2022 Polish MO Finals

Day 1

1

Let ABC be an acute triangle with AB<AC. The angle bisector of BAC intersects the side BC and the circumcircle of ABC at D and MA, respectively. Points X and Y are chosen so that MXAB, BXMB, MYAC, and CYMC. Prove that the points X,D,Y are collinear.

2

Let m,n2 be given integers. Prove that there exist positive integers a1<a2<<am so that for any 1i<jm the number ajajai is an integer divisible by n.

3

One has marked n points on a circle and has drawn a certain number of chords whose endpoints are the marked points. It turned out that the following property is satisfied: whenever any 2021 drawn chords are removed one can join any two marked points by a broken line composed of some of the remaining drawn chords. Prove that one can remove some of the drawn chords so that at most 2022n chords remain and the property described above is preserved.

Day 2

4

Find all triples (a,b,c) of real numbers satisfying the system {a3+b2c=acb3+c2a=bac3+a2b=cb

5

Let ABC be a triangle satisfying AB<AC. Let M be the midpoint of BC. A point P lies on the segment AB with AP>PB. A point Q lies on the segment AC with MPA=AQM. The perpendicular bisectors of BC and PQ intersect at S. Prove that BAC+QSP=QMP.

6

A prime number p and a positive integer n are given. Prove that one can colour every one of the numbers 1,2,,p1 using one of the 2n colours so that for any i=2,3,,n the sum of any i numbers of the same colour is not divisible by p.