For postive integers $k,n$, let $$f_k(n)=\sum_{m\mid n,m>0}m^k$$Find all pairs of positive integer $(a,b)$ such that $f_a(n)\mid f_b(n)$ for every positive integer $n$.

Given $a,b,c,d>0$, prove that: \[\sum_{cyc}\frac{c}{a+2b}+\sum_{cyc}\frac{a+2b}{c}\geq 8(\frac{(a+b+c+d)^2}{ab+ac+ad+bc+bd+cd}-1),\]where $\sum_{cyc}f(a,b,c,d)=f(a,b,c,d)+f(d,a,b,c)+f(c,d,a,b)+f(b,c,d,a)$.

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

Find all polynomials $P$ with real coefficients which satisfy \[P(x)P(x+1)=P(x^2-x+3) \quad \forall x \in \mathbb{R}\]

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be nice if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$ Proposed by Tigran Margaryan, Armenia

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

Find all injective functions $ f:\mathbb{N} \to \mathbb{N} $ such that $$ f^{f\left(a\right)}\left(b\right)f^{f\left(b\right)}\left(a\right)=\left(f\left(a+b\right)\right)^2 $$holds for all $ a,b \in \mathbb{N} $. Note that $ f^{k}\left(n\right) $ means $ \underbrace{f(f(\ldots f}_{k}(n) \ldots )) $

Two line $BC$ and $EF$ are parallel. Let $D$ be a point on segment $BC$ different from $B$,$C$. Let $I$ be the intersection of $BF$ ans $CE$. Denote the circumcircle of $\triangle CDE$ and $\triangle BDF$ as $K$,$L$. Circle $K$,$L$ are tangent with $EF$ at $E$,$F$,respectively. Let $A$ be the other intersection of circle $K$ and $L$. Let $DF$ and circle $K$ intersect again at $Q$, and $DE$ and circle $L$ intersect again at $R$. Let $EQ$ and $FR$ intersect at $M$. Prove that $I$, $A$, $M$ are collinear.

Let $n$ be an odd number larger than 1, and $f(x)$ is a polynomial with degree $n$ such that $f(k)=2^k$ for $k=0,1,\cdots,n$. Prove that there is only finite integer $x$ such that $f(x)$ is the power of two.

There are $n \geq 3$ islands in a city. Initially, the ferry company offers some routes between some pairs of islands so that it is impossible to divide the islands into two groups such that no two islands in different groups are connected by a ferry route. After each year, the ferry company will close a ferry route between some two islands $X$ and $Y$. At the same time, in order to maintain its service, the company will open new routes according to the following rule: for any island which is connected to a ferry route to exactly one of $X$ and $Y$, a new route between this island and the other of $X$ and $Y$ is added. Suppose at any moment, if we partition all islands into two nonempty groups in any way, then it is known that the ferry company will close a certain route connecting two islands from the two groups after some years. Prove that after some years there will be an island which is connected to all other islands by ferry routes.