Let $a_1<a_2< \cdots$ be a strictly increasing sequence of positive integers. Suppose there exist $N$ such that for all $n>N$, $$a_{n+1}\mid a_1+a_2+\cdots+a_n$$Prove that there exist $M$ such that $a_{m+1}=2a_m$ for all $m>M$. Proposed by Ivan Chan Kai Chin

Let $k>1$. Fix a circle $\omega$ with center $O$ and radius $r$, and fix a point $A$ with $OA=kr$. Let $AB$, $AC$ be tangents to $\omega$. Choose a variable point $P$ on the minor arc $BC$ in $\omega$. Lines $AB$ and $CP$ intersect at $X$ and lines $AC$ and $BP$ intersect at $Y$. The circles $(BPX)$ and $(CPY)$ meet at another point $Z$. Prove that the line $PZ$ always passes through a fixed point except for one value of $k>1$, and determine this value. Proposed by Ivan Chan Kai Chin

Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ such that for all integers $x$, $y$, $$f(x-f(y))=f(f(y))+f(x-2y)$$ Proposed by Ivan Chan Kai Chin

Ivan has a $n \times n$ board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let $d_n$ be the maximum number of black squares possible, prove that there exist some real constants $a$, $b$, $c\ge 0$ such that; $$an^2-bn\le d_n\le an^2+cn.$$ Proposed by Ivan Chan Kai Chin

Let $ABC$ be a scalene triangle and $D$ be the feet of altitude from $A$ to $BC$. Let $I_1$, $I_2$ be incenters of triangles $ABD$ and $ACD$ respectively, and let $H_1$, $H_2$ be orthocenters of triangles $ABI_1$ and $ACI_2$ respectively. The circles $(AI_1H_1)$ and $(AI_2H_2)$ meet again at $X$. The lines $AH_1$ and $XI_1$ meet at $Y$, and the lines $AH_2$ and $XI_2$ meet at $Z$. Suppose the external common tangents of circles $(BI_1H_1)$ and $(CI_2H_2)$ meet at $U$. Prove that $UY=UZ$. Proposed by Ivan Chan Kai Chin