Let $ABC$ be a triangle with $AB<AC$ and with its incircle touching the sides $AB$ and $BC$ at $M$ and $J$ respectively. A point $D$ lies on the extension of $AB$ beyond $B$ such that $AD=AC$. Let $O$ be the midpoint of $CD$. Prove that the points $J$, $O$, $M$ are collinear. (Proposed by Tan Rui Xuen)

Determine all infinite sequences of nonnegative integers $a_1,a_2,\ldots$ such that: 1. Every positive integer appears in the sequence at least once, and; 2. $a_i$ is the smallest integer $j$ such that $a_{j+2}=i$, for all $i\ge 1$. (Proposed by Ho Janson)

Minivan and Megavan play a game. For a positive integer $n$, Minivan selects a sequence of integers $a_1,a_2,\ldots,a_n$. An \textit{operation} on $a_1,a_2,\ldots,a_n$ means selecting an $a_i$ and increasing it by $1$. Minivan and Megavan take turns, with Minivan going first. On Minivan's turn, he performs at most $2025$ operations, and he may choose the same integer repeatedly. On Megavan's turn, he performs exactly $1$ operation instead. Megavan wins if at any point in the game, including in the middle of Minivan's operations, two numbers in the sequence are equal. (Proposed by Ho Janson)

For each positive integer $k$, find all positive integer $n$ such that there exists a permutation $a_1,\ldots,a_n$ of $1,2,\ldots,n$ satisfying $$a_1a_2\ldots a_i\equiv i^k \pmod n$$for each $1\le i\le n$. (Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)

Let $ABC$ be a scalene triangle and $I$ be its incenter. Suppose the incircle $\omega$ touches $BC$ at a point $D$, and $N$ lies on $\omega$ such that $ND$ is a diameter of $\omega$. Let $X$ and $Y$ be points on lines $AC$ and $AB$ respectively such that $\angle BIX = \angle CIY = 90^\circ$. Let $V$ be the feet of perpendicular from $I$ onto line $XY$. Prove that the points $I$, $V$, $A$, $N$ are concyclic. (Proposed by Ivan Chan Guan Yu)

Given a positive real $t$, a set $S$ of nonnegative reals is called $t$-good if for any two distinct elements $a,b$ in $S$, $\frac{a+b}2\ge\sqrt{ab}+t$. For all positive reals $N$, find the maximum number of elements a $t$-good set can have, if all elements are at most $N$. (Proposed by Ho Janson)

Let $n$ be a positive integer. Navinim writes down all positive square numbers that divide $n$ on a blackboard. For each number $k$ on the blackboard, Navagem replaces it with $d(k)$. Show that the sum of all numbers on the blackboard now is a perfect square. (Note: $d(k)$ denotes the number of divisors of $k$.) (Proposed by Ivan Chan Guan Yu)

Let $\triangle MAB$ be a triangle with circumcenter $O$. $P$ and $Q$ lie on line $AB$ (both interior or exterior) such that $\angle PMA = \angle BMQ$. Let $D$ be a point on the perpendicular line through $M$ to $AB$. $E$ is the second intersection of the two circles $(DAB)$ and $(DPQ)$. The line $MO$ intersects $AB$ at $J$. Show that the circumcenter of $\triangle EMJ$ lies on line $AB$. (Proposed by Tan Rui Xuen)

There are $n$ people arranged in a circle, and $n^{n^n}$ coins are distributed among them, where each person has at least $n^n$ coins. Each person is then assigned a random index number in $\{1,2,...n\}$ such that no two people have the same number. Then every minute, if $i$ is the number of minutes passed, the person with index number congruent to $i$ mod $n$ will give a coin to the person on his left or right. After some time, everyone has the same number of coins. For what $n$ is this always possible, regardless of the original distribution of coins and index numbers? (Proposed by Ho Janson)

Let $p$ be a fixed prime number. Jomland has $p$ cities labelled $0,1,\dots,p-1$. Navi is a traveller and JomAirlines only has flights between two cities with labels $a$ and $b$ (flights are available in both directions) iff there exist positive integers $x$ and $y$ such that \[ \begin{cases} a \equiv x^2 + 2025xy + y^2\pmod{p}\\ b \equiv 20x^2 + xy + 25y^2\pmod{p} \end{cases} \]Prove that: i) There exist infinitely many primes $p$ such that there exist $2$ cities where Navi cannot start from one city and get to the other through a sequence of flights; ii) There exist infinitely many primes $p$ such that for any $2$ cities, Navi can start from one city and get to the other through a sequence of flights. (Proposed by Ivan Chan Guan Yu)

Is it possible for Pingu to choose $2025$ positive integers $a_1, ..., a_{2025}$ such that: 1. The sequence $a_i$ is increasing; 2. $\gcd(a_1,a_2)>\gcd(a_2,a_3)>...>\gcd(a_{2024},a_{2025})>\gcd(a_{2025},a_1)>1$? (Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)

Fix $n$. Given $n$ points on Cartesian plane such that no pair of points forms a segment that is parallel to either axes, a pair of points is said to be good if their segment gradient is positive. For which $k$ can there exist a set of $n$ points with exactly $k$ good pairs? (Proposed by Ivan Chan Kai Chin)

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(x)^2+f(2y+1)=x^2+f(y)+y+1\]for all reals $x$, $y$. (Proposed by Lim Yun Zhe)

Let $ABC$ be a triangle and $E$ and $F$ lie on $AC$ and $AB$ such that $AE=AF$. $EF$ intersects $BC$ at $D$ and $(BDF)$ intersects $(CDE)$ at $X$. Let $O_1$ be the center of $(BDF)$ and $O_2$ be the center of $(CDE)$. Let $O$ be the center of $ABC$. Suppose that $XD$ intersects $(XO_1O_2)$ at $Z$. Show that $OZ\parallel BC$. (Proposed by Tan Rui Xuen and Yeoh Yi Shuen)

There are $n>1$ cities in Jansonland, with two-way roads joining certain pairs of cities. Janson will send a few robots one-by-one to build more roads. The robots operate as such: 1. Janson first selects an integer $k$ and a list of cities $a_0, a_1, \dots, a_k$ (cities can repeat). 2. The robot begins at $a_0$ and goes to $a_1$, then $a_2$, and so on until $a_k$. 3. When the robot goes from $a_i$ to $a_{i+1}$, if there is no road then the robot builds a road, but if there is a road then the robot destroys the road. In terms of $n$, determine the smallest constant $k$ such that Janson can always achieve a configuration such that every pair of cities has a road connecting them using no more than $k$ robots. (Proposed by Ho Janson)