Let $k>1$ be an integer. The sequence $a_1,a_2, \cdots$ is defined as: $a_1=1, a_2=k$ and for all $n>1$ we have: $a_{n+1}-(k+1)a_n+a_{n-1}=0$ Find all positive integers $n$ such that $a_n$ is a power of $k$. [i]Proposed by Amirhossein Pooya[/i]

Suppose that $x$ and $y$ are complex numbers such that \[\frac{x^{n}-y^{n}}{x-y}\] are integers for some four consecutive positive integers $n$. Prove that it is an integer for all positive integers $n$.

Show that $(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$ , for all real positive numbers $x, y $ and $z$.

Let $f: \mathbb{R}^{2} \to \mathbb{R}^{+}$such that for every rectangle $A B C D$ one has $$ f(A)+f(C)=f(B)+f(D). $$ Let $K L M N$ be a quadrangle in the plane such that $f(K)+f(M)=f(L)+f(N)$, for each such function. Prove that $K L M N$ is a rectangle. [i]Proposed by Navid.[/i]

Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$ .

Prove that the sum of the $ \text{2005-th} $ powers of three pairwise distinct complex numbers is the imaginary unit if their modulus are equal and the sum of these numbers is the imaginary unit.

Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[xy+x+y = 71\qquad\text{and}\qquad x^2y+xy^2 = 880.\]

Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$ $$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$ $$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$ [i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]

$p(x)$ is a polynomial such that $p(y^2+1) = 6y^4 - y^2 + 5$. Find $p(y^2-1)$.

A contractor has two teams of workers: team $A$ and team $B$. Team $A$ can complete a job in $12$ days and team $B$ can do the same job in $36$ days. Team $A$ starts working on the job and team $B$ joins team $A$ after four days. The team $A$ withdraws after two more days. For how many more days should team $B$ work to complete the job?

Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ with the following properties: 1) For every natural number $n\geq 3$, $\gcd(f(n),n)\neq 1$. 2) For every natural number $n\geq 3$, there exists $i_n\in\mathbb{Z}_{>0}$, $1\leq i_n\leq n-1$, such that $f(n)=f(i_n)+f(n-i_n)$. [i]Proposed by Pavel Ciurea[/i]

An occasionally unreliable professor has devoted his last book to a certain binary operation $*$. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms: $\text{a})\ x*(x*y)=y$ for all $x,y\in\mathbb{Z}$; $\text{b})\ (x*y)*y=x$ for all $x,y\in\mathbb{Z}$. The professor claims in his book that $1.$ The operation $*$ is commutative: $x*y=y*x$ for all $x,y\in\mathbb{Z}$. $2.$ The operation $*$ is associative: $(x*y)*z=x*(y*z)$ for all $x,y,z\in\mathbb{Z}$. Which of these claims follow from the stated axioms?

Prove that \[ (x^4+y)(y^4+z)(z^4+x) \geq (x+y^2)(y+z^2)(z+x^2) \] for all positive real numbers $x, y, z$ satisfying $xyz \geq 1$.

There is a sequence of integers $ a_1, a_2, \ldots, a_{2n+1} $ with the following property: after eliminating any term, the remaining ones can be divided into two groups of $ n $ terms such that the sum of the terms in the first group is equal to the sum words in the second. Prove that all terms of the sequence are equal.

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

The number $2023$ is written $2023$ times on a blackboard. On one move, you can choose two numbers $x, y$ on the blackboard, delete them and write $\frac{x+y} {4}$ instead. Prove that when one number remains, it is greater than $1$.

Prove that if $ c, d $ are integers with $ c \neq d $, $ d > 0 $ then the equation $$ x^3 - 3cx^2 - dx + c = 0$$ has no more than one rational root.

Let $n$ be a positive integer and $a_1,a_2,\dots,a_n$ be integers. Function $f: \mathbb{Z} \rightarrow \mathbb{R}$ is such that for all integers $k$ and $l$, $l \neq 0$, $$\sum_{i=1}^n f(k+a_il)=0.$$ Prove that $f \equiv 0$.

There are $2500$ people in Lexington High School, who all start out healthy. After $1$ day, $1$ person becomes infected with coronavirus. Each subsequent day, there are twice as many newly infected people as on the previous day. How many days will it be until over half the school is infected?

Given $n > 2$ nonnegative distinct integers $a_1,...,a_n$, find all nonnegative integers $y$ and $x_1,...,x_n$ satisfying $gcd(x_1,...,x_n) = 1$ and $$\begin{cases} a_1x_1 + a_2x_2 +...+ a_nx_n = yx_1 \\ a_2x_1 + a_3x_2 +...+ a_1x_n = yx_2 \\ ... \\ a_nx_1 + a_1x_2 +...+ a_{n-1}x_n = yx_n \end{cases}$$

Within an arithmetic progression of length $ 2005, $ find the number of arithmetic subprogressions of length $ 501 $ that don't contain the $ \text{1000-th} $ term of the progression.

Let $f$ be a primitive polynomial with integral coefficients (their highest common factor is $1$ ) such that $f$ is irreducible in $\mathbb{Q}[X]$ , and $f(X^2)$ is reducible in $\mathbb{Q}[X]$ . Show that $f= \pm(u^2-Xv^2)$ for some polynomials $u$ and $v$ with integral coefficients.

$\frac1{1+\sqrt3}+\frac1{\sqrt3+\sqrt5}+\frac1{\sqrt5+\sqrt7}+\ldots+\frac1{\sqrt{2017}+\sqrt{2019}}=?$ $\textbf{(A)}~\frac{\sqrt{2019}-1}2$ $\textbf{(B)}~\frac{\sqrt{2019}+1}2$ $\textbf{(C)}~\frac{\sqrt{2019}-1}4$ $\textbf{(D)}~\text{None of the above}$

Solve in the set $R$ the equation $$\frac{3x+3}{\sqrt{x}}-\frac{x+1}{\sqrt{x^2-x+1}}=4$$

For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$. Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$. [i]Proposed by Georgia[/i]