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For all composite integers $n$, what is the largest integer that always divides into the difference between $n$ and the cube of $n$? | nvidia |
Let $x$ and $y$ be positive real numbers. Find the maximum value of $x + y + 1/x + 1/y$ given that $xy \leq 1$. | nvidia |
Find the number of positive integers less than 1000 that have a remainder of 7 when divided by 11. | nvidia |
You have 5 different colored scarves, 6 pairs of gloves, and 8 hats, all of which are distinct. How many outfits can you make consisting of one scarf, one pair of gloves, and one hat, such that the color of the scarf is different from the color of the hat, and the color of the gloves is the same as the color of the sca... | nvidia |
Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| nvidia |
Let's consider a function with a similar trigonometric theme and domain restriction.
Find the smallest value of the positive integer $k$ such that the function $g(x) = \arccos(\sqrt{kx})$ has a domain that includes the interval $[0, \frac{1}{2}]$. | nvidia |
Determine the smallest positive integer $n$ such that the product of its prime factors is equal to $3\times5^2$. | nvidia |
Determine the sum of all roots of the equation:
\[z^5 = i.\] | nvidia |
Find the largest integer k for which there exists a polynomial P(x) with integer coefficients such that the product of the leading coefficient and the constant term is equal to k and P(7) = 340. | nvidia |
Find the maximum value of $10^x - 100^x,$ over all real numbers $x.$ | nvidia |
Determine the number of ways to arrange the letters of the word PROOF such that the two O's are not adjacent. | nvidia |
What is the smallest positive integer $n$ such that $\frac{2n+3}{n+1} > \frac{n+1}{n-2}$? | nvidia |
Evaluate the product \[ (n-1) \cdot n \cdot (n+1) \cdot (n+2) \cdot (n+3), \] where $n=2$. | nvidia |
A cryptographer is trying to decode a message that consists of three distinct positive integers, each less than 10000. The only clue she has is that the sum of the digits of each integer is 7. How many different possible combinations of integers could be the decoded message? | nvidia |
For a positive integer $n$ and positive integers $a_1$, $a_2$, $b$, and $c$, let $A_{n, 1}$ be the $n$-digit integer each of whose digits is equal to $a_1$, let $A_{n, 2}$ be the $n$-digit integer each of whose digits is equal to $a_2$, let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C... | nvidia |
Determine the smallest positive integer $n$ such that the polynomial $P(x) = x^n + 12^{n-1}x^{n-1} + \cdots + 12x + 8$ has a root that is an integer multiple of 8. | nvidia |
Find the smallest value of $x$ that satisfies the equation $|3x+7|=26$. | nvidia |
What is the positive difference between the sum of the first 20 positive even integers and the sum of the first 15 positive odd integers? | nvidia |
Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.) | nvidia |
Find the value of $x$ such that $\sqrt{x^2+4x+5}=3x+1$ | nvidia |
Find the sum of the values of $y$ that satisfy the equation $|3y| - 2 = 23$. | nvidia |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | nvidia |
How many positive integers $n$ satisfy $200 < n^2 < 900$? | nvidia |
Determine the number of distinct positive integers $N$, such that the sum of $k$ consecutive positive prime numbers from exactly 3 different values of $k \ge 1$ is equal to $N$. | nvidia |
Determine the maximum possible value of the expression \[\frac{a^2 + b^2 + 4ab}{a^2 + b^2 + 2ab},\] where $a$ and $b$ are positive real numbers. | nvidia |
Find the smallest positive integer that has exactly 12 positive integer factors. | 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 |
Let $f$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2f(a^2 + b^2) = [f(a)]^2 + [f(b)]^2\]for all nonnegative integers $a$ and $b.$
Let $n$ be the number of possible values of $f(25),$ and let $s$ be the sum of the possible values of $f(25).$ Find $n \times s.$ | nvidia |
Let $(x,y)$ be an ordered pair of real numbers that satisfies the equation $x^2+y^2=14x+48y$. What is the minimum value of $y$?
Here is another problem inspired by this one:
Find the maximum value of $x$ in the equation $x^2+y^2=14x+48y$, where $x$ and $y$ are real numbers. | nvidia |
What is the largest integer $n$ such that $2^n$ divides $100!$? | nvidia |
Find the largest value of x such that the expression \[\dfrac{x+1}{8x^2-65x+8}\] is not defined.
The given problem involves determining the largest value of $x$ for which the expression $\frac{x+1}{8x^2-65x+8}$ is undefined. This problem focuses on finding the values of $x$ that make the denominator zero, hence undefi... | nvidia |
In a triangle, the sum of the squares of the lengths of the sides is equal to twice the sum of the squares of the lengths of the medians. A "nice" triangle has integer side lengths and satisfies this property. What is the sum of all the different possible perimeters of nice triangles? | nvidia |
Let $a$ be a real number. For what values of $b$ does the equation
$$x-\sqrt{x^2-a}=b$$ have at least one real solution? | nvidia |
Find the sum of all the positive integers that are divisors of 48. | nvidia |
Determine a positive integer $k$ such that $6\times7\times8\times k$ is equal to the sum of all integers from $1$ to $99$. | nvidia |
Find the smallest positive integer $n$ such that the polynomial $x^n - 2$ has at least $4$ distinct integer roots.
Challenge: Can you generalize your solution to find the smallest positive integer $n$ such that the polynomial $x^n - k$ has at least $m$ distinct integer roots for any positive integers $k$ and $m$? | nvidia |
Find the value of $c$ such that the equation $x^2-4x+c=0$ has two real roots and the sum of their squares is 32. | nvidia |
Given the graph of a line that passes through the points (2,3) and (4,5), find the equation of a line that is perpendicular to this line and passes through the point (3,4). | nvidia |
How many positive integers less than $10^6$ have the property that their squares are divisible by $27$? | nvidia |
Let $b$ be the base $5$ representation of the decimal number $243_{10}$. Let $x$ be the number of digits in the base $5$ representation of $b$ that are greater than $2$ and $y$ be the number of digits in the base $5$ representation of $b$ that are less than or equal to $2$. What is the value of $x+y$? | nvidia |
Given that $f(x) = ax^3 + bx^2 + cx + d$ is a cubic function with real coefficients, find the value of $a$ such that $f(x)$ has exactly two real roots, one of which is a double root. | nvidia |
Let $g(x)$ be a function such that $g(x) = \frac{1}{ax^2 + bx + c}$. Given that the line $y = x + 2$ is tangent to the graph of $g(x)$ at two points, determine the value of $a$ such that the sum of the roots of the equation $g(x) = x + 2$ is equal to the product of the roots of the same equation. | nvidia |
Given that $n$ is an integer and $0 < 4n <30$, what is the sum of all possible integer values of $n$? | nvidia |
A cube is painted so that one side is blue, two sides are red, and three sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube.
New Problem:
A hexagon is painted so that two adjacent sides are red, two adjacent si... | nvidia |
Let $g$ be a function taking the positive integers to the positive integers such that
$$g(m) g(n) = g(m+n) g(\gcd(m,n)) + 1$$
for all positive integers $m$ and $n.$
Find the number of possible values of $g(3).$ | nvidia |
Let $A$ be the set of positive integers less than or equal to $100$ that are divisible by $6$, and let $B$ be the set of positive integers less than or equal to $100$ that are divisible by $7$. Determine the percentage of elements of $B$ that are also elements of $A$. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a+b+c=1.$ Find the maximum value of
\[\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}.\] | nvidia |
A set of 10 points are chosen on a circle centered at the origin of the coordinate plane, such that the sum of the squares of the x-coordinates of the points is 256. These points are then connected in order to form a decagon. The midpoints of the sides of this decagon are then connected to form a new decagon, and the p... | nvidia |
The function $g(z)$ satisfies
\[g(zw) = \frac{g(z)}{w^2}\]for all positive real numbers $z$ and $w.$ If $g(10) = 5,$ find $g(15).$ | nvidia |
How many positive seven-digit integers can be made that are not palindromes, using the digits 7, 8, and 9? | nvidia |
Let $x$ and $y$ be positive real numbers such that $xy = 2x + 2y.$ Find the minimum value of $\frac{x}{y} + \frac{y}{x}.$ | nvidia |
A sector with acute central angle $\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is
$\textbf{(A)}\ 3\cos\theta \qquad \textbf{(B)}\ 3\sec\theta \qquad \textbf{(C)}\ 3 \cos \frac12 \theta \qquad \textbf{(D)}\ 3 \sec \frac12 \theta \qquad \textbf{(E)}\ 3$
| nvidia |
The number of solutions of $2^{2x}-3^{2y}=55$, in which $x$ and $y$ are integers, is:
\[\textbf{(A)} \ 0 \qquad\textbf{(B)} \ 1 \qquad \textbf{(C)} \ 2 \qquad\textbf{(D)} \ 3\qquad \textbf{(E)} \ \text{More than three, but finite}\]
| nvidia |
What is the largest positive integer that has exactly three divisors? | nvidia |
Find the smallest positive integer that has exactly three distinct prime factors, and when divided by the sum of these prime factors, the quotient is a prime number. | nvidia |
Let $x$ and $y$ be real numbers such that $|x+y|=1$ and $|x-y|=p$. If $x>y$, determine $\frac{x}{y}$ in terms of $p$. | nvidia |
If $a$ and $b$ are positive real numbers, determine $a$ and $b$ in the equation $a^2 \cdot b^{-3}=\frac{1}{2}$ if it is given that $a^3 \cdot b^{-2}=2$. | nvidia |
In the triangle $ABC$ points $D$ and $E$ are the midpoints of the sides $AB$ and $AC$, respectively. Point $F$ is the midpoint of the side $DE$. Find the ratio of the area of triangle $ABC$ to the area of triangle $AEF$. | nvidia |
Evaluate the sum of the products of each pair of elements in a row of the matrix:
\[\begin{pmatrix} x & x+1 & x+2 \\ x+3 & x+4 & x+5 \\ x+6 & x+7 & x+8 \end{pmatrix}.\] | 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 |
Determine the number of different complex numbers $z$ with the properties that $|z|=\sqrt{2}$ and $\ds\frac{1}{z^{3}}+\ds\frac{1}{z^{2}}+z^{2}+z^{3}$ is a real number. | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(xy) = g(x) + g(y)\]for all positive integers $x$ and $y.$ Given $g(15) = 12$ and $g(90) = 18,$ find $g(1350).$ | nvidia |
Find the number of different positive integers that can be expressed as the sum of two distinct members of the set $\{1, 2, 3, \ldots, 18, 19, 20\}$. | nvidia |
A polynomial with real coefficients has the form $x^2 + ax + b$, where $a$ and $b$ are real numbers. Two of its roots $x_1$ and $x_2$ are such that $x_1 - x_2 = 2$ and $x_1 x_2 = 3$. Determine $a$ and $b$ and then compute the sum $a+b$. | nvidia |
Suppose $x$, $y$, and $z$ are positive real numbers such that $x>y>z$. Which of the following inequalities must hold?
$x^2>y^2+z^2$
$x+y>y+z$
$xy>xz$
$x^2+y^2>x^2+z^2$
$\frac{x}{z}>1$
Enter your answer as a list of those options that are always true. | nvidia |
Find the area of the triangle with vertices $(2,3),$ $(-1,2),$ and $(4,1).$ | nvidia |
Find the largest integer $n$ such that $n^2+7n+9$ is divisible by $n+3$. | nvidia |
Find the value of $c$ such that the equation $x^2 + 4x + c = 0$ has roots whose product is equal to the sum of the squares of the roots. | nvidia |
How many odd three-digit numbers are there such that the sum of the tens and units digits is 12? | nvidia |
In a set of five distinct integers, the product of the greatest three integers is 840 and the product of the smallest two integers is 14. What are the five integers? | nvidia |
The midpoints of the sides of triangle $ABC$ are $D,$ $E,$ and $F,$ and $G$ is the point where the medians meet. The circle with center $G$ and radius $GD$ meets the line through $A$ and $G$ at $H$ and $I.$ If the area of $\triangle ABC$ is $180,$ then find the area of quadrilateral $AIGE.$ | nvidia |
Let $x,y,z$ be three positive real numbers such that $x,y,z$ is a geometric progression and $x,z,y$ is an arithmetic progression. Find the largest possible value of $\frac{x+y+z}{xyz}$. | nvidia |
Find the value of $x$ such that $\sqrt{2x + 5} = 8$. | nvidia |
Determine the smallest non-negative integer $n$ such that $3n \equiv 5 \pmod{7}$. | nvidia |
A cube with side length 3 has vertices labeled with numbers. Each vertex is connected to three edges, and the numbers on the vertices at the endpoints of each edge add up to the same total value, which is an integer $N$. Find the sum of all possible values of $N$. | nvidia |
Let $g : \mathbb{C} \to \mathbb{C} $ be defined by $ g(z) = z^3 - 2iz + 3 $. Find all complex numbers $z $ such that $ \text{Re}(z) < 0 $ and both the real and the imaginary parts of $g(z)$ are rational numbers with denominators at most $5$. | nvidia |
Find the positive integer $n$ such that $|(n+i)^2|=|(n-i)^3|$. | nvidia |
Find the sum of all the solutions of the equation $x(x+1)(x+2)(x+3)=24$, where $x$ is a real number. | nvidia |
The expression $729x^3+8$ can be written as $(ax+b)(cx^2+dx+e)$. Find $a+b+c+d+e$.
Write another problem inspired by this one:
Find all values of $a$ such that the expression $a^3+8$ can be written as the product of three consecutive positive integers. | nvidia |
Find the sum of all values of $y$ such that $3^{y^2-2y-1} = 9^{y - 3}$. | nvidia |
Find the smallest integer $m$, greater than $2$, such that $m^{-1}\pmod{1350}$ is defined. | nvidia |
If the inverse tangent of the product of two positive numbers is equal to the sum of their inverse tangents, find the ratio of these two numbers. | nvidia |
Determine the value of $\left|z^4\right|$ if $z$ is a complex number such that $|z|=2$. | nvidia |
Find the number of different positive integers that can be expressed as a sum of two distinct members of the set {3, 5, 7, 9, 11, 13}. | nvidia |
Find the number of positive divisors of 2160 that are not multiples of 6. | 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 |
The terms $a_1, a_2, a_3, \dots$ of a sequence are all positive integers such that for each $n \ge 2$, $a_n$ is the harmonic mean of $a_1, a_2, \dots, a_{n-1}$. Given that $a_1=2$, find $a_{2008}$. | nvidia |
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is t... | nvidia |
Calculate the sum of the arithmetic series $1+2+3+4+\dots$, where the sum is taken over all positive integers less than or equal to $n$, where $n$ is a positive integer. Express your answer in terms of $n$. | nvidia |
The lengths of two sides of a triangle are in the ratio of 3 to 4. The length of the shorter side is increased by 15%. By what percent must the length of the longer side be decreased so that the perimeter of the triangle remains the same? | nvidia |
Given an 8-digit integer $ABCDEFGH$ where each letter represents a distinct digit between 1 and 9, find the number of ways to arrange the digits so that the numbers formed by $AB$, $CD$, $EF$, and $GH$ are all distinct and represent valid dates in the month-day format, where none of the formed dates are in the future a... | nvidia |
Let $a$, $b$, and $c$ be complex numbers such that $a+b+c=0$. Prove that
\[a^3+b^3+c^3=3abc.\] | nvidia |
Find the number of natural numbers between 1 and 100 that are not perfect squares, but the sum of their digits is a perfect square. | nvidia |
Let $r_1,$ $r_2,$ and $r_3$ be the roots of the equation $x^3-3x^2+5x-7=0$. Find the value of $r_1r_2+r_1r_3+r_2r_3$ if it is known that $r_1+r_2+r_3$ and $r_1r_2r_3$ are integers. | nvidia |
A sequence of real numbers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n - 1}^2 - a_{n - 2}$ for each $n \ge 3.$ Given that the sum of the first $1200$ terms of this sequence is $0$ and the sum of the first $1500$ terms is $1$, determine the sum of the first $2500$ terms. | nvidia |
Let $a$, $b$ and $c$ be positive integers such that $\gcd(a,b) = \gcd(a,c) = \gcd(b,c) = 1$.
Determine the largest possible value of $\gcd(a^3,b^2c)$. | nvidia |
Three points $A$, $B$, and $C$ are on a circle with center $O$ such that $O$, $A$, and $B$ are collinear and $\angle AOC=120^{\circ}$.
Determine the measure of $\angle CBA$ in degrees if $CA=CB$. | nvidia |
Find the product of all positive integer values of $c$ such that $8x^2+15x+c=0$ has two real roots. | nvidia |
If $n$ is a positive integer, then let $f(n)$ be the sum of the digits of $\frac{1}{5^{{}^n}}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $f(n) > 10$? | nvidia |
Find the length of the side of the largest regular hexagon that can be inscribed in a circle of radius 6. | nvidia |
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