4

Part of 1972 Bundeswettbewerb Mathematik

Problems(2)

Greatest integer function

Source:

Which natural numbers cannot be presented in that way:

[n+\sqrt{n}+\frac{1}{2}]

n\in\mathbb{N}

[y]

is the greatest integer function.

functionalgebrafloor function

p players, n games, out of 3, 2 haven't played => n<=p²/4

Source: Bundeswettbewerb Mathematik 1972, round 2, problem 4

p>2

persons participate at a chess tournament, two players play at most one game against each other. After

n

games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other. Show that

n \leq \frac{p^{2}}4

combinatorics proposedcombinatorics