Problem 1
Let H be the orthocenter of an acute-angled triangle ABC.
The circle G_A centered at the midpoint of BC and passing
through H intersects the sideline BC at points A_1 and A_2.
Similarly, define the points B_1, B_2, C_1, C_2.
Prove that six points A_1, A_2, B_1, B_2, C_1, C_2 are concyclic.
Problem 2
(i) If x, y and z are three real numbers, all different
from 1, such that xyz=1, then prove that Σ(x^2/(x-1)^2)>=1
(ii) Prove that equality is achieved for infinitely many
triples of rational numbers x, y and z.
Problem 3
Prove that there are infinitely many positive integers n
such that n^2+1 has a prime divisor greater than 2n+sqrt(2n)