INMO 2024

In triangle $A B C$ with $C A=C B$, point $E$ lies on the circumcircle of $A B C$ such that $\angle E C B=90^{\circ}$. The line through $E$ parallel to $C B$ intersects $C A$ in $F$ and $A B$ in $G$. Prove that the centre of the circumcircle of triangle $E G B$ lies on the circumcircle of triangle $E C F$.

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

$$a^{2023}+b^{2023}, b^{2024}+c^{2024}, c^{2025}+a^{2025}$$

are all divisible by $p$. Prove that $p$ divides each of $a, b$, and $c$.

A finite set $S$ of positive integers is called cardinal if $S$ contains the integer $|S|$, where $|S|$ denotes the number of distinct elements in $S$. Let $f$ be a function from the set of positive integers to itself, such that for any cardinal set $S$, the set $f(S)$ is also cardinal. Here $f(S)$ denotes the set of all integers that can be expressed as $f(a)$ for some $a$ in $S$. Find all possible values of $f(2024)$.

Note: As an example, $\{1,3,5\}$ is a cardinal set because it has exactly 3 distinct elements, and the set contains 3 .

Let points $A_{1}, A_{2}$, and $A_{3}$ lie on the circle $\Gamma$ in counter-clockwise order, and let $P$ be a point in the same plane. For $i \in\{1,2,3\}$, let $\tau_{i}$ denote the counter-clockwise rotation of the plane centred at $A_{i}$, where the angle of the rotation is equal to the angle at vertex $A_{i}$ in $\triangle A_{1} A_{2} A_{3}$. Further, define $P_{i}$ to be the point $\tau_{i+2}\left(\tau_{i}\left(\tau_{i+1}(P)\right)\right)$, where indices are taken modulo 3 (i.e., $\tau_{4}=\tau_{1}$ and $\tau_{5}=\tau_{2}$ ).

Prove that the radius of the circumcircle of $\triangle P_{1} P_{2} P_{3}$ is at most the radius of $\Gamma$.

For each positive integer $n \geq 3$, define $A_{n}$ and $B_{n}$ as

$$\begin{gathered} A_{n}=\sqrt{n^{2}+1}+\sqrt{n^{2}+3}+\cdots+\sqrt{n^{2}+2 n-1} \\ B_{n}=\sqrt{n^{2}+2}+\sqrt{n^{2}+4}+\cdots+\sqrt{n^{2}+2 n} \end{gathered}$$

Determine all positive integers $n \geq 3$ for which $\left\lfloor A_{n}\right\rfloor=\left\lfloor B_{n}\right\rfloor$.

Note. For any real number $x,\lfloor x\rfloor$ denotes the largest integer $N$ such that $N \leq x$.