VMO 2020

Let a sequence $(x_n)$ satisfy : $x_1=1$ and $x_{n+1}=x_n+3\sqrt{x_n} + \frac{n}{\sqrt{x_n}}$ , $\forall$ n $\ge1$ a) Prove lim $\frac{n}{x_n}=0$ b) Find lim $\frac{n^2}{x_n}$

a)Let $a,b,c\in\mathbb{R}$ and $a^2+b^2+c^2=1$ .Prove that: $|a-b|+|b-c|+|c-a|\le2\sqrt{2}$ b) Let $a_1,a_2,..a_{2019}\in\mathbb{R}$ and $\sum_{i=1}^{2019}a_i^2=1$ .Find the maximum of: $S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|$

Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$ a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$ b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$ , if p is a prime factor of $a_{2^k}$

Let a non-isosceles acute triangle ABC with the circumscribed cycle (O) and the orthocenter H. D, E, F are the reflection of O in the lines BC, CA and AB. a) $H_a$ is the reflection of H in BC, A' is the reflection of A at O and $O_a$ is the center of (BOC). Prove that $H_aD$ and OA' intersect on (O). b) Let X is a point satisfy AXDA' is a parallelogram. Prove that (AHX), (ABF), (ACE) have a comom point different than A

Let a system of equations: $\left\{\begin{matrix}x-ay=yzy-az=zxz-ax=xy\end{matrix}\right.$ a)Find (x,y,z) if a=0 b)Prove that: the system have 5 distinct roots $\forall$ a>1,a $\in\mathbb{R}.$

Let a non-isosceles acute triangle ABC with tha attitude AD, BE, CF and the orthocenter H. DE, DF intersect (AD) at M, N respectively. $P\in AB,Q\in AC$ satisfy $NP\perp AB,MQ\perp AC$ a) Prove that EF is the tangent line of (APQ) b) Let T be the tangency point of (APQ) with EF,.DT $\cap$ MN={K}. L is the reflection of A in MN. Prove that MN, EF ,(DLK) pass through a piont

Given a positive integer $n>1$ . Denote $T$ a set that contains all ordered sets $(x;y;z)$ such that $x,y,z$ are all distinct positive integers and $1\leq x,y,z\leq 2n$ . Also, a set $A$ containing ordered sets $(u;v)$ is called "connected" with $T$ if for every $(x;y;z)\in T$ then $\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing$ . a) Find the number of elements of set $T$ . b) Prove that there exists a set "connected" with $T$ that has exactly $2n(n-1)$ elements. c) Prove that every set "connected" with $T$ has at least $2n(n-1)$ elements.