MathDB

2

Part of 2017 Auckland Mathematical Olympiad

Problems(2)

2 player game with divisors of naturals (2017 Auckland MO J2)

Source:

11/14/2021
Two players take turns to write natural numbers on a board. The rules forbid writing numbers greater than pp and also divisors of previously written numbers. The player who has no move loses. Determine which player has a winning strategy for p=10p = 10 and describe this strategy.
combinatoricsnumber theorygamewinning strategy
(1 - x_1)(1 - x_2)(1 - x_3) >= 1/2 if x_1+x_2+x_3<=1/2 (2017 Auckland MO S2)

Source:

11/15/2021
The sum of the three nonnegative real numbers x1,x2,x3 x_1, x_2, x_3 is not greater than 12\frac12. Prove that (1−x1)(1−x2)(1−x3)≥12(1 - x_1)(1 - x_2)(1 - x_3) \ge \frac12
algebrainequalities