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Let $g(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the largest possible integer such that $k!$ is not divisible by $n$. Given that $n$ is a prime number, what is the largest value of $n$ such that $g(n) < 10$? | nvidia |
Problem:
Find the rectangular coordinates of the point on the surface of the sphere with equation $x^2 + y^2 + z^2 = 16$, such that the point lies in the first octant and makes an angle of $\frac{\pi}{4}$ with the $xy$-plane and an angle of $\frac{2\pi}{3}$ with the $yz$-plane. | nvidia |
The real numbers $a$, $b$, $c$, and $d$ are such that
$$\gcd(a,b)=\gcd(b,c)=\gcd(c,d)=\gcd(d,a)=\gcd(a,c)=5$$
and $\gcd(a,b,c,d)=1$.
Find the smallest possible value of $\lceil a\rceil\lceil b\rceil\lceil c\rceil\lceil d\rceil$. | nvidia |
A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The prob... | nvidia |
Let f be a function defined on the set of positive integers such that f(1)=1, f(2n)=f(n)+n for n>=1, and f(2n+1)=f(n)+n+1 for n>=1. Determine f(2017). | nvidia |
The number of ways to arrange 5 men and 5 women in a circle such that no two men are adjacent is 2880. What is the number of ways to arrange 6 men and 6 women in a circle such that no two men are adjacent? | nvidia |
Find the smallest positive integer $n$ such that $\frac{1}{n}$ is a repeating decimal and $n$ is a palindrome number. | nvidia |
Compute the smallest positive integer $n$ such that
\[\sum_{k = 0}^n \log_2 \left( 1 + \frac{1}{2^{2^k}} \right) \ge 1 + \log_2 \frac{2014}{2015}.\] | nvidia |
A set of positive integers is called "balanced" if the sum of the digits of each number is a multiple of 3. How many balanced numbers are there between 100 and 200? | nvidia |
There are five positive integers that are divisors of each number in the list $$20, 40, 80, 60, 140.$$Find the sum of these five positive integers. | nvidia |
Find the only positive real number $x$ for which $\displaystyle \frac{x-4}{9} = \frac{4}{x-9}$. | nvidia |
The number $0.\overline{162}$ can be written as a fraction in the form $m/n$ where $m$ and $n$ are positive integers and $n<81$. Find the greatest possible value of $n+m$. | nvidia |
A right triangle with integer leg lengths is called "nice'' if the number of square units in its area is equal to three times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of nice right triangles? | nvidia |
Find $t$ such that $(t,5)$ lies on the line through $(0,3)$ and $(-8,0)$. | nvidia |
Find the smallest five-digit positive integer with distinct digits that is divisible by the sum of its digits. | nvidia |
Someone observed that $8! = 10 \cdot 11 \cdot 12 \cdot 13$. Find the largest positive integer $m$ for which $m!$ can be expressed as the product of $m - 4$ consecutive positive integers. | nvidia |
The function g is defined on the set of real numbers and satisfies:
g(x) = {
2x if x is rational,
g(g(1/x)) if x is irrational.
}
Find g(√2) | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(xz) - f(x) f(yz) \ge 1\]for all real numbers $x,$ $y,$ and $z.$
Find the number of functions $g : \mathbb{R} \to \mathbb{R}$ such that
\[g(x^2) + g(y^2) - g(x) g(y) \le 0\]for all real numbers $x$ and $y.$ | nvidia |
Find the largest value of $c$ such that the function $f(x) = x^3 - 2x^2 + cx + 1$ has at least one real root. | nvidia |
The roots of the equation $x^3+px^2+qx+r=0$ are in arithmetic progression. If the sum of the squares of the roots is 45, find the largest possible value of $p$. | nvidia |
Find all positive integers that can be expressed as the sum of three distinct primes, where each prime is one less than a multiple of 5. | nvidia |
The function $f(x)$ satisfies
\[f(x^2) + xf(x^{-2}) = 2\]for all real numbers $x.$ Find $f(-1).$ | nvidia |
Let $g$ be a function taking the integers to the integers such that
\[g(mn) + g(m + n) = g(m) g(n) + m + n\]for all integers $m$ and $n.$
Let $k$ be the number of possible values of $g(1),$ and let $t$ be the sum of all possible values of $g(1).$ Find $k + t.$ | nvidia |
Find the sum of the areas of all squares with side lengths between 5 and 15, inclusive, if the side lengths are limited to positive integer values. | nvidia |
Find the smallest positive integer $n$ such that the sum of all positive divisors of $n!$ is divisible by $n+1$. | nvidia |
Consider the equation $x^3 \cdot x^{-3} = 1$. Prove that this equation holds for any non-zero value of $x$. | nvidia |
Find the smallest positive integer $n$ such that $n$ is not divisible by the product of its distinct prime divisors, but $n+1$ is. | nvidia |
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot \{x\} = 4.5$, where $\{x\}$ is the fractional part of $x$. Express $x$ as a decimal. | nvidia |
When four standard dice are tossed, the numbers $a,b,c,d$ are obtained. Find the probability that $abcd = 8$. | nvidia |
The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $10$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle. | 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 |
Let $A$, $B$, and $C$ be points on the parabola $y^2=x$ such that the circle with diameter $\overline{AB}$ passes through $C$. Find the smallest possible area of triangle $ABC$. | nvidia |
How many four-digit positive integers have at least one digit that is a 2 or a 3? | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $xyz=1$. Find the minimum value of
\[x^2 + y^2 + z^2 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.\] | nvidia |
Find the sum of the squares of the reciprocals of the positive divisors of 36 that are divisible by 3. | nvidia |
In a convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $O$. Circles $\omega_1$ and $\omega_2$ are tangent to sides $AB$ and $BC$ at points $E$ and $F$, respectively, and both circles are also tangent to $AC$ at points $G$ and $H$, respectively. Given that the radius of $\omega_1$ is twice the radius... | nvidia |
Find the smallest five-digit positive integer, with all distinct digits, that is divisible by the sum of its digits. | nvidia |
Let $ABCDEF$ be a convex hexagon with $AB \parallel CE, BC \parallel AD, CD \parallel EF, \angle ABC=120^\circ, AB=3, BC=5,$ and $EF = 15.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $EFD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | 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 the number of integers that can be written as the difference between two squares, with the smaller square being less than or equal to 16. | nvidia |
In a 300-seat cinema, the seats are numbered from 1 to 300. Each seat number is written on a label, and all the labels are placed in a box. If a label is randomly selected from the box, what is the probability that the sum of the digits of the seat number on the label is a multiple of 3? | nvidia |
Find the smallest positive integer $k$ such that $3^k$ is a factor of the sum of the squares of the first 20 positive integers. | nvidia |
Let $f$ be a function from the positive integers to the positive real numbers such that
$$f(n+2)-f(n)=\ln(n+1).$$Find $f(13)-f(4)$. | nvidia |
Determine the sum of all real numbers $x$ satisfying
\[(x^2-4x+2)^{x^2-5x+2} = 1.\] | nvidia |
Find the value of $x$ such that $\sin(x) = \cos(\frac{\pi}{6})$. | nvidia |
Let $x, y, z$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}x^2+y^2+z^2&=&4,\\ xyz&=&1.\end{array}\]
Find the value of $\lfloor x+y+z\rfloor$. | nvidia |
Let $a$, $b$, and $c$ be the roots of the polynomial $x^3+px+q=0$, where $p$ and $q$ are integers. Suppose that the equation $x^2+ax+b=0$ has two integer roots, and the equation $x^2+bx+c=0$ has only one integer root. What is the sum $|a|+|b|+|c|$? | nvidia |
Determine the value of $x$ such that $(\frac{1}{2})^{\frac{x}{7}} \cdot 8^{\frac{1}{5}} = \frac{1}{16}$ | nvidia |
Find the number of distinct positive integers that can be formed by taking the sum of two or more different integers from the set $\{2, 5, 7, 11, 13\}$. | 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 |
Problem:
Suppose that the number $e^{\frac{1}{4}} - \frac{1}{2}$ can be expressed in the form $(e^x - y)^2,$ where $x$ and $y$ are positive real numbers. Find $x+y.$ | nvidia |
Find the smallest positive integer greater than $100$ that has exactly $5$ positive factors. | nvidia |
Given that $a$ and $b$ are positive integers such that $a^2 + b^2 = 16$ and $ab = 2$, what is the value of $a + b$? | nvidia |
Find the largest negative integer $m$ such that $\frac{m}{m-27}$ is equal to a recurring decimal with only two distinct digits after the decimal point. | nvidia |
What is the sum of all real numbers that are tripled when added to their squares? | nvidia |
Find the largest possible six-digit palindrome that is a perfect square. | nvidia |
Consider a function $f(x)$ defined on the set of all positive integers, satisfying the following conditions:
(i) $f(1)=1$
(ii) For any $m$ and $n$ such that $\gcd(m,n)=1$, $f(m)+f(n)=f(mn)$
Determine the possible values of $f(6)$ and the sum of all possible values of $f(6)$ | nvidia |
Find a number that is a factor of every sum of four consecutive even integers. | nvidia |
Given that the graph of the function $y=\frac{3x^2+7x-5}{x^2+2x+1}$ has a horizontal asymptote at $y = a$, find the equation of the oblique asymptote (if any) that the graph of $y=\frac{3x^2+7x-5}{x^2+2x+1}+x$ approaches as $x$ goes to infinity. | nvidia |
Find $t$ such that $(t,7)$ lies on the line through $(2,1)$ and $(5,9)$. | nvidia |
Find the sum of the positive divisors of $28$ that are not multiples of $3$. | nvidia |
Let $P(z)$ be a polynomial of degree $4$ with integer coefficients such that $P(3i)=P(-3i)=0$, and $P(1)=P(-1)=0$. Determine the value of $P(0)$. | nvidia |
Find the minimum value of
\[2a^2b + 2ab^2 + 3b^2c + 3bc^2\]
over all nonnegative real numbers $a,$ $b,$ and $c$ such that $a + b + c = 1.$ | nvidia |
What is the sum of all real numbers that are tripled when subtracted from their cubes? | nvidia |
Given a hyperbola defined by the equation $xy = 1$, and a circle $\mathcal{C}$ intersecting the hyperbola at four distinct points. If three of these points are $(\frac{1}{2}, 2)$, $(1, 1)$, and $(2, \frac{1}{2})$, find the sum of the distances from the center of the hyperbola to all four of the intersection points. | nvidia |
Let $\mathcal{S}$ be the set of all subsets of $\{1, 2, \dots, 100\}$ that contain $k$ distinct elements, where $k$ is an integer between $1$ and $100$ inclusive. Define the "magnitude" of a subset to be the sum of its elements. Determine the number of possible magnitudes of subsets in $\mathcal{S}$. | nvidia |
What is the sum of all integer values of $n$ such that $\frac{20}{2n - 1}$ is an integer? | nvidia |
Determine all real numbers $a$ such that the function
\[f(x) = f(x^2) + 2ax\]
has a unique solution for all $x.$ | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x + f(y)) = x + y\]for all real numbers $x$ and $y.$ | nvidia |
What is the maximum perimeter of a triangle with integer side lengths such that the sum of the squares of its sides is equal to 200? | nvidia |
Determine the smallest positive integer $k$ such that $7k \equiv 3\pmod{11}$ and $4k \equiv 5\pmod{13}$ simultaneously. | nvidia |
Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\] | nvidia |
Let $n$ be the first integer of a sequence of six consecutive integers. Consider the polynomial
$$P(x)=\prod_{k=n}^{n+5}(x-k).$$
Determine the smallest integer value of $n$ for which $P(x)$ has no real root between $n$ and $n+5$. | nvidia |
Let $N$ be the smallest positive integer such that the sum of the digits of $\underbrace{77\ldots7}_{2n\text{ times}}$ is divisible by 49 for every positive integer $n$ greater than or equal to $N$. Determine $N$. | nvidia |
Find all integer values of $b$ so that the polynomial
\[x^4 + 2x^3 + bx^2 + 11x + 1 = 0\]has at least one rational root. Enter all possible values of $b,$ separated by commas. | nvidia |
Find the smallest positive integer $b$ for which $x^2+bx+2008$ factors into a product of two binomials, each having integer coefficients. | nvidia |
In a regular icosahedron, all the vertices are connected by line segments to form a network of triangles. Find the sum of the areas of all these triangles, if each edge of the icosahedron is of unit length. | nvidia |
Determine the base for which the representation of the decimal number 164 as a three-digit number has a middle digit that is twice the last digit. | nvidia |
Find the largest three-digit number that is divisible by each of its non-zero digits. | nvidia |
Determine the largest positive integer that always divides the product of any three consecutive terms of an arithmetic sequence whose terms are positive integers. | nvidia |
The sum of the lengths of all the edges of a cube is 60 cm. Find the number of cubic centimeters in the volume of the cube. | nvidia |
Find the largest possible value of $x$ if $x$ and $y$ are positive real numbers such that $x^2+xy+2y^2=120$ and $x+2y=15$. | nvidia |
Find $s$ such that $(s,4)$ lies on the line through $(2,1)$ and $(-6,-3)$. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xf(y)) = x + y\]for all real numbers $x$ and $y.$ | nvidia |
The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:2:3$ that can be written in the form\[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]Find the sum of all possible values of $n+k$.
| nvidia |
Find the maximum value of
\[\frac{x}{y + 1} + \frac{y}{x + 1}\]
for real numbers $x > 0$ and $y > 0$ such that $x^2 + y^2 = 1$. | nvidia |
A bookstore has 300 books on its shelves. 120 of the books are fiction and 150 are non-fiction. If the number of books that are both fiction and non-fiction (e.g., historical novels) is 5 more than the number of books that are neither fiction nor non-fiction (e.g., dictionaries, atlases), how many books are both fictio... | nvidia |
Find the number of positive and negative integer divisors of 42. | nvidia |
Find the number of solutions to the equation $x_1 + x_2 + \cdots + x_k = n$ where $x_i$ are positive integers and $n$ and $k$ are given positive integers. | nvidia |
Ava wrote 4 different numbers, one on each side of 2 cards, and placed the cards on a table. The sums of the two numbers on each of the two cards are equal. The two numbers on the hidden sides are prime numbers, and the visible numbers are 27 and 41. What is the average of the hidden prime numbers? | nvidia |
Consider the function $f(n)$ defined as the highest power of $2$ that divides $n!$, where $n$ is a positive integer. For example, $f(5) = 3$ because the highest power of $2$ that divides $5! = 120$ is $2^3$.
Find the smallest value of $n$ such that $f(n) > 10$. | nvidia |
The sequences of positive integers $1,x_2, x_3, \dots$ and $1,y_2, y_3, \dots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $z_n=x_n+y_n$. There is an integer $m$ such that $z_{m-1}=50$ and $z_{m+1}=500$. Find $z_m$. | nvidia |
Let $z$ be a complex number such that
$\left| \frac{z^{2008}-1}{z^{2007}-1} \right|= \sqrt{2}$.
Determine the greatest integer that does not exceed the modulus of the imaginary part of $z$. | nvidia |
Let $x$ be a real number. Consider the following five statements:
$0 < x^2 < 1$
$x^2 > 1$
$-1 < x < 0$
$0 < x < 1$
$0 < x - x^2 < 1$
What is the maximum number of these statements that can be true for any value of $x$? | nvidia |
Given a tetrahedron $ABCD$ such that every face is an equilateral triangle and the edge length of the tetrahedron is 6 cm, calculate the height of the tetrahedron if $A$ is taken as the apex. | nvidia |
A five-digit number has the property that its square ends with the same last three digits. Determine these last three digits. | nvidia |
Find the real number $x$ such that $|\cos x + i \sin x - 2i| = 1$. | nvidia |
Find the smallest value of $c$ such that $|c-3|+|c-5| \le 4$. | nvidia |
Let $g(x, y)$ be a function defined on the set of ordered pairs of positive integers such that
$$g(x, y) + g(y, z) = g(x, z)$$
for all positive integers $x, y, z$. Suppose $g(1, n) = n^2$ for all positive integers $n$. Calculate $g(15, 100)$. | nvidia |
Find the sum of all $x$ such that the graph of the function $f(x) = \frac{x^2-4x+4}{x^2-4x-3}$ has a hole. | nvidia |
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