19th Central American and Caribbean
Mathematical Olympiad 2017
San Ignacio, El Salvador
Day 2 (June 19, 2017)
Problem 4
Let ABC be a triangle with a right angle at B. Let B 0 be the reflection of B with respect to the line
AC, and M be the midpoint of AC. The segment BM is extended beyond M to a point D such that
BD = AC. Show that B 0 C is the bisector of 6 M B 0 D.
Problem 5
Susan and Brenda play a game writing polynomials, taking turns starting with Susan.
On the preparatory turn (0th turn), Susan chooses a positive integer n0 and writes the polynomial
P0 (x) = n0 .
On the 1st turn, Brenda chooses a positive integer n1 distinct from n0 and writes either
P1 (x) = n1 x − P0 (x)
or P1 (x) = n1 x + P0 (x).
On the kth turn, the corresponding player chooses a positive integer nk distinct from n0 , n1 , . . . , nk−1
and writes either
Pk (x) = nk xk − Pk−1 (x)
or Pk (x) = nk xk + Pk−1 (x).
The winner is the first player to write a polynomial with at least one integer root. Determine which
player has a winning strategy and describe it.
Problem 6
Let k be an integer greater than 1. Initially, Tita the frog is sitting at the point k on the number line.
In one movement, if Tita is located on the point n, then she jumps to the point f (n)+g(n), where f (n)
and g(n) are the greatest and the least prime divisors (both positive) of n, respectively. Determine all
the values of k for which Tita will visit infinitely many distinct points on the number line.
Time: four and one-half hours
Each problem is worth seven points