Singapore Math Olympiad Senior Section Rd 2

I think I'm quite screwed.

1. (x - y) / (2 + xy) + (y - z) / (2 + yz) + (z - x) / (2 + zx) = 0. Must two out of the three numbers be the same? Justify your answer.
Can do. But it wasted about 1 of my 3 hours on the paper.

2. f(n) accepts positive integers and returns the nth nonsquare positive integer. Prove that f(n) = n + {sqrt n} where {n} is the nearest positive integer.
Well... Easy. Mathematical Induction + defining the set of numbers before {sqrt n} increases.

3. Equilateral triangle ABC, midpoints of AB, AC are M,N respectively. MN intersects with the circumcircle such that the intersections are P and Q. Prove PA^2 x QB = QA^2 x PB.
Interesting. I constructed it but definitely did not get PA^2 x QB = QA^2 x PB. I got something like PA^2 x QB = 1.8 x QA^2 x PB, so I probably got this wrong.

4. There are 64 identical twins arranged in an 8x8 formation. Show that it is possible to select 32 people, one from each pair of twins, s.t. there is at least 1 in each row and column.
I did not really understand this question, as there has to be a reason why they said 32 people and not 8. When I asked the invigilator, he said he'd come back to me wrt that question... and nothing happened after that.

5. Find the maximum x + y such that x + y = SQRT (2x - 1) + SQRT (4y + 3) or something like that.
I couldn't get anywhere with this one. I wrote about two pages on treating this as a parametric equation as well as implicit differentiation but then realised that getting a maximum x or a maximum y may not result in getting the maximum x + y. So I was stuck.

jk