TH-TMO 2016

Let $ABC$ be a triangle with $AB \ne AC$ . Let the angle bisector of $\angle BAC$ intersects $BC$ at $P$ and intersects the perpendicular bisector of segment $BC$ at $Q$ . Prove that $\frac{PQ}{AQ} =\left( \frac{BC}{AB + AC}\right)^2$

Let $M$ be a positive integer, and $A = \{1, 2,... , M + 1\}$ . Show that if $f$ is a bijection from $A$ to $A$ then $\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} > \frac{M}{M + 3}$

Determine all functions $f : R \to R$ satisfying $f (f(x)f(y) + f(y)f(z) + f(z)f(x))= f(x) + f(y) + f(z)$ for all real numbers $x, y, z$ .

Each point on the plane is colored either red, green, or blue. Prove that there exists an isosceles triangle whose vertices all have the same color.

given $p_1,p_2,...$ be a sequence of integer and $p_1=2$ , for positive integer $n$ , $p_{n+1}$ is the least prime factor of $np_1^{1!}p_2^{2!}...p_n^{n!}+1$ prove that all primes appear in the sequence

(Proposed by Beatmania)

Let $m$ and $n$ be positive integers. Prove that if $m^{4^n+1} - 1$ is a prime number, then there exists an integer $t \ge 0$ such that $n = 2^t$ .

Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$ be a polynomial with real coefficients and $a_{2016} \neq 0$ satisfies $|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$ Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also)

edited: complex roots

Let $\vartriangle ABC$ be an acute triangle with incenter $I$ . The line passing through $I$ parallel to $AC$ intersects $AB$ at $M$ , and the line passing through $I$ parallel to $AB$ intersects $AC$ at $N$ . Let the line $MN$ intersect the circumcircle of $\vartriangle ABC$ at $X$ and $Y$ . Let $Z$ be the midpoint of arc $BC$ (not containing $A$ ). Prove that $I$ is the orthocenter of $\vartriangle XY Z$

A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$ .

A Pattano coin is a coin which has a blue side and a yellow side. A positive integer not exceeding $100$ is written on each side of every coin (the sides may have different integers). Two Pattano coins are identical if the number on the blue side of both coins are equal and the number on the yellow side of both coins are equal. Two Pattano coins are pairable if the number on the blue side of both coins are equal or the number on the yellow side of both coins are equal. Given $2559$ Pattano coins such that no two coins are identical. Show that at least one Pattano coin is pairable with at least $50$ other coins