4

Part of 2003 Irish Math Olympiad

Problems(2)

Chess Tournament

Source: 2003 IrMO Paper 1 Problem 4

Eight players, Ann, Bob, Con, Dot, Eve, Fay, Guy and Hal compete in a chess tournament. No pair plays together more than once and there is no group of five people in which each one plays against all of the other four.
(a) Write down an arrangement for a tournament of

24

games satisfying these conditions.
(b) Show that it is impossible to have a tournament of more than

24

games satisfying these conditions.

Source: 16th Irish mo , 2003

Given real positive a,b , find the larget real c such that

c\leq max(ax+\frac{1}{ax},bx+\frac{1}{bx})

for all positive ral x. There is a solution here,,,, http://www.kalva.demon.co.uk/irish/soln/sol039.html but im wondering if there is a better one . Thank you.

inequalitiesinequalities proposed