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. Proposed by Anant Mudgal and Rohan Goyal

Let $x_1, x_2 \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2\cdots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that \[\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.\] Proposed by Shantanu Nene

Let $P(x) \in \mathbb{Q}[x]$ be a polynomial with rational coefficients and degree $d\ge 2$. Prove there is no infinite sequence $a_0, a_1, \ldots$ of rational numbers such that $P(a_i)=a_{i-1}+i$ for all $i\ge 1$. Proposed by Pranjal Srivastava and Rohan Goyal

Let $n$ be a positive integer. Let $s: \mathbb N \to \{1, \ldots, n\}$ be a function such that $n$ divides $m-s(m)$ for all positive integers $m$. Let $a_0, a_1, a_2, \ldots$ 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 $(n+1)^2$. Proposed by N.V. Tejaswi

Let $ABC$ be an acute angled triangle with $AC>AB$ 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 $Z\ne A$. Proposed by Atul Shatavart Nadig and Shantanu Nene

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: 1. $A_1$ is junior, 2. $A_m$ is senior, and 3. $A_i$ and $A_{i+1}$ are friends, and $A_{i+1}$ is older than $A_i$ for all $1 \leq i \leq m-1$. 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. Proposed by Shantanu Nene

Let $ABC$ be an acute-angled triangle with $AB<AC$, incentre $I$, 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 $\triangle PIQ$. Proposed by Pranjal Srivastava and Rohan Goyal

Let $a$ and $n$ be positive integers such that: 1. $a^{2^n}-a$ is divisible by $n$, 2. $\sum\limits_{k=1}^{n} k^{2024}a^{2^k}$ is not divisible by $n$. Prove that $n$ has a prime factor smaller than $2024$. Proposed by Shantanu Nene

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real numbers $a, b, c$, we have \[ f(a+b+c)f(ab+bc+ca) - f(a)f(b)f(c) = f(a+b)f(b+c)f(c+a). \] Proposed by Mainak Ghosh and Rijul Saini

Let $r>0$ 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 $r>1$, prove that there exists an $r$-good polynomial that does not have a primordial multiple. Proposed by Pranjal Srivastava

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 $s$. Proposed by N.V. Tejaswi

Let $ABC$ be an acute-angled triangle with $AB<AC$, 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. Proposed by Kazi Aryan Amin and K.V. Sudharshan

Find all functions $f:\mathbb R \to \mathbb R$ such that \[ xf(xf(y)+yf(x))= x^2f(y)+yf(x)^2, \]for all real numbers $x,y$. Proposed by B.J. Venkatachala

Let $ABCD$ be a convex cyclic quadrilateral with circumcircle $\omega$. Let $BA$ produced beyond $A$ meet $CD$ produced beyond $D$, at $L$. Let $\ell$ be a line through $L$ meeting $AD$ and $BC$ at $M$ and $N$ respectively, so that $M,D$ (respectively $N,C$) are on opposite sides of $A$ (resp. $B$). Suppose $K$ and $J$ are points on the arc $AB$ of $\omega$ not containing $C,D$ so that $MK, NJ$ are tangent to $\omega$. Prove that $K,J,L$ are collinear. Proposed by Rijul Saini

In a conference, mathematicians from $11$ different countries participate and they have integer-valued ages between $27$ and $33$ years (including $27$ and $33$). There is at least one mathematician from each country, and there is at least one mathematician of each possible age between $27$ and $33$. Show that we can find at least five mathematicians $m_1, \ldots, m_5$ such that for any $i \in \{1, \ldots, 5 \}$ there are more mathematicians in the conference having the same age as $m_i$ than those having the same nationality as $m_i$. Proposed by S. Muralidharan

There are $n$ cities in a country, one of which is the capital. An airline operates bi-directional flights between some pairs of cities such that one can reach any city from any other city. The airline wants to close down some (possibly zero) number of flights, such that the number of flights needed to reach any particular city from the capital does not increase. Suppose that there are an odd number of ways that the airline can do this. Prove that the set of cities can be divided into two groups, such that there is no flight between two cities of the same group. Proposed by Pranjal Srivastava

Fix a positive integer $a > 1$. Consider triples $(f(x), g(x), h(x))$ of polynomials with integer coefficients, such that 1. $f$ is a monic polynomial with $\deg f \ge 1$. 2. There exists a positive integer $N$ such that $g(x)>0$ for $x \ge N$ and for all positive integers $n \ge N$, we have $f(n) \mid a^{g(n)} + h(n)$. Find all such possible triples. Proposed by Mainak Ghosh and Rijul Saini

Let $ABCD$ be a convex quadrilateral which admits an incircle. Let $AB$ produced beyond $B$ meet $DC$ produced towards $C$, at $E$. Let $BC$ produced beyond $C$ meet $AD$ produced towards $D$, at $F$. Let $G$ be the point on line $AB$ so that $FG \parallel CD$, and let $H$ be the point on line $BC$ so that $EH \parallel AD$. Prove that the (concave) quadrilateral $EGFH$ admits an excircle tangent to $\overline{EG}, \overline{EH}, \overrightarrow{FG}, \overrightarrow{FH}$. Proposed by Rijul Saini

Denote by $\mathbb{S}$ the set of all proper subsets of $\mathbb{Z}_{>0}$. Find all functions $f : \mathbb{S} \mapsto \mathbb{Z}_{>0}$ that satisfy the following: ___1. For all sets $A, B \in \mathbb{S}$ we have \[f(A \cap B) = \text{min}(f(A), f(B)).\]___2. For all positive integers $n$ we have \[\sum \limits_{X \subseteq [1, n]} f(X) = 2^{n+1}-1.\] (Here, by a proper subset $X$ of $\mathbb{Z}_{>0}$ we mean $X \subset \mathbb{Z}_{>0}$ with $X \ne \mathbb{Z}_{>0}$. It is allowed for $X$ to have infinite size.) Proposed by MV Adhitya, Kanav Talwar, Siddharth Choppara, Archit Manas

A circus act consists of $2024$ bamboo sticks of pairwise different heights placed in some order, with a monkey standing atop one of them. The circus master can then give commands to the monkey as follows: ___$\bullet$ Left! : When given this command, the monkey locates the closest bamboo stick to the left taller than the one it is currently atop, and jumps to it. If there is no such stick, the monkey stays put. ___$\bullet$ Right! : When given this command, the monkey locates the closest bamboo stick to the right taller than the one it is currently atop, and jumps to it. If there is no such stick, the monkey stays put. The circus master claims that given any two bamboo sticks, if the monkey is originally atop the shorter stick, then after giving at most $c$ commands he can reposition the monkey atop the taller stick. What is the smallest possible value of $c$? Proposed by Archit Manas

Let $\Delta_0$ be an equilateral triangle with incircle $\omega$. A point on $\omega$ is reflected in the sides of $\Delta_0$ to obtain a new triangle $\Delta_1$. The same point is then reflected over the sides of $\Delta_1$ to obtain another triangle $\Delta_2$. Prove that the circumcircle of $\Delta_2$ is tangent to $\omega$. Proposed by Siddharth Choppara

Let $ABC$ be a triangle with circumcenter $O$ and $\angle BAC = 60^{\circ}$. The internal angle bisector of $\angle BAC$ meets line $BC$ and the circumcircle of $\triangle ABC$ in points $M,L$ respectively. Let $K$ denote the reflection of $BL\cap AC$ over the line $BC$. Let $D$ be on the line $CO$ with $DM$ perpendicular to $KL$. Prove that points $K,A,D$ are collinear. Proposed by Sanjana Philo Chacko

Prove that there exists a function $f : \mathbb{N} \mapsto \mathbb{N}$ that satisfies the following: ___1. For all positive integers $m, n$ we have \[\gcd(|f(m)-f(n)|, f(mn)) = f(\gcd(m, n))\]___2. For all positive integers $m$, we have $f(f(m)) = f(m)$. ___3. For all positive integers $k$, there exists a positive integer $n$ with $2024^{k} \mid f(n)$. Proposed by MV Adhitya, Archit Manas

There are $n > 1$ distinct points marked in the plane. Prove that there exists a set of circles $\mathcal C$ such that ___$\bullet$ Each circle in $\mathcal C$ has unit radius. ___$\bullet$ Every marked point lies in the (strict) interior of some circle in $\mathcal C$. ___$\bullet$ There are less than $0.3n$ pairs of circles in $\mathcal C$ that intersect in exactly $2$ points. Note: Weaker results with $\it{0.3n}$ replaced by $\it{cn}$ may be awarded points depending on the value of the constant $\it{c > 0.3}$. Proposed by Siddharth Choppara, Archit Manas, Ananda Bhaduri, Manu Param