SMO 2007 Senior Round 2

It is given that $x,y,z$ are three real numbers such that

$$\frac{x-y}{2+xy}+\frac{y-z}{2+yz}+\frac{z-x}{2+zx}=0.$$

Is it true that at least two of the three numbers must be equal? Justify your answer.

For any positive integer $n$, let $f(n)$ denote the $n$th positive nonsquare integer, i.e., $f(1)=2$, $f(2)=3$, $f(3)=5$, $f(4)=6$, etc. Prove that

$$f(n)=n+\{\sqrt n\},$$

where $\{x\}$ denotes the integer closest to $x$. (For example, $\{\sqrt1\}=1$, $\{\sqrt2\}=1$, $\{\sqrt3\}=2$, $\{\sqrt4\}=2$.)

In the equilateral triangle $ABC$, $M,N$ are the midpoints of the sides $AB,AC$, respectively. The line $MN$ intersects the circumcircle of $\triangle ABC$ at $K$ and $L$ and the lines $CK$ and $CL$ meet the line $AB$ at $P$ and $Q$, respectively. Prove that $PA^2\cdot QB=QA^2\cdot PB$.

Thirty-two pairs of identical twins are lined up in an $8\times8$ formation. Prove that it is possible to choose $32$ persons, one from each pair of twins, so that there is at least one chosen person in each row and in each column.

Find the maximum and minimum of $x+y$ such that

$$x+y=\sqrt{2x-1}+\sqrt{4y+3}.$$