MathDB

9.8

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(2)

card with 10 numbered cells in a lottery (III Soros Olympiad 1996-97 R1 9.8)

Source:

5/29/2024
Some lottery is played as follows. A lottery participant buys a card with 1010 numbered cells. He has the right to cross out any 44 of these 1010 cells. Then a drawing occurs, during which some 77 out of 1010 cells become winning. The player wins when all 44 squares he crosses out are winning. The question arises, what is the smallest number of cards that can be used so that, if filled out correctly, at least one of these cards will win in any case? We do not suggest that you answer this question (we ourselves do not know the answer), although, of course, we will be very glad if you do and will evaluate this achievement accordingly. The task is; to indicate a certain number nn and a method of filling n cards that guarantees at least one win. The smaller nn, the higher the rating of the work.
combinatorics
min max cos in triangle with sides 1-x (III Soros Olympiad 1996-97 R3 9.8)

Source:

5/31/2024
The two sides of the triangle are equal to 11 and xx, and x≥1 x \ge 1. The values aa and bb are the largest and smallest angles of this triangle, respectively. Find the greatest value of cos⁡a\cos a and the smallest value of cos⁡b\cos b.
trigonometryalgebrageometric inequality