VN-TST 2018

Let $ABC$ be a acute, non-isosceles triangle. $D,\ E,\ F$ are the midpoints of sides $AB,\ BC,\ AC$ , resp. Denote by $(O),\ (O')$ the circumcircle and Euler circle of $ABC$ . An arbitrary point $P$ lies inside triangle $DEF$ and $DP,\ EP,\ FP$ intersect $(O')$ at $D',\ E',\ F'$ , resp. Point $A'$ is the point such that $D'$ is the midpoint of $AA'$ . Points $B',\ C'$ are defined similarly. a. Prove that if $PO=PO'$ then $O\in(A'B'C')$ ; b. Point $A'$ is mirrored by $OD$ , its image is $X$ . $Y,\ Z$ are created in the same manner. $H$ is the orthocenter of $ABC$ and $XH,\ YH,\ ZH$ intersect $BC, AC, AB$ at $M,\ N,\ L$ resp. Prove that $M,\ N,\ L$ are collinear.

For every positive integer $m$ , a $m\times 2018$ rectangle consists of unit squares (called "cell") is called complete if the following conditions are met: i. In each cell is written either a " $0$ ", a " $1$ " or nothing; ii. For any binary string $S$ with length $2018$ , one may choose a row and complete the empty cells so that the numbers in that row, if read from left to right, produce $S$ (In particular, if a row is already full and it produces $S$ in the same manner then this condition ii. is satisfied). A complete rectangle is called minimal, if we remove any of its rows and then making it no longer complete.

a. Prove that for any positive integer $k\le 2018$ there exists a minimal $2^k\times 2018$ rectangle with exactly $k$ columns containing both $0$ and $1$ . b. A minimal $m\times 2018$ rectangle has exactly $k$ columns containing at least some $0$ or $1$ and the rest of columns are empty. Prove that $m\le 2^k$ .

For every positive integer $n\ge 3$ , let $\phi_n$ be the set of all positive integers less than and coprime to $n$ . Consider the polynomial: $$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$ a. Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial). b. Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$ .

Let $a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n),\ (v_n)$ are defined as follows: $$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$ a. Prove that $${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$ b. Find all values of $a$ in the equality case.

In a $m\times n$ square grid, with top-left corner is $A$ , there is route along the edges of the grid starting from $A$ and visits all lattice points (called "nodes") exactly once and ending also at $A$ .

a. Prove that this route exists if and only if at least one of $m,\ n$ is odd. b. If such a route exists, then what is the least possible of turning points?

*A turning point is a node that is different from $A$ and if two edges on the route intersect at the node are perpendicular.

Triangle $ABC$ circumscribed $(O)$ has $A$ -excircle $(J)$ that touches $AB,\ BC,\ AC$ at $F,\ D,\ E$ , resp.

a. $L$ is the midpoint of $BC$ . Circle with diameter $LJ$ cuts $DE,\ DF$ at $K,\ H$ . Prove that $(BDK),\ (CDH)$ has an intersecting point on $(J)$ . b. Let $EF\cap BC =\{G\}$ and $GJ$ cuts $AB,\ AC$ at $M,\ N$ , resp. $P\in JB$ and $Q\in JC$ such that $$\angle PAB=\angle QAC=90{}^\circ .$$ $PM\cap QN=\{T\}$ and $S$ is the midpoint of the larger $BC$ -arc of $(O)$ . $(I)$ is the incircle of $ABC$ . Prove that $SI\cap AT\in (O)$ .