TR-MO 2020

Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$ . If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$ .

Let $P$ be an interior point of acute triangle $\Delta ABC$ , which is different from the orthocenter. Let $D$ and $E$ be the feet of altitudes from $A$ to $BP$ and $CP$ , and let $F$ and $G$ be the feet of the altitudes from $P$ to sides $AB$ and $AC$ . Denote by $X$ the midpoint of $[AP]$ , and let the second intersection of the circumcircles of triangles $\Delta DFX$ and $\Delta EGX$ lie on $BC$ . Prove that $AP$ is perpendicular to $BC$ or $\angle PBA = \angle PCA$ .

If $x, y, z$ are positive real numbers find the minimum value of $$2\sqrt{(x+y+z) \left( \frac{1}{x}+ \frac{1}{y} + \frac{1}{z} \right)} - \sqrt{ \left( 1+ \frac{x}{y} \right) \left( 1+ \frac{y}{z} \right)}$$

Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$ .

Find all polynomials with real coefficients such that one can find an integer valued series $a_0, a_1, \dots$ satisfying $\lfloor P(x) \rfloor = a_{ \lfloor x^2 \rfloor}$ for all $x$ real numbers.

$2021$ points are given on a circle. Each point is colored by one of the $1,2, \cdots ,k$ colors. For all points and colors $1\leq r \leq k$ , there exist an arc such that at least half of the points on it are colored with $r$ . Find the maximum possible value of $k$ .