TH-TMO 2017

Let $p$ be a prime. Show that $\sqrt[3]{p} +\sqrt[3]{p^5}$ is irrational.

A cyclic quadrilateral $ABCD$ has circumcenter $O$ , its diagonals $AC$ and $BD$ intersect at $G$ . Let $P, Q, R, S$ be the circumcenters of $\vartriangle AGB, \vartriangle BGC, \vartriangle CGD, \vartriangle DGA$ respectively. Lines $P R$ and $QS$ intersect at $M$ . Show that $M$ is the midpoint of $OG$ .

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

In a math competition, $14$ schools participate, each sending $14$ students. The students are separated into $14$ groups of $14$ so that no two students from the same school are in the same group. The tournament organizers noted that, from the competitors, exactly $15$ have participated in the competition before. The organizers want to select two representatives, with the conditions that they must be former participants, must come from different schools, and must also be in different groups. It turns out that there are $n$ ways to do this. What is the minimum possible value of $n$ ?

Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$ ? Justify your answer.

In an acute triangle $\vartriangle ABC$ , $D$ is the foot of altitude from $A$ to $BC$ . Suppose that $AD = CD$ , and define $N$ as the intersection of the median $CM$ and the line $AD$ . Prove that $\vartriangle ABC$ is isosceles if and only if $CN = 2AM$ .

Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$ . .

Let $a, b, c$ be side lengths of a right triangle. Determine the minimum possible value of $\frac{a^3 + b^3 + c^3}{abc}$ .

Determine all functions $f$ on the set of positive rational numbers such that $f(xf(x) + f(y)) = f(x)^2 + y$ for all positive rational numbers $x, y$ .

A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$ , there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).