THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh

THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh

THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh

THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh

3. Let n be a positive integer. A finite sequence {a1, a2, · · · , an} of positive integers ai is said to be tight if and only if 1 ≤ a1 < a2 < · · · < an, all ( n 2 ) differences aj − ai with i < j are distinct, and an is as small as possible. (a) Determine a tight sequence for n = 5. (b) Prove that there is a polynomial p(n) of degree not exceeding 3 such that an ≤ p(n) for every tight sequen...

THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh

1. Determine the supremum and the infimum of (x− 1)x−1xx (x− (1/2))2x−1 for x > 1. 2. Let n and k be integers with n ≥ 0 and k ≥ 1. Let x0, x1, · · ·, xn be n+1 distinct points in R and let y0, y1, · · ·, yn be n + 1 real numbers (not necessarily distinct). Prove that there exists a polynomial p of degree at most n in the coordinates of x with respect to the standard basis for which p(xi) = yi ...

THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION In Memory of Robert Barrington Leigh

6. Two competitors play badminton. They play two games, each winning one of them. They then play a third game to determine the overall winner of the match. The winner of a game of badminton is the first player to score at least 21 points with a lead of at least 2 points over the other player. In this particular match, it is observed that the scores of each player listed in order of the games fo...