TH-TMO 2018

In $\vartriangle ABC$ , the incircle is tangent to the sides $BC, CA, AB$ at $D, E, F$ respectively. Let $P$ and $Q$ be the midpoints of $DF$ and $DE$ respectively. Lines $P C$ and $DE$ intersect at $R$ , and lines $BQ$ and $DF$ intersect at $S$ . Prove that a) Points $B, C, P, Q$ lie on a circle. b) Points $P, Q, R, S$ lie on a circle.

Show that there are no functions $f : R \to R$ satisfying $f(x + f(y)) = f(x) + y^2$ for all real numbers $x$ and $y$

Karakade has three flash drives of each of the six capacities $1, 2, 4, 8, 16, 32$ gigabytes. She gives each of her $6$ servants three flash drives of different capacities. Prove that either there are two capacities where each servant has at most one of the two capacities, or all servants have flash drives with different sums of capacities.

Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$ . Determine the maximum possible value of $\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$ .

Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$ . What is the minimum possible value of $a + b$ ?

We color each number in the set $S = \{1, 2, ..., 61\}$ with one of $25$ given colors, where it is not necessary that every color gets used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$ ?

There are $2n + 1$ tickets, each with a unique positive integer as the ticket number. It is known that the sum of all ticket numbers is more than $2330$ , but the sum of any $n$ ticket numbers is at most $1165$ . What is the maximum value of $n$ ?

In $\vartriangle ABC$ the incircle is tangent to $AB$ at $D$ . Let $P$ be a point on $BC$ different from $B$ and $C$ , and let $K$ and $L$ be incenters of $\vartriangle ABP$ and $\vartriangle ACP$ respectively. Suppose that the circumcircle of $\vartriangle KP L$ cuts $AP$ again at $Q$ . Prove that $AD = AQ$ .

Let $a,b,c$ be non-zero real numbers.Prove that if function $f,g:\mathbb{R}\to\mathbb{R}$ satisfy $af(x+y)+bf(x-y)=cf(x)+g(y)$ for all real number $x,y$ that $y>2018$ then there exists a function $h:\mathbb{R}\to\mathbb{R}$ such that $f(x+y)+f(x-y)=2f(x)+h(y)$ for all real number $x,y$ .