VMO 2023

Given are the integers $a , b , c, \alpha, \beta$ and the sequence $(u_n)$ is defined by $u_1=\alpha, u_2=\beta, u_{n+2}=au_{n+1}+bu_n+c$ for all $n \geq 1$ .

a) Prove that if $a = 3 , b= -2 , c = -1$ then there are infinitely many pairs of integers $(\alpha ; \beta)$ so that $u_{2023}=2^{2022}$ .

b) Prove that there exists a positive integer $n_0$ such that only one of the following two statements is true:

i) There are infinitely many positive integers $m$ , such that $u_{n_0}u_{n_0+1}\ldots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$ ii) There are infinitely many positive integers $k$ so that $u_{n_0}u_{n_0+1}\ldots u_{n_0+k}-1$ is divisible by $2023$

Find the maximum value of the positive real number $k$ such that the inequality $$\frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2} \geq \frac{k+3}{a^2+b^2+c^2}$$ holds for all positive real numbers $a,b,c$ such that $a^2+b^2+c^2=2(ab+bc+ca).$

Given is a triangle $ABC$ and let $D$ be the midpoint the major arc $BAC$ of its circumcircle. Let $M , N$ be the midpoints of $AB , AC$ and $J , E , F$ are the touchpoints of the incircle $(I)$ of $\triangle ABC$ with $BC, CA, AB$ . The line $MN$ intersects $JE , JF$ at $K , H$ respectively; $IJ$ intersects the circle $(BIC)$ at $G$ and $DG$ intersects $(BIC)$ at $T$ .

a) Prove that $JA$ passes through the midpoint of $HK$ and is perpendicular to $IT$ .

b) Let $R, S$ respectively be the perpendicular projection of $D$ on $AB, AC$ . Take the points $P, Q$ on $IF , IE$ respectively such that $KP$ and $HQ$ are both perpendicular to $MN$ . Prove that the three lines $MP , NQ$ and $RS$ are concurrent .

Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $f (0)=2022$ and $f (x+g(y)) =xf(y)+(2023-y)f(x)+g(x)$ for all $x, y \in \mathbb{R}$ .

There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students in this group participate in. For any two extracurricular groups, there are no more than $b$ classes with students participating in both groups simultaneously. a) Find $m$ when $n = 8, a = 4 , b = 1$ . b) Prove that $n \geq 20$ when $m = 6 , a = 10 , b = 4$ . c) Find the minimum value of $n$ when $m = 20 , a = 4 , b = 1$ .

Let $\triangle{ABC}$ be a scalene triangle with orthocenter $H$ and circumcenter $O$ . Incircle $(I)$ of the $\triangle{ABC}$ is tangent to the sides $BC,CA,AB$ at $M,N,P$ respectively. Denote $\Omega_A$ to be the circle passing through point $A$ , external tangent to $(I)$ at $A'$ and cut again $AB,AC$ at $A_b,A_c$ respectively. The circles $\Omega_B,\Omega_C$ and points $B',B_a,B_c,C',C_a,C_b$ are defined similarly. $a)$ 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. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c,B_cB_a,C_aC_b.$ Prove $OH//IK$ .