MathDB

1

Part of 1989 Iran MO (2nd round)

Problems(2)

Floor Problem and the equation - Iran 1989 Second Round

Source:

12/6/2010
(a) Let nn be a positive integer, prove that n+1−n<12n \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}
(b) Find a positive integer nn for which ⌊1+12+13+14+⋯+1n⌋=12 \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12
floor functionfunctionlimitinequalities proposedinequalities
The old problem of tournament - [Iran Second Round 1989]

Source:

12/6/2010
In a sport competition, mm teams have participated. We know that each two teams have competed exactly one time and the result is winning a team and losing the other team (i.e. there is no equal result). Prove that there exists a team xx such that for each team y,y, either xx wins yy or there exists a team zz for which xx wins zz and zz wins y.y.
[i.e. prove that in every tournament there exists a king.]
combinatorics proposedcombinatorics