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The sum of the squares of the digits of a three-digit number is equal to the square of the sum of its digits. If the number is divisible by 3, what is the sum of its digits? | nvidia |
Find the smallest positive integer $k$ such that the product of the first $k$ positive integers is divisible by $13^3$. | nvidia |
Find the area enclosed by the region defined by the equation $x^2+y^2+6x+8y=0$, assuming that the region is rotated by $45^\circ$ counterclockwise about the origin. | nvidia |
Find the smallest positive integer such that when divided by each prime number less than 10, it leaves a remainder of 2. | nvidia |
The semi-major axis $a$ of an ellipse is equal to the square root of the sum of the squares of the distances of its foci from the center. The distance between the foci is $8$. Find the coordinates of the foci. | nvidia |
How many positive integers less than 1000 can be expressed as the sum of two different palindromes? | nvidia |
Given that the product of the solutions of the quadratic equation $ax^2 + bx + c = 0$ is $c/a$, find the product of the solutions of the equation $-32x^2 + 84x + 135=0$. | nvidia |
Let $A$ be a set of positive integers such that each element in $A$ is a divisor of at least one of the numbers 30, 60, or 90, but not all three. What is the sum of the elements in $A$? | nvidia |
Compute the number of nondecreasing sequences of size 11, such that the elements are chosen from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. | nvidia |
A circle with center $C$ passes through the points $A(3,4)$ and $B(1,6)$. The line $l$ passing through $C$ and $A$ intersects the circle at a point $D$. Find the sum of the $x$-coordinate of $D$ and the radius of the circle. | nvidia |
Find the smallest positive integer $y$ that satisfies $y + 2018 \equiv 700 \pmod{15}$. | nvidia |
In three-dimensional space, find the number of lattice points that have a distance of 3 from the origin.
Note: A point is a lattice point if all its coordinates are integers. | nvidia |
Three complex numbers $a+bi$, $c+di$, and $e+fi$ are such that $a^2+b^2=1$, $c^2+d^2=1$, and $e^2+f^2=1$. If the sum of these numbers is 0, find the value of $(a+c+e)^2+(b+d+f)^2$. | nvidia |
Determine the smallest five-digit palindrome that is a perfect square, knowing that its third digit is a prime number. | nvidia |
Let $p$ and $q$ be positive integers such that $\frac{1}{p} + \frac{1}{q} = 1$. Find the maximum value of $\frac{p^2 + q^2}{pq}$ for all such pairs $(p,q)$. | nvidia |
Let ∘ be a binary operation defined on the set of real numbers R such that
a∘b=a(a+1)(b+1)+1
for all a,b∈R. Determine the value of (2∘(5∘4)). | nvidia |
What is the least positive integer that can be expressed as a sum of two different prime numbers in exactly two ways? | nvidia |
Determine the largest possible value of a positive integer $a$ such that the number of divisors of $a$ is equal to the number of prime factors of $a$ plus two. | nvidia |
What is the smallest positive integer $n$ such that the decimal representation of $\frac{1}{n}$ has a repeating block of digits with a length greater than 5? | nvidia |
A sphere is inscribed in a cube so that it touches the sides of the cube.
Determine the radius of the sphere if the length of the diagonal of the cube is 24 cm. | nvidia |
Consider the function $g: \mathbb{R} \to \mathbb{R}$ defined by the equation
\[g(xy) = x^2g(y) + y^2g(x)\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(-1),$ and let $t$ be the product of all possible values of $g(-1).$ Find $m + t.$ | 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 |
Let $P$ be a point inside triangle $ABC$ such that the areas of triangles $PAB$, $PBC$, and $PCA$ are in the ratio $1:2:3$. Find the ratio of the area of triangle $PAB$ to the area of triangle $ABC$. | nvidia |
In the rhombus ABCD, the diagonals intersect at point O. Given that AO = 3x and OC = 5x, find the area of the rhombus. | nvidia |
The base of a triangle is of length $b$, and the altitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:
$\textbf{(A)}\ \frac{bx}{h}(h-x)\qquad \textbf{(B)}\ \frac{hx}{b}(b-x)\qquad \textbf{(C)}\ \frac{bx... | nvidia |
Given a quadratic equation $ax^2+bx+c=0$ with roots $x_1$ and $x_2$, the sum of the squares of its roots can be found using the formula $x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2$. This can be expressed in terms of the coefficients $a$, $b$, and $c$ as $x_1^2+x_2^2=\left(\frac{-b}{a}\right)^2-2\left(\frac{c}{a}\right)$.
Problem... | nvidia |
Consider the expansion of $(a + b + c)^4$, where $a, b,$ and $c$ are odd integers. Determine the sum of the coefficients of the terms that contain exactly two of the variables $a, b,$ and $c$ to an odd power. | nvidia |
Find the point on the surface of the sphere $x^2 + y^2 + z^2 = 16$ that is farthest from the point $(3,4,5)$. | nvidia |
The graphs of two functions, $f(x)$ and $g(x)$, are shown below.
[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;
real wholetickd... | nvidia |
The sum of the squares of two consecutive integers is greater than 100. How many pairs of consecutive integers satisfy the given conditions? | nvidia |
Find the largest positive integer that divides each of the numbers 3x + 2, 5x + 3, and 7x + 4 for all integers x. | nvidia |
We call a line a \emph{median bisector} of a triangle if it passes through a vertex and bisects the opposite side. If two median bisectors of a triangle are parallel to the third median bisector, then we call this triangle \emph{special}.
Let $T$ be the set of all positive real numbers such that there exist two differ... | nvidia |
Find the smallest positive integer $n$ such that the decimal representation of $13n$ has a digit $7$ in the tens place and $2$ in the units place. | nvidia |
Determine the largest possible value of $c$ such that the inequality
$$\frac{x_1^2}{a_1}+\frac{x_2^2}{a_2}+\ldots+\frac{x_{101}^2}{a_{101}}\ge cx^2$$
holds for any real numbers $x_1, x_2, \ldots, x_{101}$ satisfying $x_1+x_2+\ldots+x_{101}=0$, where $x=\frac{x_1+x_2+\ldots+x_{101}}{101}$ and $a_1, a_2, \ldots, a_{101... | nvidia |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | nvidia |
The average of two 2-digit positive integers is equal to the decimal number obtained by writing one of the two-digit integers before the decimal point and the other two-digit integer after the decimal point. What is the smaller of the two integers? | nvidia |
Problem:
A point $P$ is located inside a regular pentagon $ABCDE$ such that angles $PAB, PBC, PCD, PDE,$ and $PEA$ are all congruent. The side length of the pentagon is 10. Find the radius of the circle passing through points $P, B,$ and $D$. | nvidia |
Consider a function $g(x)$ defined for all non-negative real numbers such that $g(x)\ge 0$, and
\[g(\frac{x+y}{2})=g(x)+g(y)-\sqrt {g(x)g(y)}\]
holds for all non-negative numbers $x$ and $y$.
Determine $g(6)$ if $g(0)=1$. | nvidia |
If $a$ and $b$ are positive integers such that $2^a \cdot 5^b = 10^{10}$, determine $a + b$. | nvidia |
Find the set of all possible values of the expression
\[f(x) = \sin^2 2x + \sin^2 x,\]
as $x$ ranges over all real numbers. | nvidia |
Given a polynomial $p(x)$ with real coefficients and the fact that $p(0)=-1$, what is the probability that $p(x)$ has at least one real root in the interval $(0, 1)$, assuming $p(x)$ is chosen randomly and uniformly from the space of all polynomials of degree $n$? | nvidia |
Let $x$ be a positive real number. Find the minimum value of $4x^5 + 5x^{-4}.$ | nvidia |
Find the set of all real numbers that can be expressed as
\[g(x) = \frac{\cos^2(x)\sqrt{x+3}+\frac{\sqrt{x+7}}{1+x^2}+1}{e^x-2},\]as $x$ ranges over all real numbers such that $e^x \neq 2.$ Enter your answer using interval notation. | nvidia |
Find all real numbers $x$ such that $[2x]-[x]=1$ where $[.]$ represents the least integer function. | nvidia |
A positive integer is called "prime-like" if it has exactly three distinct prime divisors. Determine the largest "prime-like" number less than 1000. | nvidia |
Find the maximum value of $y$ such that the inequality $|y - 3x| \le 2$ holds true for all real values of $x$. | nvidia |
A bag contains $n$ red and $n$ blue marbles. Two marbles are randomly drawn without replacement. The probability that the two marbles are of the same color is $\frac{5}{12}$. Determine the value of $n$. | nvidia |
Find the equation of the line that passes through the midpoint of the segment with endpoints $(-3,5)$ and $(2,-5)$ and is perpendicular to the line with equation $y=2x-3$. | nvidia |
Let $n$ be a positive integer such that $3n+1$ is divisible by 7. Determine the remainder when $2n-1$ is divided by 7. | nvidia |
Find the largest integer $n$ such that $n^2 + 3n - 5 < 0$. | nvidia |
Let $x$ be the smallest positive integer such that $x^2$ is a multiple of 16 and $x^3$ is a multiple of 27. Find the sum of the digits of $x$. | nvidia |
One vertex of a right triangle is located at (8,6) and another vertex is at the origin. The third vertex is on the positive y-axis. Find the coordinates of the third vertex if the area of the triangle is 24 square units. | nvidia |
The blue curve shown is the graph of the equation $y = ax^3 + bx^2 + cx + d$. Find $d$. (Assume that the graph has the standard unit scale.)
[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;
rea... | nvidia |
Compute the value of $a$ in the expression $5 + 3(2a-1)^3$, given that the expression is equal to $140$. | nvidia |
Let $p(x)$ be a polynomial with integer coefficients such that for every positive integer $n$ the polynomial $n \cdot p(x) + x$ has integer coefficients and $p(1)=100$. What is the sum of all possible values of $p(10)$? | nvidia |
Find the number of integer values of $x$ that satisfy the equation $\lceil{\sqrt{x}}\rceil=15$, where $x$ is a non-negative integer.
(Note: I've added the condition that $x$ is a non-negative integer to make the problem more challenging and require a different approach to solve) | nvidia |
Let $f(n)$ be the sum of all positive divisors of $n$. What is the largest integer $n$ with $f(n)=32$? | nvidia |
A convex polygon with 12 sides is given. Determine the number of ways to choose 3 vertices out of 12, such that the triangle with these vertices contains at least one side of the polygon. | nvidia |
A portion of the graph of y = F(x) is shown below.
The distance between grid lines is 1 unit.
Compute F(F(F(F(F(2))))).
[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... | nvidia |
Find the coordinates of the point that is one-third of the way from the point (2, -3) to the point (8, 9). | nvidia |
Find the smallest positive integer that can be expressed as the sum of two cubes in three different ways. | nvidia |
Find all complex values of $k$ such that the equation \[\frac{x^2}{x+1} + \frac{x}{x^2+2} = k\]has exactly three distinct roots. | nvidia |
Find the value of $x$ such that $\left(\frac{1}{x} \right)^{2} \cdot \left(\frac{1}{3} \right) = \frac{1}{2}$. | nvidia |
Consider a sequence of fractions where each term has the form $\frac{n}{n+1}-\frac{n-1}{n}$ for positive integers $n$ starting from 2. Determine the sum of the first 200 terms of this sequence. | nvidia |
Find all the triples (a, b, c) of integers such that a + b + c = 1995 and 2ab + 2bc + 2ca is a perfect square. | nvidia |
Find the number of integers $x$ that satisfy the inequality $|x^2 - 4| \leq 3$. | nvidia |
Find a positive integer that is divisible by 10 and has a square root between 36 and 36.2. | nvidia |
In a triangle with side lengths 8, 15, and 17 inches, the three sides are cut into two equal parts each. What is the length of the longest segment that can be removed from each of the six pieces so that the remaining pieces can still form a triangle? | nvidia |
A chessboard has 64 squares, each of which can be colored black or white. A "diagonal stripe" is a set of squares that are diagonally adjacent to each other. What is the smallest number of diagonal stripes needed to cover all the squares of the chessboard, such that no two stripes overlap and each stripe contains at le... | nvidia |
In a zoo, the number of giraffes is 30% more than the number of elephants. If there are 450 elephants in the zoo, calculate the total number of elephants and giraffes that are there. | nvidia |
Find the largest possible value of $|x - 2y|$ for all real numbers $x$ and $y$ that satisfy the equation $\frac{x^2}{4} + y^2 = 1$. | nvidia |
Let $\alpha$ be the sum of the squares of four distinct positive integers $a,b,c,d$ in a geometric progression, that is, $d=r^3a$, $c=r^2a$, $b=ra$, where $r$ is a rational number such that $r>1$. Given that $a+b+c+d=400$, determine $\alpha$. | nvidia |
In the parabola $y=ax^2+bx+c$, the sum of the x-coordinates of the vertices and the two x-intercepts is 0, while the product of the two x-intercepts is $-4$. Find a. | nvidia |
The real function $f$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a + b = 2^n,$ the equation
\[f(a) + f(b) = n^2\]holds. What is $f(2002)$? | nvidia |
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are... | nvidia |
A positive integer $n$ has a factorization into primes that can be written as $n = p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$. If $f(n) = p_1^{2a_1}p_2^{2a_2}\cdots p_k^{2a_k}$, find the least positive integer $n$ such that $f(n) \equiv 1 \pmod{117}$. | nvidia |
Let $\mathbf{B}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{B}^2 = \mathbf{B}.$ Find the number of possible matrices that $\mathbf{B}$ can be such that all of its eigenvalues are 0 or 1. | nvidia |
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 |
Let $x$, $y$, and $z$ be positive real numbers such that $x+y+z=3$. What is the minimum value of \[\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}?\] | nvidia |
A complex number $z$ has the property that when multiplied by itself, the result is equal to the complex conjugate of $z$ plus a real number $r$. Find all possible values of $r$ for which this is true. | nvidia |
Let $x$ and $y$ be two numbers such that $xy=25$ and $\sqrt{x}+\sqrt{y}=7$. Find the value of $\frac{x}{y}+\frac{y}{x}$. | nvidia |
Let $r$ be a real number such that the polynomial $x^{3}-r$ is divisible by the polynomial $x^{2}+2ax+a^{2}$. Determine $r$. | nvidia |
The sum of the infinite series $1+2x+3x^2+4x^3+\cdots$ can be expressed as $\frac{a}{(1-x)^b}$, where $a$ and $b$ are positive integers. Find $\frac{a}{b}$. | nvidia |
Determine all real numbers x that satisfy the inequality
$$|x^2-2x+2|<2x.$$ | nvidia |
Find three consecutive integers such that their sum is equal to the product of their squares. | nvidia |
Find the greatest common divisor of $7!$ and $(5!)^2.$
Find the smallest positive integer that is a multiple of the sum of the divisors of $7!$ and the sum of the divisors of $(5!)^2$. | nvidia |
A certain number of people are standing in a circle. Each person has a different height. What is the minimum number of people that need to be removed from the circle so that no two adjacent people have heights that are relatively prime? | nvidia |
Prove that the expression \[\frac{\cos 3\theta}{\cos\theta}+\frac{\sin 3\theta}{\sin\theta}\] is always equal to 4 for all values of $\theta$ where it is defined. | nvidia |
The difference between the squares of the solutions of $y^2+|y|=3$ is equal to 5. Find the sum of the solutions. | nvidia |
A square is drawn such that one of its sides coincides with the line $y = 5$, and so that the endpoints of this side lie on the parabola $y = x^2 + 3x + 2$. What is the area of the square? | nvidia |
Determine the greatest real number $x$ such that the inequality $x^3-3x^2+x-1 \leq 0$ is true. | nvidia |
Determine all real values of x such that the equation $$(x^2 + 4x - 7)^2 = (2x + 5)^2$$ has no real solutions for x. | nvidia |
Find the number of positive integer solutions to the equation $2^a \cdot 3^b \cdot 5^c = 3600$, where $a, b,$ and $c$ are natural numbers. | nvidia |
Nine tiles are numbered $1, 2, 3, \cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
| nvidia |
Find all real values of $a$ such that the equation $x^4 + (2 - x)^4 + a(x^2 + 1) = 34$ has exactly two distinct real solutions. | nvidia |
Find the smallest multiple of $7$ that is greater than the sum of its digits. | nvidia |
Determine the number of ways to arrange 5 people of different heights in a line, such that no two people of the same height are adjacent to each other. | nvidia |
There are 2 complex numbers $x+yi$ and $z+wi$. If $x=2$, $z=-x$, and the sum of the numbers is $3i$, find $y+w$. | nvidia |
Find the number of four-digit positive integers that are divisible by 5 and have a digit sum equal to 20. | nvidia |
Find the sum of all the values of $y$ that satisfy the equation:
$$y = \sqrt{3|y| - |2y - 9|}$$ | nvidia |
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