positive integer $m$ such that for all positive integers $n \ge m$, there exists a complex number $z \in T$ such that $z^n = i$. | nvidia |
A rectangular garden has a fixed perimeter of 24 meters, and its sides are all integers. What is the maximum area that can be enclosed by the garden, given that no two adjacent sides have the same length? | nvidia |
Consider the three lines $2y-x=3$, $x-3y=-1$, and $6x-4y=11$. Determine the number of points that lie at the intersection of at least two of these three lines. | nvidia |
Two of the roots of the equation \[ax^3+bx^2+cx+d=0\]are $3$ and $-2.$ Given that $a \neq 0,$ compute $\frac{b+c}{a}.$ | nvidia |
Consider a function $f(x) = \prod_{k=1}^{m} (x - a_k)$, where $a_k$ are distinct positive integers. Determine the maximum value of $m$ such that the graph of $y = f(x)$ has exactly two intersections with the x-axis in the interval $[1, a_m]$. | nvidia |
On square $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$.
| nvidia |
Find all differentiable functions $f$ defined on the positive real numbers such that for every $x,$ $$f\left(\frac{x}{2}\right)+f\left(2x\right)=2f(x),$$ and for every $\lambda>0,$ $$f(f(\lambda x))=f(\lambda)f(x).$$ Determine $f(x)$ as a function of $x$. | nvidia |
In the 3-dimensional space consider the set $C$ of points $(x,y,z)$ such that $x$, $y$, $z$ are non-negative real numbers and the following conditions hold:
\begin{align*}
&2x^2+3y^2+z^2=1\\
&2x^2+6y^2+3z^2=1
\end{align*}
For every natural $n$, denote by $t_n$ the minimum number such that for every point $M\in... | nvidia |
The product of three consecutive positive integers is 504. What is the sum of the fourth powers of the three original integers? | nvidia |
A pair of standard 6-sided dice is rolled. Let the sum of the numbers on the upper faces be S. Determine the conditional probability that S = 4 if it is known that S is an even number. | nvidia |
A circle with center at $(2,5)$ passes through the point $(3,4)$. Find the sum of the coordinates of the other endpoint of the diameter of the circle. | nvidia |
Find the value of $x$ such that $\left(\frac{a^3}{b^2}\right)^x = \frac{a^{2x}}{b^{3x}}$, where $a$ and $b$ are positive integers. Express your answer in terms of $a$ and $b$. | nvidia |
Given that
\[g(x) = \left\{
\begin{array}{cl}
\sin x & \text{if } x \le \pi/2, \\
\cos (x-\pi/2) & \text{if } x > \pi/2,
\end{array}
\right.\]
find all values of $x$ for which $g(x) = \frac{\sqrt{2}}{2}$. | nvidia |
In a triangle $ABC$, the lengths of the sides $AB$ and $AC$ are equal to $8$ and $6$ respectively, and the angle $A$ is a right angle. The triangle is inscribed in a circle with the center $O$.
Find the length of the diameter of the circle, given that the line segment $BC$ is a diameter of a smaller circle that passes... | nvidia |
Given that $A$, $B$, $C$ and $D$ are points on a circle, such that $ABCD$ is a rectangle with $AB=4$ and $BC=3$. Let $E$ be the point where the diagonals of the rectangle intersect. If the acute angle $BEC$ is $30^\circ$, determine the measure of the angle $AEC$. | nvidia |
Suppose $m$ and $n$ are positive integers such that $\gcd(m,n)$ is divisible by exactly $5$ distinct primes and $\mathop{\text{lcm}}[m,n]$ is divisible by exactly $25$ distinct primes.
If $m$ has fewer distinct prime factors than $n$, then $m$ has at most how many distinct prime factors? | nvidia |
Find the area of the triangle with vertices $(2,3),$ $(-3,0),$ and $(5,8).$ | nvidia |
Given the positive real numbers $x,$ $y,$ and $m,$ and the equation
\[4=m^2\left(\dfrac{x^3}{y^3}+\dfrac{y^3}{x^3}\right)+m\left(\dfrac{x^2}{y^2}+\dfrac{y^2}{x^2}\right),\]
find the minimum possible value of $m.$ | nvidia |
The graph of the equation $\frac{x^2}{4}-\frac{y^2}{k}=1$ has asymptotes $y=\pm mx$ where $m$ is positive and $k>0$. Find the value of $k$ so that the vertex of the parabola $y^2=2px$ ($p>0$) lies on one of the asymptotes of the hyperbola. | nvidia |
In a grid of squares, each square can be colored either red or blue. A sequence of 20 squares is chosen such that each square is adjacent to the previous one (horizontally, vertically, or diagonally). What is the smallest number of colors that can be used to color the sequence such that no two consecutive squares have ... | nvidia |
Determine $a$ such that the difference of the roots of the equation $2x^2-ax+1=0$ be equal to unity. | nvidia |
The sum of the digits of a number $n$ is denoted by $S(n)$. Find the sum of all prime numbers $p$ such that $S(p)=S(p^2)$. | nvidia |
Find the largest integer $n$ such that $\frac{123}{n}$ is an integer and $\frac{123}{n} < 10$. | nvidia |
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