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Find the number of positive integers less than or equal to 30 that are relatively prime to 30. | nvidia |
Find the largest three-digit number that is a multiple of 7 and has exactly three distinct digits. | nvidia |
Determine all pairs $(n, m)$ of positive integers, where $n > m$, such that the number $\frac{n^3-m^3}{n-m}$ is divisible by 1106. | nvidia |
How many three-digit numbers with digits from the set {3, 5, 6, 8, 9} can be formed such that each digit appears only once and the sum of the digits is a multiple of 7? | nvidia |
A library has 12 books on a shelf, each with a different title. In how many ways can the books be arranged so that all the fiction books are on the left side of the shelf and all the non-fiction books are on the right side, given that there are 4 fiction books and 8 non-fiction books? | nvidia |
Find the smallest positive integer n, such that 3^n ends in the digits ...9999. | nvidia |
Find the positive real value of $t$ that satisfies $|t+2i\sqrt{3}| |6-4i| = 26$. | nvidia |
Find the eccentricity of the ellipse with the equation \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\], where $a$ and $b$ are positive real numbers, if the area of the ellipse is $\pi$ and the length of the major axis is $2\sqrt{2}$. | nvidia |
Let \(P\) be a point inside a rectangle. Determine the minimum number of different rectangles that are similar to the given one and have a side lying on a side of the given rectangle, to cover the entire given rectangle so that at least one of them contains \(P\). | nvidia |
Let $x$ and $y$ be two positive numbers such that $x^2 + y^2 = 20$ and $x^2y^2 = 16$. Find the value of $x^4 + y^4$. | nvidia |
A mouse starts at a corner of a square. On each move, it randomly selects one of the three corners where it is not currently located, and runs along the edges of the square to that corner. Given that the probability that the mouse moves to its starting corner on its fifteenth move is $p/q,$ where $p$ and $q$ are relati... | nvidia |
The function $g(x)$ satisfies
\[g(xy) = \frac{g(x)}{\sqrt{y}}\]for all positive real numbers $x$ and $y.$ If $g(20) = 15,$ find $g(35)$. | nvidia |
Find the number of positive divisors of 2018. | nvidia |
A sphere is inscribed in a cube with edge length $s$. For any point $X$ on the surface of the cube, let $g(X)$ be the sum of the distances from $X$ to the four vertices of the face of the cube that is closest to $X$. Find the maximum possible value of $g(X)$ in terms of $s$. | nvidia |
Consider the set of all integers that are multiples of either 4 or 7. Determine the largest integer N such that no element of this set has exactly N positive divisors. | nvidia |
Find all real numbers $x$ such that
\[\left(\frac{x^3-3x^2+2x+1}{x^2-2x-1}\right)^{x^2-4x+1} = \frac{x^3-3x^2+2x+1}{x^2-2x-1}.\] | nvidia |
Find the number of distinct ordered-pair solutions $(m, n)$ of the equation $m^2 + n^2 = 100$, where $m$ and $n$ are non-negative integers. | nvidia |
Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[f(x - y) = \sqrt{f(xy) + 2}\]for all $x > y > 0.$ Determine $f(2009).$ | nvidia |
The number of divisors of a positive integer n is equal to 12. What is the sum of all possible values of n? | nvidia |
How many three-digit positive integers exist, all of whose digits are 2's and/or 5's? | nvidia |
Let $a$ and $b$ be positive integers such that $a + b = 12$ and $\text{lcm}(a, b) = 15$. Find the smallest possible value of $a$. | nvidia |
Let $P(z)=x^3+ax^2+bx+c$, where $a,$ $b,$ and $c$ are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $a+b+c$. | nvidia |
Consider the sequence $a_n$ such that $a_0=5$ and $a_{n+1}=f(a_n)$, where $f(x)$ is a function that returns the sum of the digits of $x$. Find the smallest positive integer $n$ such that $a_n=5$. | nvidia |
Given the cubic equation
$$x^3-2ax^2+bx+a^2=0$$
with roots $x_1$, $x_2$, and $x_3$ such that
$$x_1x_2+x_2x_3+x_3x_1=a(x_1+x_2+x_3).$$
Find the value of $b$. | nvidia |
What is the smallest number of terms that can be removed from the sequence of positive integers so that no three consecutive terms are greater than 1000? | nvidia |
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors such that
$\mathbf{u}\cdot \mathbf{v}=8.$ Find the vector $\mathbf{x}$ such that for any vector $\mathbf{y},$
\[(\mathbf{x}-\mathbf{y})\cdot(\mathbf{u}+\mathbf{v})=(\mathbf{x}+\mathbf{y})\cdot(\mathbf{u}-\mathbf{v}).\] | nvidia |
Find the sum of the positive divisors that all the numbers in the list $\frac{15}{2}, \frac{20}{3}, \frac{12}{5}, \frac{10}{3}$ can divide into evenly, given that these divisors are also factors of $15$ and are greater than $1$. | nvidia |
Find the real value of $z$ such that the complex number $2+3iz$ has an argument of $\frac{\pi}{6}$. | nvidia |
Find a positive integer $N$ with three or fewer digits such that $N$ is the multiplicative inverse of $63$ modulo 1000. | nvidia |
In the set $T=\{1,2,3,...,98,99,100\}$, after the perfect squares and the numbers that are one more than a perfect square are removed, how many integers remain in the set $T$? | nvidia |
What is the smallest positive integer that can be expressed as the sum of two distinct perfect squares in more than one way? | nvidia |
Consider the function $f(x) = |x^2 - 6x| + k$, where $k$ is a constant. Find the largest value of $k$ such that the equation $f(x) = 2$ has exactly three distinct real solutions. | nvidia |
If $x$ and $y$ are positive integers such that $5x+3y=100$, what is the greatest possible value of $xy$? | nvidia |
The arithmetic mean of five numbers is 24. Three of the numbers are 20, 25, and 30, and the other two are equal. What is the sum of the two equal numbers? | nvidia |
In the set $\{0, 1, c, d, 1/d\}$, where $c$ is a positive number and $d$ is a nonzero integer such that $cd^2=\sqrt{d}$, find the mean of the set. | nvidia |
Find the number of positive divisors of 240 that are perfect cubes. | nvidia |
Determine the maximum value of the expression
$\ds\frac{(a+b+c+d)^5}{a^2(b+c+d)^3}$, where $a$, $b$, $c$, $d$ are real non-negative numbers. | nvidia |
Find the value of $a$ such that the solutions of the equation
$$\frac{x^2+ax+2}{x-1}=x+a+1$$
are reciprocals of each other. | nvidia |
Points $C$ and $D$ are selected on the graph of $y = -\frac{1}{3}x^2$ so that triangle $CDO$ is isosceles with a base $CD$ of length 6 units. Find the length of one of the equal sides of triangle $CDO$. | nvidia |
The graph of $y=g(x)$ is shown below, with $1$ unit between grid lines. Assume $g(x)$ is defined only on the domain shown.
What is the sum of all integers $d$ for which the equation $g(x)=d$ has exactly $4$ solutions?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm... | nvidia |
When two fair 6-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that the two digit number $ab$ (where $a$ and $b$ are digits) and $a$ and $b$ are all divisible by 3? | nvidia |
A hexagon inscribed in a circle has three consecutive sides, each of length 3, and three consecutive sides, each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides, each of length 3, and the other with three sides, each of length 5, has length equal to $m/n$, where $... | nvidia |
Find the sum of all the even divisors of $360$. | nvidia |
Find the largest value of $n$ such that $5x^2+nx+48$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Given that $a$ and $b$ are the roots of $x^2 + x - 2 = 0,$ and that $P(x)$ is a polynomial such that $P(a) = 1$ and $P(b) = 3,$ find $P(1).$ | nvidia |
What is the smallest positive integer $n$ such that $17n \equiv 1234 \pmod{7}?$ | nvidia |
Find the largest of three consecutive integers that are not multiples of 3, knowing that they are all 4-digit numbers and each is a multiple of 7. | nvidia |
Given a convex quadrilateral, its interior angles are such that if we group them into two pairs, with each pair consisting of adjacent angles, the sum of the measures of the angles in each pair is equal to $165^\circ$. If one of the angles is $95^\circ$, determine the measure of the angle adjacent to it. | nvidia |
Let $a,b,c$ be positive integers such that $a+b+c=2abc+1$ and $ab+bc+ca=2abc$. Compute $abc$. | nvidia |
Determine the maximum value of $k$ such that it is possible to complete a triangle with side lengths $k$, $k+3$, and $k+6$. | nvidia |
In the complex plane, there exists a number $\omega$ with $|\omega| = 2$ and a real number $\mu$ such that the points $\omega$, $\omega^3$, and $\mu \overline{\omega}$ form an isosceles triangle with base $\overline{\omega \omega^3}$. Find the value of $\mu$. | nvidia |
Compute the value of $S$, where $S$ is the sum of all complex numbers $z$ satisfying the equation $z^{10} = 1$.
There are exactly ten solutions $z$ of the equation $z^{10}=1$, two of which are real, namely, $1$ and $-1$, and eight of which are complex. Each of the eight complex solutions is a primitive 10th root of u... | nvidia |
Let $f(x)$ be a quadratic function such that $f(x+1)-f(x)=2x+1$ for all real numbers $x$. Find $f(5)-f(2)$. | nvidia |
Let $f(x)$ be a function defined for all positive real numbers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[f(x - y) = \sqrt{f(xy) + 2}\]for all $x > y > 0.$ Determine $f(2009).$ | nvidia |
A probability experiment is conducted by tossing two fair dice. The number of times that the product of the two numbers that appear is even is recorded. If the experiment is conducted 120 times, determine the expected number of times the product of the numbers is even. | nvidia |
How many positive integers between $100$ and $500$ have at least one digit that is a multiple of $3$, but no digits that are multiples of $9$? | nvidia |
Determine the number of integers from 1 to 1000 that are not divisible by 7, 11, or 13. | nvidia |
A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}$, where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) + y) = g(x) + g(g(y) + g(x)) - 2x\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$ Find $m \times t.$ | nvidia |
Find the number $c$ such that for any real number $a$, the equation
$$(x^2 + ax + 2)^2 + c = x^4 + 5x^2 + 8$$
has at least one real solution. | nvidia |
Simplify $t^{a-2} \cdot t^{3a+1}$, where $a$ is a positive integer. | nvidia |
Let $a$ be a real number such that $\tan a=1+\tan 2a$.
Determine $\cos 3a$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a + b + c = \pi$. Compute
\[\begin{vmatrix} \sin a & \sin b & \sin c \\ \cos a & \cos b & \cos c \\ \sin 2a & \sin 2b & \sin 2c \end{vmatrix}.\] | nvidia |
Find the largest number of consecutive even integers whose sum is 100. | nvidia |
Let $P$ be the product of the first $150$ positive even integers. Find the largest integer $k$ such that $P$ is divisible by $2^k$. | nvidia |
Consider a function $g : \mathbb{Z}^+ \to \mathbb{Z}$ defined for all positive integers $m$ and $n$ such that
\[g(m+n) = g(m) + g(n) + m \cdot n.\]
Let $\alpha$ be the minimum value that $g(6)$ can attain and $\beta$ be the maximum value that $g(6)$ can attain. Find $\alpha + \beta$. | nvidia |
Find the value of $x$ that satisfies the equation:
$$x^8 = 2^{16} \cdot 3^8$$
Can you find the value of $x$ that satisfies this equation? | nvidia |
What is the sum of all the distinct digits that can appear as the last digit of a number that is a perfect square? | nvidia |
Determine the smallest whole number greater than 50 that has an even number of positive divisors. | nvidia |
Find the smallest positive integer $k$ such that the product of the $k$-th roots of the $k$-th degree monic polynomial $P(x) = x^k + x^{k-1} + \dots + x + 1$ is equal to 1.
Note: A monic polynomial is a polynomial with the leading coefficient equal to 1. | nvidia |
Find the smallest positive integer that is not the difference of two positive square integers greater than 1. | nvidia |
$n$ coins are simultaneously flipped. The probability that at most one of them shows tails is $\frac{3}{16}$. Find $n$. | nvidia |
Find the number of positive integers less than 1000 that are divisible by 7 and have exactly 3 digits that are all different. | nvidia |
What is the smallest positive integer $n$ such that $n^2 + 1$ is divisible by 5 and $n$ is a multiple of 7? | nvidia |
Determine the sum of the digits of the number $6^{6!}$. | nvidia |
Let $x$ be a digit such that $0 \le x \le 9$. If the number $7129x436_8$ is divisible by 10, find the value of $x$. | nvidia |
Let $x$, $y$, and $z$ be positive integers such that
\begin{align*}
(x+y)^2+(y+z)^2+(z+x)^2&=108,\\
x^2+y^2+z^2&=18.
\end{align*}Find $xyz$. | nvidia |
Find the number of three-digit numbers that are equal to the sum of their digits cubed. | nvidia |
Find the point $P$ that minimizes the sum of its distances to the vertices $A$ and $B$ of the triangle with vertices $A(1,2)$, $B(6,4)$, and $C(3,1)$. If the coordinates of $P$ are $(m,n)$, where $m$ and $n$ are integers, what is $m+n$? | nvidia |
Determine the number of different prime divisors of the sum of all even positive integers between 10 and 100. | nvidia |
Find the number of non-congruent triangles whose side lengths are consecutive positive integers in geometric progression. | nvidia |
For all positive integers $n$, let $g(n) = \sqrt{n} + \sqrt{n+1}$. Find $g(16)+g(25)+g(30)$. | nvidia |
What is the sum of all values of $x$ such that $|2x-1|=|3x-2|$? | nvidia |
A circle is inscribed in a regular heptagon. How many distinct chords of the circle can be drawn such that they are also diagonals of the heptagon? | nvidia |
A rectangular prism has six faces, each with a number assigned to it, and each edge has a length of 1 unit. The numbers on the faces are all different prime numbers less than 20. For each of the eight vertices of the prism, a product of three numbers is calculated, where the three numbers are the numbers assigned to th... | nvidia |
A positive integer has the property that the units digit of its cube is 7. Determine the possible values of the units digit of the number itself. | nvidia |
In the expansion of $(1+x)^n$, the coefficients of the terms form a triangular pattern similar to Pascal's Triangle, where each number is the sum of the number just above it and to the left and the number just above it and to the right. Prove that the sum of the squares of the coefficients in the $n$th row of this tria... | nvidia |
Determine the sum of all values of $y$ that satisfy the equation
$$\sqrt{(y-2)^2} + \sqrt{(y+2)^2} = 8.$$ | nvidia |
Determine the number of different diagonals that can be formed by connecting the vertices of a regular tetrahedron. | nvidia |
Let $C$ be the number of three-digit numbers that have at least one digit repeated. Let $D$ be the number of three-digit numbers that are divisible by 3. Find $C+D$. | nvidia |
The sum of the areas of two squares with sides $x$ and $y$ is equal to the area of a rectangle with sides $z$ and $x+y$. Determine $x+y$ when $z=10$ and the perimeter of the rectangle is 36 units. | nvidia |
Let $a$ and $b$ be real numbers. If the polynomial $x^2+ax+b$ has exactly one real root and $a = b - 2$, find the sum of all possible values of $b$. | nvidia |
Find the smallest positive integer that is both an integer power of 12 and has a digit sum that is a perfect square. | nvidia |
An urn contains 100 colored balls, some of which are white, while the rest are either red or blue. It is known that if you pick two balls at random (without replacement), the probability that both will be white is less than $\frac{1}{2}$. What is the minimum number of white balls that can be in the urn? | nvidia |
Find the number of positive integers less than or equal to 200 that are divisible by either 5 or 7, but not by 35. | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $x + y + z = 2$. Compute the maximum value of
\[xyz(x^2 + y^2 + z^2).\] | nvidia |
Let $P$ be a point inside the triangle $ABC$ such that the sum of the distances from $P$ to the sides of the triangle is equal to half of the perimeter of the triangle. Find the maximum value of the ratio of the areas of the triangle $ABC$ and the triangle formed by the reflections of point $P$ across the sides of the ... | nvidia |
Determine the number of 5-digit positive integers that can be formed using the digits 1, 2, 3, 4, and 5, with the condition that the sum of the digits in each integer must be exactly 15. | nvidia |
Find the smallest positive integer $x$ such that it has exactly 12 positive factors and is divisible by 3 and 7. | nvidia |
Find the greatest positive integer whose sum of divisors is equal to 18. | nvidia |
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