TH-TMO 2019

Let $ABCDE$ be a convex pentagon with $\angle AEB=\angle BDC=90^o$ and line $AC$ bisects $\angle BAE$ and $\angle DCB$ internally. The circumcircle of $ABE$ intersects line $AC$ again at $P$ . (a) Show that $P$ is the circumcenter of $BDE$ . (b) Show that $A, C, D, E$ are concyclic.

Let $a,b$ be two different positive integers. Suppose that $a,b$ are relatively prime. Prove that $\dfrac{2a(a^2+b^2)}{a^2-b^2}$ is not an integer.

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that $f(x+yf(x)+y^2) = f(x)+2y$ for every $x,y\in\mathbb{R}^+$ .

A rabbit initially stands at the position $0$ , and repeatedly jumps on the real line. In each jump, the rabbit can jump to any position corresponds to an integer but it cannot stand still. Let $N(a)$ be the number of ways to jump with a total distance of $2019$ and stop at the position $a$ . Determine all integers $a$ such that $N(a)$ is odd.

Let $a,b,c$ be positive reals such that $abc=1$ . Prove the inequality $$\frac{4a-1}{(2b+1)^2} + \frac{4b-1}{(2c+1)^2} + \frac{4c-1}{(2a+1)^2}\geqslant 1.$$

Determine all function $f:\mathbb{R}\to\mathbb{R}$ such that $xf(y)+yf(x)\leqslant xy$ for all $x,y\in\mathbb{R}$ .

Let $A=\{-2562,-2561,...,2561,2562\}$ . Prove that for any bijection (1-1, onto function) $f:A\to A$ , $$\sum_{k=1}^{2562}\left\lvert f(k)-f(-k)\right\rvert\text{ is maximized if and only if } f(k)f(-k)<0\text{ for any } k=1,2,...,2562.$$

Let $ABC$ be a triangle such that $AB\ne AC$ and $\omega$ be the circumcircle of this triangle. Let $I$ be the center of the inscribed circle of $ABC$ which touches $BC$ at $D$ . Let the circle with diameter $AI$ meets $\omega$ again at $K$ . If the line $AI$ intersects $\omega$ again at $M$ , show that $K, D, M$ are collinear.

A chaisri figure is a triangle which the three vertices are vertices of a regular $2019$ -gon. Two different chaisri figure may be formed by different regular $2019$ -gon. A thubkaew figure is a convex polygon which can be dissected into multiple chaisri figure where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon.

Determine the maximum number of vertices that a thubkaew figure may have.

Prove that there are infinitely many positive odd integer $n$ such that $n!+1$ is composite number.