[Tony]

British Mathematical Olympiad
Round 2 : Tuesday, 1 February 2005

1. The integer N is positive. There are exactly 2005 ordered pairs (x,y)
of positive integers satisfying
\[ \frac{1}{x}+\frac{1}{y} = \frac{1}{N} \]
Prove that N is a perfect square.

2. In triangle ABC,ะBAC=120ฐ. Let the angle bisectors of angles
A;B and C meet the opposite sides in D;E and F respectively.
Prove that the circle on diameter EF passes through D.

3. 3. Let a; b; c be positive real numbers. Prove that
\[ \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^2 ณ \left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \]

4. Let X = {A₁,A₂ ... A_n} be a set of distinct 3-element subsets of { 1,2, ... , 36 } such that
i ) A_i and A_j have non-empty intersection for every i, j.
ii ) The intersection of all the elements of X is the empty sets
Show that nฃ100. How many such sets X are there when n = 100 ?

ใช้เวลา 3 ชั่วโมงครึ่ง และแต่ละข้อ 10 คะแนน