Consider the sequence $(a_n)$ satisfying
$$a_1=\frac12,\qquad a_{n+1}=\sqrt[3]{3a_{n+1}-a_n},\qquad0\le a_n\le1\quad(n\ge1).$$
a) Prove that the sequence is uniquely determined and has a finite limit.
b) Define $b_n=(1+2a_1)(1+2^2a_2)\cdots(1+2^na_n)$ for $n\ge1$. Prove that $(b_n)$ has a finite limit.
Let $a,b,c,\alpha,\beta$ be integers and define $u_1=\alpha$, $u_2=\beta$, $u_{n+2}=au_{n+1}+bu_n+c$ for $n\ge1$.
a) Prove that for $a=3,b=-2,c=-1$, there are infinitely many integer pairs $(\alpha,\beta)$ such that $u_{2023}=2^{2022}$.
b) Prove that some positive integer $n_0$ has exactly one of the following properties:
i) There are infinitely many positive integers $m$ such that $u_{n_0}u_{n_0+1}\cdots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$.
ii) There are infinitely many positive integers $k$ such that $u_{n_0}u_{n_0+1}\cdots u_{n_0+k}-1$ is divisible by 2023.
Find the largest positive real number $k$ such that
$$\frac1{kab+c^2}+\frac1{kbc+a^2}+\frac1{kca+b^2}\ge\frac{k+3}{a^2+b^2+c^2}$$
for every positive triple $(a,b,c)$ satisfying $a^2+b^2+c^2=2(ab+bc+ca)$.
Let $ABCD$ be a cyclic quadrilateral with $DB=DC$. Let $M,N$ be the midpoints of $AB,AC$, and let $J,E,F$ be the contact points of the incircle $(I)$ of triangle $ABC$ with $BC,CA,AB$, respectively. Line $MN$ meets $JE,JF$ at $K,H$, respectively. Line $IJ$ meets circle $(IBC)$ again at $G$, and $DG$ meets $(IBC)$ again at $T$.
a) Prove that $JA$ passes through the midpoint of $HK$ and is perpendicular to $IT$.
b) Let $R,S$ be the perpendicular projections of $D$ on $AB,AC$. Choose $P\in IF$, $Q\in IE$ so that $KP,HQ$ are perpendicular to $MN$. Prove that $MP,NQ,RS$ are concurrent.
Let $f,g:\mathbb R\to\mathbb R$ satisfy $f(0)=2022$ and
$$f(x+g(y))=xf(y)+(2023-y)f(x)+g(x)\quad(x,y\in\mathbb R).$$
a) Prove that $f$ is surjective and $g$ is injective.
b) Find all pairs of functions $f,g$ satisfying the conditions.
There are $n\ge2$ classes organizing $m\ge1$ extracurricular clubs. Every class has students participating in at least one club. Each club has participants from exactly $a$ classes. For any two clubs, at most $b$ classes have participants in both.
a) Find $m$ when $n=8,a=4,b=1$.
b) Prove $n\ge20$ when $m=6,a=10,b=4$.
c) Find the smallest $n$ when $m=20,a=4,b=1$.
Let $ABC$ be an acute scalene triangle, with orthocenter $H$ and circumcenter $O$. Its incircle $(I)$ touches $BC,CA,AB$ at $M,N,P$, respectively. Let $\Omega_A$ be a circle through $A$, externally tangent to $(I)$ at $A'$, and meeting $AB,AC$ again at $A_b,A_c$, respectively. Define $\Omega_B,\Omega_C$ and $B',B_a,B_c,C',C_a,C_b$ analogously.
a) Prove
$$B_cC_b+C_aA_c+A_bB_a\ge NP+PM+MN.$$
b) Suppose $A',B',C'$ lie on $AM,BN,CP$, respectively. Let $K$ be the circumcenter of the triangle whose side lines are $A_bA_c,B_cB_a,C_aC_b$. Prove that $OH\parallel IK$.