VMO 2021

Let $\bigtriangleup ABC$ is not an isosceles triangle and is an acute triangle, $AD,BE,CF$ be the altitudes and $H$ is the orthocenter .Let $I$ is the circumcenter of $\bigtriangleup HEF$ and let $K,J$ is the midpoint of $BC,EF$ respectively.Let $HJ$ intersects $(I)$ again at $G$ and $GK$ intersects $(I)$ at $L\neq G$ . a) Prove that $AL$ is perpendicular to $EF$ . b) Let $AL$ intersects $EF$ at $M$ , the line $IM$ intersects the circumcircle $\bigtriangleup IEF$ again at $N$ , $DN$ intersects $AB,AC$ at $P$ and $Q$ respectively then prove that $PE,QF,AK$ are concurrent.

For an integer $n \geq 2$ , let $s (n)$ be the sum of positive integers not exceeding $n$ and not relatively prime to $n$ . a) Prove that $s (n) = \dfrac {n} {2} \left (n + 1- \varphi (n) \right)$ , where $\varphi (n)$ is the number of integers positive cannot exceed $n$ and are relatively prime to $n$ . b) Prove that there is no integer $n \geq 2$ such that $s (n) = s (n + 2021)$

Let the polynomial $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0$ where $1011\leq a_i\leq 2021$ for all $i=0,1,2,...,21.$ Given that $P(x)$ has an integer root and there exists an positive real number $c$ such that $|a_{k+2}-a_k|\leq c$ for all $k=0,1,...,19.$ a) Prove that $P(x)$ has an only integer root.

b) Prove that $$\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.$$

A student divides all $30$ marbles into $5$ boxes numbered $1, 2, 3, 4, 5$ (after being divided, there may be a box with no marbles). a) How many ways are there to divide marbles into boxes (are two different ways if there is a box with a different number of marbles)? b) After dividing, the student paints those $30$ marbles by a number of colors (each with the same color, one color can be painted for many marbles), so that there are no $2$ marbles in the same box. have the same color and from any $2$ boxes it is impossible to choose $8$ marbles painted in $4$ colors. Prove that for every division, the student must use no less than $10$ colors to paint the marbles. c) Show a division so that with exactly $10$ colors the student can paint the marbles that satisfy the conditions in question b).

Let $ABC$ be an inscribed triangle in circle $(O)$ . Let $D$ be the intersection of the two tangent lines of $(O)$ at $B$ and $C$ . The circle passing through $A$ and tangent to $BC$ at $B$ intersects the median passing $A$ of the triangle $ABC$ at $G$ . Lines $BG, CG$ intersect $CD, BD$ at $E, F$ respectively. a) The line passing through the midpoint of $BE$ and $CF$ cuts $BF, CE$ at $M, N$ respectively. Prove that the points $A, D, M, N$ belong to the same circle. b) Let $AD, AG$ intersect the circumcircle of the triangles $DBC, GBC$ at $H, K$ respectively. The perpendicular bisectors of $HK, HE$ , and $HF$ cut $BC, CA$ , and $AB$ at $R, P$ , and $Q$ respectively. Prove that the points $R, P$ , and $Q$ are collinear.