MathDB

1

Part of 2024 Caucasus Mathematical Olympiad

Problems(2)

Boxes and colored balls

Source: Caucasus MO 2024, Juniors P1

3/15/2024
Balls of 33 colours — red, blue and white — are placed in two boxes. If you take out 33 balls from the first box, there would definitely be a blue one among them. If you take out 44 balls from the second box, there would definitely be a red one among them. If you take out any 55 balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.
combinatorics
Easy algebra on 4 positive reals

Source: Caucasus MO 2024, Seniors P1

3/15/2024
Let a,b,c,da, b, c, d be positive real numbers. It is given that at least one of the following two conditions holds: ab>min⁡(cd,dc),cd>min⁡(ab,ba).ab >\min(\frac{c}{d}, \frac{d}{c}), cd >\min(\frac{a}{b}, \frac{b}{a}). Show that at least one of the following two conditions holds: bd>min⁡(ca,ac),ca>min⁡(db,bd).bd>\min(\frac{c}{a}, \frac{a}{c}), ca >\min(\frac{d}{b}, \frac{b}{d}).
algebra