The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. Each of the thirteen marked points is assigned a color, green or red. Prove that there are always three points of the same color, which are the vertices of an equilateral triangle.
2022 Spain Mathematical Olympiad
Day 1
Let a,b,c,d be four positive real numbers. If they satisfy a+b+1ab=c+d+1cdand1a+1b+ab=1c+1d+cdthen prove that at least two of the values a,b,c,d are equal.
Let ABC be a triangle, with AB<AC, and let Γ be its circumcircle. Let D, E and F be the tangency points of the incircle with BC, CA and AB respectively. Let R be the point in EF such that DR is an altitude in the triangle DEF, and let S be the intersection of the external bisector of ∠BAC with Γ. Prove that AR and SD intersect on Γ.
Day 2
Let P be a point in the plane. Prove that it is possible to draw three rays with origin in P with the following property: for every circle with radius r containing P in its interior, if P1, P2 and P3 are the intersection points of the three rays with the circle, then |PP1|+|PP2|+|PP3|≤3r.
Given is a simple graph G with 2022 vertices, such that for any subset S of vertices (including the set of all vertices), there is a vertex v with degS(v)≤100. Find χ(G) and the maximal number of edges G can have.
Find all triples (x,y,z) of positive integers, with z>1, satisfying simultaneously that x divides y+1,y divides z−1,z divides x2+1.