The Alberta High School Mathematics Competition
Part II, February 2nd , 2022
Problem 1. Find all positive integers n such that n 2 + n + 19 is the square of an integer.
Problem 2. Ajooni and Sagal live beside a lake that has a 10 km long circular path around it. One day they start
from their house at the same time, Ajooni biking around the lake in one direction, and Sagal walking around the
lake in the same direction. Ajooni’s biking speed is b km/hour, while Sagal’s walking speed is w km/hour, where
b > w are positive integers. So Ajooni goes faster than Sagal and will pass him over and over, if they keep going
around the lake. They agree to stop whenever they both arrive back at their house at the same time.
(a) Prove that eventually Ajooni and Sagal will meet back at their house, regardless of the values of b and w.
(b) Suppose that b = 15 and w = 6, and that Ajooni and Sagal leave their house at 9 AM. At what time do they
meet back at their house and stop going around the lake?
Problem 3. Show that for any positive integer n the number
111
| {z. . . 1}
3n digits
consisting of 3n 1’s, is divisible by 3n .
Problem 4. Let B be a point on the segment AC such that B 6= A and B A < BC . Point M is on the perpendicular
bisector of AC such that ∠ AM B is as large as possible. Find ∠B MC .
Problem 5.
The numbers 1, 2, ..., 63 are placed on a 7 by 9 grid randomly, one number in each little square. Prove that you can
always draw five L-shaped trominos on the grid so that no two of them overlap (but are allowed to touch along an
edge), and so that, for each of the five trominos, the sum of the three numbers in it is at least 78. (An L-shaped
tromino is a polygon made of three squares of the grid, connected edge-to-edge and having the shape of an L, but
in any orientation. An example of a L-shaped tromino on the grid is given below.)
L-shaped tromino:
A 7 by 9 grid:
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