2019 Puerto Rico Team Selection Test

Day 1

1

A square is divided into 25 unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?

2

Let ABCD be a square. Let M and K be points on segments BC and CD respectively, such that MC=KD. Let P be the intersection of the segments MD and BK. Prove that AP is perpendicular to MK.

3

Find the largest value that the expression can take a3b+b3a where a,b are non-negative real numbers, with a+b=3.

Day 2

4

Rectangle ABCD has sides AB=3, BC=2. Point P lies on side AB is such that the bisector of the angle CDP passes through the midpoint M of BC. Find BP.

5

The wizard Gandalf has a necklace that is shaped like a row of magic pearls. The necklace has 2019 pearls, 2018 are black and the last one is white. Everytime that the magician Gandalf touches the necklace, the following occurs: the pearl in position i is move to position i1, for 1<i<2020, furthermore the pearl in position 1 moves to position 2019. But something else happens, if the pearl in position 1 now is white, then the last pearl turns white without the need for Gandalf to touch the necklace again. How many times does Gandalf have to touch the necklace to be sure that all pearls are white?

6

Starting from a pyramid T0 whose edges are all of length 2019, we construct the Figure T1 when considering the triangles formed by the midpoints of the edges of each face of T0, building in each of these new pyramid triangles with faces identical to base. Then the bases of these new pyramids are removed. Figure T2 is constructed by applying the same process from T1 on each triangular face resulting from T1, and so on for T3,T4,... Let D0=max, where x and y are vertices of T_0 and d(x,y) is the distance between x and y. Then we define D_{n + 1} = \max \{d (x, y) |d (x, y) \notin \{D_0, D_1,...,D_n\}, where x, y are vertices of T_{n+1}. Find the value of D_n for all n.