2014 Canadian MO Problems
Let be positive real numbers whose product is . Show that the sum is greater than or equal to .
Let and be odd positive integers. Each square of an by board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of and .
Let be a fixed odd prime. A -tuple of integers is said to be good if
(i) for all , and (ii) is not divisible by , and (iii) is divisible by .
Determine the number of good -tuples.
The quadrilateral is inscribed in a circle. The point lies in the interior of , and . The lines and meet at , and the lines and meet at . Prove that the lines and form the same angle as the diagonals of .
Fix positive integers and . A list of n integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add to all of them or subtract from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least of the numbers on the blackboard are all simultaneously divisible by .