2019 Canadian Mathematical Olympiad Qualification

1

A function f is called injective if when f(n)=f(m), then n=m. Suppose that f is injective and 1f(n)+1f(m)=4f(n)+f(m). Prove m=n

2

Rosemonde is stacking spheres to make pyramids. She constructs two types of pyramids Sn and Tn. The pyramid Sn has n layers, where the top layer is a single sphere and the ith layer is an i×i square grid of spheres for each 2in. Similarly, the pyramid Tn has n layers where the top layer is a single sphere and the ith layer is i(i+1)2 spheres arranged into an equilateral triangle for each 2in.

3

Let f(x)=x3+3x21 have roots a,b,c. (a) Find the value of a3+b3+c3 (b) Find all possible values of a2b+b2c+c2a

4

Let n be a positive integer. For a positive integer m, we partition the set {1,2,3,...,m} into n subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of n, find the largest positive integer m for which such a partition exists.

5

Let (m,n,N) be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. The players alternate turns, with Bruce going first. On Bruce's turn, he picks a row and either adds 1 to all of the entries in the row or subtracts 1 from all the entries in the row. On Duncan's turn, he picks a column and either adds 1 to all of the entries in the column or subtracts 1 from all of the entries in the column. Bruce wins if at some point there is an entry x with |x|N. Find all triples (m,n,N) such that no matter how Duncan plays, Bruce has a winning strategy.

6

Pentagon ABCDE is given in the plane. Let the perpendicular from A to line CD be F, the perpendicular from B to DE be G, from C to EA be H, from D to AB be I,and from E to BC be J. Given that lines AF,BG,CH, and DI concur, show that they also concur with line EJ.

7

There are n passengers in a line, waiting to board a plane with n seats. For 1kn, the kth passenger in line has a ticket for the kth seat. However, the rst passenger ignores his ticket, and decides to sit in a seat at random. Thereafter, each passenger sits as follows: If his/her assigned is empty, then he/she sits in it. Otherwise, he/she sits in an empty seat at random. How many different ways can all n passengers be seated?

8

For t2, define S(t) as the number of times t divides into t!. We say that a positive integer t is a peak if S(t)>S(u) for all values of u<t. Prove or disprove the following statement: For every prime p, there is an integer k for which p divides k and k is a peak.