Let $\mathbb{N}_0$ denote the set of non-negative integers. Determine all non-negative integers $k$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and $f(f(n)) \leq f(n+1) - f(n)$ for all non-negative integers $n$.

There is a rectangular sheet of paper on an infinite blackboard. Marvin secretly chooses a convex $2024$-gon $P$ that lies fully on the piece of paper. Tigerin wants to find the vertices of $P$. In each step, Tigerin can draw a line $g$ on the blackboard that is fully outside the piece of paper, then Marvin replies with the line $h$ parallel to $g$ that is the closest to $g$ which passes through at least one vertex of $P$. Prove that there exists a positive integer $n$, independent of the choice of the polygon, such that Tigerin can always determine the vertices of $P$ in at most $n$ steps.

Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects the segments $AB$ and $AC$ at the interior points $D$ and $E$, respectively. The lines $BE$ and $CD$ intersects at $F$. Let $G$ be a point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$ and let $H$ be a point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T\neq A$, independent of the choice of $\omega$, such that the circumcircle of triangle $AGH$ passes through $T$.

Determine all polynomials $P(x)$ with integer coefficients such that $P(n)$ is divisible by $\sigma(n)$ for all positive integers $n$. (As usual, $\sigma(n)$ denotes the sum of all positive divisors of $n$.)

Consider two infinite sequences $a_0,a_1,a_2,\dots$ and $b_0,b_1,b_2,\dots$ of real numbers such that $a_0=0$, $b_0=0$ and \[a_{k+1}=b_k, \quad b_{k+1}=\frac{a_kb_k+a_k+1}{b_k+1}\]for each integer $k \ge 0$. Prove that $a_{2024}+b_{2024} \ge 88$.

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[yf(x+1)=f(x+y-f(x))+f(x)f(f(y))\]for all $x,y \in \mathbb{R}$.

There are $2024$ mathematicians sitting in a row next to the river Tisza. Each of them is working on exactly one research topic, and if two mathematicians are working on the same topic, everyone sitting between them is also working on it. Marvin is trying to figure out for each pair of mathematicians whether they are working on the same topic. He is allowed to ask each mathematician the following question: “How many of these 2024 mathematicians are working on your topic?” He asks the questions one by one, so he knows all previous answers before he asks the next one. Determine the smallest positive integer $k$ such that Marvin can always accomplish his goal with at most $k$ questions.

A finite sequence $x_1,\dots,x_r$ of positive integers is a palindrome if $x_i=x_{r+1-i}$ for all integers $1 \le i \le r$. Let $a_1,a_2,\dots$ be an infinite sequence of positive integers. For a positive integer $j \ge 2$, denote by $a[j]$ the finite subsequence $a_1,a_2,\dots,a_{j-1}$. Suppose that there exists a strictly increasing infinite sequence $b_1,b_2,\dots$ of positive integers such that for every positive integer $n$, the subsequence $a[b_n]$ is a palindrome and $b_{n+2} \le b_{n+1}+b_n$. Prove that there exists a positive integer $T$ such that $a_i=a_{i+T}$ for every positive integer $i$.

Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Let $D$ be a point on the line $AC$ such that $AB = AD$ and $A$ lies between $C$ and $D$. Suppose that there are two points $E \ne F$ on the circumcircle of the triangle $DBC$ such that $AE = AF = BC$. Prove that the line $EF$ passes through the circumcenter of $ABC$.

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of the segment $BC$. Let $I, J, K$ be the incenters of triangles $ABC$, $ABM$, $ACM$, respectively. Let $P, Q$ be points on the lines $MK$, $MJ$, respectively, such that $\angle AJP=\angle ABC$ and $\angle AKQ=\angle BCA$. Let $R$ be the intersection of the lines $CP$ and $BQ$. Prove that the lines $IR$ and $BC$ are perpendicular.

Define glueing of positive integers as writing their base ten representations one after another and interpreting the result as the base ten representation of a single positive integer. Find all positive integers $k$ for which there exists an integer $N_k$ with the following property: for all $n \ge N_k$, we can glue the numbers $1,2,\dots,n$ in some order so that the result is a number divisible by $k$. Remark. The base ten representation of a positive integer never starts with zero. Example. Glueing $15, 14, 7$ in this order makes $15147$.

Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that \[a_ia_{i+1} \mid k-a_i^2\]for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all integers $n \ge M$.