Problem 3 is interesting...
Not too hard I think,
that is, after someone showed me the answer.
※ 引述《boggart0803 (幻形怪)》之銘言：
: 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)