Find all real $x, y$, satisfying $$(x+1)^2(y+1)^2=27xy$$and $$(x^2+1)(y^2+1)=10xy.$$

Let $p>q$ be primes, such that $240 \nmid p^4-q^4$. Find the maximal value of $\frac{q} {p}$.

Let $ABC$ be a triangle, satisfying $2AC=AB+BC$. If $O$ and $I$ are its circumcenter and incenter, show that $\angle OIB=90^{\circ}$.

There are $11$ points equally spaced on a circle. Some of the segments having endpoints among these vertices are drawn and colored in two colors, so that each segment meets at an internal for it point at most one other segment from the same color. What is the greatest number of segments that could be drawn?

Find all positive integers $k$ for which there exist positive integers $x, y$, such that $\frac{x^ky}{x^2+y^2}$ is a prime.

Let $n \geq 3$ be a positive integer. Find the smallest positive real $k$, satisfying the following condition: if $G$ is a connected graph with $n$ vertices and $m$ edges, then it is always possible to delete at most $k(m-\lfloor \frac{n} {2} \rfloor)$ edges, so that the resulting graph has a proper vertex coloring with two colors.

Let $q>3$ be a rational number, such that $q^2-4$ is a perfect square of a rational number. The sequence $a_0, a_1, \ldots$ is defined by $a_0=2, a_1=q, a_{i+1}=qa_i-a_{i-1}$ for all $i \geq 1$. Is it true that there exist a positive integer $n$ and nonzero integers $b_0, b_1, \ldots, b_n$ with sum zero, such that if $\sum_{i=0}^{n} a_ib_i=\frac{A} {B}$ for $(A, B)=1$, then $A$ is squarefree?

Let $n, k$ be positive integers with $k \geq 3$. The edges of of a complete graph $K_n$ are colored in $k$ colors, such that for any color $i$ and any two vertices, there exists a path between them, consisting only of edges in color $i$. Prove that there exist three vertices $A, B, C$ of $K_n$, such that $AB, BC$ and $CA$ are all distinctly colored.

Maria and Bilyana play the following game. Maria has $2024$ fair coins and Bilyana has $2023$ fair coins. They toss every coin they have. Maria wins if she has strictly more heads than Bilyana, otherwise Bilyana wins. What is the probability of Maria winning this game?

Let $ABC$ be scalene and acute triangle with $CA>CB$ and let $P$ be an internal point, satisfying $\angle APB=180^{\circ}-\angle ACB$; the lines $AP, BP$ meet $BC, CA$ at $A_1, B_1$. If $M$ is the midpoint of $A_1B_1$ and $(A_1B_1C)$ meets $(ABC)$ at $Q$, show that $\angle PQM=\angle BQA_1$.

Let $n$ be a positive integer and let $\mathcal{A}$ be a family of non-empty subsets of $\{1, 2, \ldots, n \}$ such that if $A \in \mathcal{A}$ and $A$ is subset of a set $B\subseteq \{1, 2, \ldots, n\}$, then $B$ is also in $\mathcal{A}$. Show that the function $$f(x):=\sum_{A \in \mathcal{A}} x^{|A|}(1-x)^{n-|A|}$$is strictly increasing for $x \in (0,1)$.

Call a positive integer $m$ $\textit{good}$ if there exist integers $a, b, c$ satisfying $m=a^3+2b^3+4c^3-6abc$. Show that there exists a positive integer $n<2024$, such that for infinitely many primes $p$, the number $np$ is $\textit{good}$.