VN-TST 2013

The $ABCD$ is a cyclic quadrilateral with no parallel sides inscribed in circle $(O, R)$ . Let $E$ be the intersection of two diagonals and the angle bisector of $AEB$ cut the lines $AB, BC, CD, DA$ at $M, N, P, Q$ respectively . a) Prove that the circles $(AQM), (BMN), (CNP), (DPQ)$ are passing through a point. Call that point $K$ . b) Denote $min \,\{AC, BD\} = m$ . Prove that $OK \le \dfrac{2R^2}{\sqrt{4R^2-m^2}}$ .

Attachments:

a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares. b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$ .

Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$ . Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously:

i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$ ; ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$ .

Let $F$ be the set of such functions. For each $f\in F$ , let $v(f)$ be the set of values of $f$ .

a) Proof that $|F|=\infty$ .

b) Proof that for each $f\in F$ then $|v(f)|<\infty$ .

c) Find the maximum value of $|v(f)|$ for $f\in F$ .

Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4}$$

Let $ABC$ be a triangle with $\angle BAC= 45^o$ . Altitudes $AD, BE, CF$ meet at $H$ . $EF$ cuts $BC$ at $P$ . $I$ is the midpoint of $BC$ , $IF$ cuts $PH$ in $Q$ . a) Prove that $\angle IQH = \angle AIE$ . b) Let $(K)$ be the circumcircle of triangle $ABC$ , $(J)$ be the circumcircle of triangle $KPD$ . $CK$ cuts circle $(J)$ at $G$ , $IG$ cuts $(J)$ at $M$ , $JC$ cuts circle of diameter $BC$ at $N$ . Prove that $G, N, M, C$ lie on the same circle.

A cube with size $10\times 10\times 10$ consists of $1000$ unit cubes, all colored white. $A$ and $B$ play a game on this cube. $A$ chooses some pillars with size $1\times 10\times 10$ such that no two pillars share a vertex or side, and turns all chosen unit cubes to black. $B$ is allowed to choose some unit cubes and ask $A$ their colors. How many unit cubes, at least, that $B$ need to choose so that for any answer from $A$ , $B$ can always determine the black unit cubes?