2021 Bolivia Ibero TST

Day 1

1

Let n be a posititve integer. On a n×n grid there are n2 unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue. a) Find the maximun number of blue unit sides we can have on the n×n grid. b) Find the minimun number of blue unit sides we can have on the n×n grid.

2

Let f:Z+Z be a function such that a) f(p)=1 for every prime p. b) f(xy)=xf(y)+yf(x) for every pair of positive integers x,y Find the least number n2021 such that f(n)=n

Day 2

3

Let p=ab+bc+ac be a prime number where a,b,c are different two by two, show that a3,b3,c3 gives different residues modulo p

4

On a isosceles triangle ABC with AB=BC let K,M be the midpoints of AB,AC respectivily. Let (CKB) intersect BM at NM, the line through N parallel to AC intersects (ABC) at A1,C1. Show that A1BC1 is equilateral.