PL-OM 2021

Let $p_i$ for $i=1,2,..., k$ be a sequence of smallest consecutive prime numbers ( $p_1=2$ , $p_2=3$ , $p_3=3$ etc. ). Let $N=p_1\cdot p_2 \cdot ... \cdot p_k$ . Prove that in a set $\{ 1,2,...,N \}$ there exist exactly $\frac{N}{2}$ numbers which are divisible by odd number of primes $p_i$ .

Let $n$ be an integer. For pair of integers $0 \leq i,$ $j\leq n$ there exist real number $f(i,j)$ such that:

1. $f(i,i)=0$ for all integers $0\leq i \leq n$ 2) $0\leq f(i,l) \leq 2\max \{ f(i,j), f(j,k), f(k,l) \}$ for all integers $i$ , $j$ , $k$ , $l$ satisfying $0\leq i\leq j\leq k\leq l\leq n$ .

Prove that $$f(0,n) \leq 2\sum_{k=1}^{n}f(k-1,k)$$

Let $\omega$ be the circumcircle of a triangle $ABC$ . Let $P$ be any point on $\omega$ different than the verticies of the triangle. Line $AP$ intersects $BC$ at $D$ , $BP$ intersects $AC$ at $E$ and $CP$ intersects $AB$ at $F$ . Let $X$ be the projection of $D$ onto line passing through midpoints of $AP$ and $BC$ , $Y$ be the projection of $E$ onto line passing through $BP$ and $AC$ and let $Z$ be the projection of $F$ onto line passing through midpoints of $CP$ and $AB$ . Let $Q$ be the circumcenter of triangle $XYZ$ . Prove that all possible points $Q$ , corresponding to different positions of $P$ lie on one circle.

Prove that for every pair of positive real numbers $a, b$ and for every positive integer $n$ , $$(a+b)^n-a^n-b^n \ge \frac{2^n-2}{2^{n-2}} \cdot ab(a+b)^{n-2}.$$

A convex hexagon $ABCDEF$ is given where $\measuredangle FAB + \measuredangle BCD + \measuredangle DEF = 360^{\circ}$ and $\measuredangle AEB = \measuredangle ADB$ . Suppose the lines $AB$ and $DE$ are not parallel. Prove that the circumcenters of the triangles $\triangle AFE, \triangle BCD$ and the intersection of the lines $AB$ and $DE$ are collinear.

Given an integer $d \ge 2$ and a circle $\omega$ . Hansel drew a finite number of chords of circle $\omega$ . The following condition is fulfilled: each end of each chord drawn is at least an end of $d$ different drawn chords. Prove that there is a drawn chord which intersects at least $\tfrac{d^2}{4}$ other drawn chords. Here we assume that the chords with a common end intersect.

Note: Proof that a certain drawn chord crosses at least $\tfrac{d^2}{8}$ other drawn chords will be awarded two points.