2

Part of 2011 Irish Math Olympiad

Problems(2)

Median Inequalities

Source: 2011 IrMO Paper 1 Problem 2

ABC

be a triangle whose side lengths are, as usual, denoted by

a=|BC|,

b=|CA|,

c=|AB|.

m_a,m_b,m_c

, respectively, the lengths of the medians which connect

A,B,C

, respectively, with the centers of the corresponding opposite sides.
(a) Prove that

2m_a<b+c

m_a+m_b+m_c<a+b+c

. (b) Give an example of (i) a triangle in which

m_a>\sqrt{bc}

; (ii) a triangle in which

m_a\le \sqrt{bc}

geometryinequalities

Combinatorics question

Source:

In a tournament with

n

n

< 10, each player plays once against each other player scoring 1 point for a win and 0 points for a loss. Draws do not occur. In a particular tournament only one player ended with an odd number of points and was ranked fourth. Determine whether or not this is possible. If so, how many wins did the player have?