Let $n$ be a positive integer. Let $s: \mathbb{N} \to \{1, \dots, n\}$ be a function such that $n$ divides $m - s(m)$ for all positive integers $m$. Let $a_0, a_1, a_2, \dots$ be a sequence such that $a_0 = 0$ and
$$
a_k = a_{k-1} + s(k) \text{ for all } k \ge 1.
$$
Find all $n$ for which this sequence contains all the residues modulo $(n+1)^2$.
Let $\triangle ABC$ be an acute angled triangle with $AC \gt AB$ and incircle $\omega$. Let $\omega$ touch the sides $BC, CA$, and $AB$ at $D, E$, and $F$ respectively. Let $X$ and $Y$ be points outside $\triangle ABC$ satisfying
$$
\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^\circ.
$$
Prove that the circumcircles of $\triangle AXY$, $\triangle AEF$ and $\triangle ABC$ meet at a point $Z \neq A$.
At an IMOTC party, all people have pairwise distinct ages. Some pairs of people are friends and friendship is mutual. Call a person *junior* if they are younger than all their friends, and *senior* if they are older than all their friends. A person with no friends is both *junior* and *senior*. A sequence of pairwise distinct people $A_1, \dots, A_m$ is called *photogenic* if:
* $A_1$ is junior,
* $A_m$ is senior, and
* $A_i$ and $A_{i+1}$ are friends, and $A_{i+1}$ is older than $A_i$ for all $1 \le i \le m-1$.
Let $k$ be a positive integer such that for every *photogenic* sequence $A_1, \dots, A_m$, $m$ is not divisible by $k$. Prove that the people at the party can be partitioned into $k$ groups so that no two people in the same group are friends.
Let $n$ and $a$ be positive integers such that
* $a^{2^n} - a$ is divisible by $n$,
* $\sum_{k=1}^{n} k^{2024} a^{2^k}$ is not divisible by $n$.
Prove that $n$ has a prime factor *smaller* than $2024$.
There are $n \ge 3$ particles on a circle situated at the vertices of a regular $n$-gon. All these particles move on the circle with the same constant speed. One of the particles moves in the clockwise direction while all others move in the anti-clockwise direction. When particles collide, that is, they are all at the same point, they all reverse the direction of their motion and continue with the same speed as before.
Let $s$ be the smallest number of collisions after which all particles return to their original positions. Find $s$.
A sleeping rabbit lies in the interior of a convex $2024$-gon. A hunter picks three vertices of the polygon and he lays a trap which covers the interior and the boundary of the triangular region determined by them. Determine the minimum number of times he needs to do this to guarantee that the rabbit will be trapped.
Let $P(x) \in \mathbb{Q}[x]$ be a polynomial with rational coefficients and degree $d \ge 2$. Prove there is no infinite sequence $a_0, a_1, \dots$ of rational numbers such that $P(a_i) = a_{i-1} + i$ for all $i \ge 1$.
Let $ABC$ be an acute-angled triangle with $AB \lt AC$, incenter $I$, and let $M$ be the midpoint of major arc $BAC$. Suppose the perpendicular line from $A$ to segment $BC$ meets lines $BI$, $CI$, and $MI$ at points $P$, $Q$, and $K$ respectively. Prove that the $A$-median line in $\triangle AIK$ passes through the circumcentre of $\triangle PIQ$.
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real numbers $a, b, c$, we have
$$
f(a + b + c)f(ab + bc + ca) - f(a)f(b)f(c) = f(a + b)f(b + c)f(c + a).
$$
Let $r \gt 0$ be a real number. We call a monic polynomial with complex coefficients $r$-good if all of its roots have absolute value at most $r$. We call a monic polynomial with complex coefficients *primordial* if all of its coefficients have absolute value at most 1.
a. Prove that any 1-good polynomial has a primordial multiple.
b. If $r \gt 1$, prove that there exists an $r$-good polynomial that does not have a primordial multiple.
Let $ABC$ be an acute-angled triangle with $AB \lt AC$, and let $O, H$ be its circumcentre and orthocentre respectively. Points $Z, Y$ lie on segments $AB, AC$ respectively, such that
$$
\angle ZOB = \angle YOC = 90^\circ.
$$
The perpendicular line from $H$ to line $YZ$ meets lines $BO$ and $CO$ at $Q, R$ respectively. Let the tangents to the circumcircle of $\triangle AYZ$ at points $Y$ and $Z$ meet at point $T$. Prove that $Q, R, O, T$ are concyclic.
In triangle $ABC$ with $CA = CB$, point $E$ lies on the circumcircle of $ABC$ such that $\angle ECB = 90^\circ$. The line through $E$ parallel to $CB$ intersects $CA$ in $F$ and $AB$ in $G$. Prove that the centre of the circumcircle of triangle $EGB$ lies on the circumcircle of triangle $ECF$.
All the squares of a $2024 \times 2024$ board are coloured white. In one move, Mohit can select one row or column whose every square is white, choose exactly $1000$ squares in this row or column, and colour all of them red. Find the maximum number of squares that Mohit can colour red in a finite number of moves.
In triangle $ABC$ with $CA = CB$, point $E$ lies on the circumcircle of $ABC$ such that $\angle ECB = 90^\circ$. The line through $E$ parallel to $CB$ intersects $CA$ in $F$ and $AB$ in $G$. Prove that the centre of the circumcircle of triangle $EGB$ lies on the circumcircle of triangle $ECF$.
All the squares of a $2024 \times 2024$ board are coloured white. In one move, Mohit can select one row or column whose every square is white, choose exactly $1000$ squares in this row or column, and colour all of them red. Find the maximum number of squares that Mohit can colour red in a finite number of moves.
In triangle $ABC$ with $CA = CB$, point $E$ lies on the circumcircle of $ABC$ such that $\angle ECB = 90^\circ$. The line through $E$ parallel to $CB$ intersects $CA$ in $F$ and $AB$ in $G$. Prove that the centre of the circumcircle of triangle $EGB$ lies on the circumcircle of triangle $ECF$.
All the squares of a $2024 \times 2024$ board are coloured white. In one move, Mohit can select one row or column whose every square is white, choose exactly $1000$ squares in this row or column, and colour all of them red. Find the maximum number of squares that Mohit can colour red in a finite number of moves.
Let $p$ be an odd prime number and $a, b, c$ be integers so that the integers
$$
a^{2023} + b^{2023}, \quad b^{2024} + c^{2024}, \quad c^{2025} + a^{2025}
$$
are all divisible by $p$. Prove that $p$ divides each of $a, b$, and $c$.