For a positive integer $n$, let $f(n)$ be equal to $n$ if there is an integer $x$ such that $x^2-n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when $$\sum^{2^{12}-1}_{n=0}f(n)$$ is divided by $2^{12}$.

Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]

Let $A$ be the set of odd integers $\leq 2n-1.$ For a positive integer $m$, let $B=\{a+m\,|\, a\in A \}.$ Determine for which positive integers $n$ there exists a positive integer $m$ such that the product of all elements in $A$ and $B$ is a square.

Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$

Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ that satisfy the following two conditions: [list]$\bullet\ f(n)$ is a perfect square for all $n\in\mathbb{Z}_{>0}$ $\bullet\ f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}_{>0}$.[/list]

Find all primes $ p,q $ such that $ \alpha^{3pq} -\alpha \equiv 0 \pmod {3pq} $ for all integers $ \alpha $.

Let $p, q$ be two different odd prime numbers and $n$ an integer such that $pq$ divides $n^{pq} + 1$. Prove that if $p^3q^3$ divides $n^{pq} + 1$ then either $p^2$ divides $n + 1$ or $q^2$ divides $n + 1$. Malik Talbi

Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of $80$ Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations: 1. Every Pokemon in Victory Road is enemies with exactly two other Pokemon. 2. Due to her inability to catch Pokemon that are enemies with one another, the maximum number of the Pokemon she can catch is equal to $n$. What is the sum of all possible values of $n$?

Consider the expansion \[(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.\] [b](a)[/b] Determine the greatest common divisor of the coefficients $a_3, a_8, a_{13}, \ldots , a_{1983}.$ [b](b)[/b] Prove that $10^{340 }< a_{992} < 10^{347}.$

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

[b]p1.[/b] Give a fake proof that $0 = 1$ on the back of this page. The most convincing answer to this question at this test site will receive a point. [b]p2.[/b] It is often said that once you assume something false, anything can be derived from it. You may assume for this question that $0 = 1$, but you can only use other statements if they are generally accepted as true or if your prove them from this assumption and other generally acceptable mathematical statements. With this in mind, on the back of this page prove that every number is the same number. [b]p3.[/b] Suppose you write out all integers between $1$ and $1000$ inclusive. (The list would look something like $1$, $2$, $3$, $...$ , $10$, $11$, $...$ , $999$, $1000$.) Which digit occurs least frequently? [b]p4.[/b] Pick a real number between $0$ and $1$ inclusive. If your response is $r$ and the standard deviation of all responses at this site to this question is $\sigma$, you will receive $r(1 - (r - \sigma)^2)$ points. [b]p5.[/b] Find the sum of all possible values of $x$ that satisfy $243^{x+1} = 81^{x^2+2x}$. [b]p6.[/b] How many times during the day are the hour and minute hands of a clock aligned? [b]p7.[/b] A group of $N + 1$ students are at a math competition. All of them are wearing a single hat on their head. $N$ of the hats are red; one is blue. Anyone wearing a red hat can steal the blue hat, but in the process that person’s red hat disappears. In fact, someone can only steal the blue hat if they are wearing a red hat. After stealing it, they would wear the blue hat. Everyone prefers the blue hat over a red hat, but they would rather have a red hat than no hat at all. Assuming that everyone is perfectly rational, find the largest prime $N$ such that nobody will ever steal the blue hat. [b]p8.[/b] On the back of this page, prove there is no function f$(x)$ for which there exists a (finite degree) polynomial $p(x)$ such that $f(x) = p(x)(x + 3) + 8$ and $f(3x) = 2f(x)$. [b]p9.[/b] Given a cyclic quadrilateral $YALE$ with $Y A = 2$, $AL = 10$, $LE = 11$, $EY = 5$, what is the area of $YALE$? [b]p10.[/b] About how many pencils are made in the U.S. every year? If your answer to this question is $p$, and our (good) estimate is $\rho$, then you will receive $\max(0, 1 -\frac 12 | \log_{10}(p) - \log_{10}(\rho)|)$ points. [b]p11.[/b] The largest prime factor of $520, 302, 325$ has $5$ digits. What is this prime factor? [b]p12.[/b] The previous question was on the individual round from last year. It was one of the least frequently correctly answered questions. The first step to solving the problem and spotting the pattern is to divide $520, 302, 325$ by an appropriate integer. Unfortunately, when solving the problem many people divide it by $n$ instead, and then they fail to see the pattern. What is $n$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that \[ ab\mid (cd)^{max(a,b)} \]

Define a sequence of integers $a_0=m, a_1=n$ and $a_{k+1}=4a_k-5a_{k-1}$ for all $k \ge 1$. Suppose $p>5$ is a prime with $p \equiv 1 \pmod{4}$. Prove that it is possible to choose $m,n$ such that $p \nmid a_k$ for any $k \ge 0$.

In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules: $\bullet$ If $p$ is a prime number, we place it in box $1$. $\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$. Find all positive integers $n$ that are placed in the box labeled $n$.

How many solutions does the equation $ \frac{[x]}{\{ x\}} =\frac{2007x}{2008} $ have? [i]Mihail Bălună[/i]

Find the last two digits of each of the numbers $3^{1974}$ and $7^{1974}$.

What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$

Prove that the number $$\bigg\lfloor\frac{2024}{1}\bigg\rfloor+\bigg\lfloor\frac{2023}{2}\bigg\rfloor+\bigg\lfloor\frac{2022}{3}\bigg\rfloor+\dots +\bigg\lfloor\frac{1013}{1012}\bigg\rfloor$$ is even.

Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]

[b]p1.[/b] The total cost of $1$ football, $3$ tennis balls and $7$ golf balls is $\$14$ , while that of $1$ football, $4$ tennis balls and $10$ golf balls is $\$17$.If one has $\$20$ to spend, is this sufficient to buy a) $3$ footballs and $2$ tennis balls? b) $2$ footballs and $3$ tennis balls? [b]p2.[/b] Let $\overline{AB}$ and $\overline{CD}$ be two chords in a circle intersecting at a point $P$ (inside the circle). a) Prove that $AP \cdot PB = CP\cdot PD$. b) If $\overline{AB}$ is perpendicular to $\overline{CD}$ and the length of $\overline{AP}$ is $2$, the length of $\overline{PB}$ is $6$, and the length of $\overline{PD}$ is $3$, find the radius of the circle. [b]p3.[/b] A polynomial $P(x)$ of degree greater than one has the remainder $2$ when divided by $x-2$ and the remainder $3$ when divided by $x-3$. Find the remainder when $P(x)$ is divided by $x^2-5x+6$. [b]p4.[/b] Let $x_1= 2$ and $x_{n+1}=x_n+ (3n+2)$ for all $n$ greater than or equal to one. a) Find a formula expressing $x_n$ as a function of$ n$. b) Prove your result. [b]p5.[/b] The point $M$ is the midpoint of side $\overline{BC}$ of a triangle $ABC$. a) Prove that $AM \le \frac12 AB + \frac12 AC$. b) A fly takes off from a certain point and flies a total distance of $4$ meters, returning to the starting point. Explain why the fly never gets outside of some sphere with a radius of one meter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] If $x + z = v$, $w + z = 2v$, $z - w = 2y$, and $y \ne 0$, compute the value of $$\left(x + y +\frac{x}{y} \right)^{101}.$$ [b]p2. [/b]Every minute, a snail picks one cardinal direction (either north, south, east, or west) with equal probability and moves one inch in that direction. What is the probability that after four minutes the snail is more than three inches away from where it started? [b]p3.[/b] What is the probability that a point chosen randomly from the interior of a cube is closer to the cube’s center than it is to any of the cube’s eight vertices? [b]p4.[/b] Let $ABCD$ be a rectangle where $AB = 4$ and $BC = 3$. Inscribe circles within triangles $ABC$ and $ACD$. What is the distance between the centers of these two circles? [b]p5.[/b] $C$ is a circle centered at the origin that is tangent to the line $x - y\sqrt3 = 4$. Find the radius of $C$. [b]p6.[/b] I have a fair $100$-sided die that has the numbers $ 1$ through $100$ on its sides. What is the probability that if I roll this die three times that the number on the first roll will be greater than or equal to the sum of the two numbers on the second and third rolls? [b]p7. [/b] List all solutions $(x, y, z)$ of the following system of equations with x, y, and z positive real numbers: $$x^2 + y^2 = 16$$ $$x^2 + z^2 = 4 + xz$$ $$y^2 + z^2 = 4 + yz\sqrt3$$ [b]p8.[/b] $A_1A_2A_3A_4A_5A_6A_7$ is a regular heptagon ($7$ sided-figure) centered at the origin where $A_1 = (\sqrt[91]{6}, 0)$. $B_1B_2B_3... B_{13}$ is a regular triskaidecagon ($13$ sided-figure) centered at the origin where $B_1 =(0,\sqrt[91]{41})$. Compute the product of all lengths $A_iB_j$ , where $i$ ranges between $1$ and $7$, inclusive, and $j$ ranges between $1$ and $13$, inclusive. [b]p9.[/b] How many three-digit integers are there such that one digit of the integer is exactly two times a digit of the integer that is in a different place than the first? (For example, $100$, $122$, and $124$ should be included in the count, but $42$ and $130$ should not.) [b]p10.[/b] Let $\alpha$ and $\beta$ be the solutions of the quadratic equation $$x^2 - 1154x + 1 = 0.$$ Find $\sqrt[4]{\alpha}+\sqrt[4]{\beta}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.

After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$