VMO 2022 — Academic year 2021–2022, both days

Let $a\ge0$ be real and define
$$u_1=6,\qquad u_{n+1}=\frac{2n+a}{n}+\sqrt{\frac{n+a}{n}u_n+4}\quad(n\ge1).$$
(a) For $a=0$, prove that $(u_n)$ has a finite limit and determine it.
(b) For every $a\ge0$, prove that $(u_n)$ has a finite limit.

Find all functions $f:(0,+\infty)\to(0,+\infty)$ such that
$$f\left(\frac{f(x)}x+y\right)=1+f(y)\qquad(x,y\gt 0).$$

Let $ABC$ be acute. Points $E,F$ vary on the rays opposite to $BA,CA$, respectively, with $BF=CE$, $E\ne B$, $F\ne C$. Let $M,N$ be the midpoints of $BE,CF$, and $D=BF\cap CE$.
(a) Let $I,J$ be the circumcenters of $DBE,DCF$, respectively. Prove $MN\parallel IJ$.
(b) Let $K$ be the midpoint of $MN$, and let $H$ be the orthocenter of $AEF$. Prove that $HK$ passes through a fixed point.

For positive integers $n\lt m$, let $s(n,m)$ be the number of positive integers in $[n,m]$ coprime to $m$. Find all integers $m\ge2$ satisfying both:
(i) $\dfrac{s(n,m)}{m-n}\ge\dfrac{s(1,m)}m$ for every $n=1,2,\ldots,m-1$;
(ii) $m^2$ divides $2022^m+1$.

Let $P,Q$ be nonconstant polynomials with nonnegative integer coefficients. Every coefficient of $P$ is at most 2021, while at least one coefficient of $Q$ exceeds 2021. Suppose $P(2022)=Q(2022)$ and $P,Q$ have a common nonzero rational root $p/q$, where $p,q\in\mathbb Z$ are coprime. Prove that
$$|p|+n|q|\le Q(n)-P(n)\qquad(n=1,2,\ldots,2021).$$

Four fair, identical dice are rolled. Let $x_i\in\{1,2,\ldots,6\}$ be the result of die $i$, for $i=1,2,3,4$.
(a) How many ordered outcomes $(x_1,x_2,x_3,x_4)$ are possible?
(b) Find the probability that one result equals the sum of the other three.
(c) Find the probability that the four results can be divided into two groups of equal sum.

Let $B,C$ be fixed points on a circle $(O)$, with $BC$ not passing through $O$. Let $A$ vary on the major arc $BC$, with $AB\ne AC$. The incircle $(I)$ of $ABC$ touches $BC$ at $D$. Let $I_a$ be the $A$-excenter, let $L=I_aD\cap OI$, and choose $E$ on $(I)$ such that $DE\parallel AI$.
(a) Let $F=LE\cap AI$. Prove that $AF=AI$.
(b) On the circumcircle $(J)$ of $I_aBC$, choose $M$ such that $I_aM\parallel AD$, and let $MD$ meet $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.