The positive integer $N$ is said amiable if the set $\{1,2,\ldots,N\}$ can be partitioned into pairs of elements, each pair having the sum of its elements a perfect square. Prove there exist infinitely many amiable numbers which are themselves perfect squares. (Dan Schwarz)

Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$. (Dan Schwarz)

For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality $$ \left | \dfrac {a} {b-c} \right | + \left | \dfrac {b} {c-a} \right | + \left | \dfrac {c} {a-b} \right | \geq 2$$ and determine all cases of equality. Prove that if we also impose $a,b,c$ positive, then all equality cases disappear, but the value $2$ remains the best constant possible. (Dan Schwarz)

The cells of some rectangular $M \times n$ array are colored, each by one of two colors, so that for any two columns the number of pairs of cells situated on a same row and bearing the same color is less than the number of pairs of cells situated on a same row and bearing different colors. i) Prove that if $M=2011$ then $n \leq 2012$ (a model for the extremal case $n=2012$ does indeed exist, but you are not asked to exhibit one). ii) Prove that if $M=2011=n$, each of the colors appears at most $1006\cdot 2011$ times, and at least $1005\cdot 2011$ times. iii) Prove that if however $M=2012$ then $n \leq 1007$. (Dan Schwarz)

Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ so that $AQ=PQ$ and $\angle PAQ = 45^{\circ}$. (Dan Schwarz)

Prove the value of the expression $$\displaystyle \dfrac {\sqrt{n + \sqrt{0}} + \sqrt{n + \sqrt{1}} + \sqrt{n + \sqrt{2}} + \cdots + \sqrt{n + \sqrt{n^2-1}} + \sqrt{n + \sqrt{n^2}}} {\sqrt{n - \sqrt{0}} + \sqrt{n - \sqrt{1}} + \sqrt{n - \sqrt{2}} + \cdots + \sqrt{n - \sqrt{n^2-1}} + \sqrt{n - \sqrt{n^2}}}$$ is constant over all positive integers $n$. (Folklore (also Philippines Olympiad))

For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality $$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | \geq 2$$ and determine all cases of equality. Prove that if we also impose $a,b,c \geq 0$, then $$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | > 3,$$ with the value $3$ being the best constant possible. (Dan Schwarz)

Consider a set $X$ with $|X| = n\geq 1$ elements. A family $\mathcal{F}$ of distinct subsets of $X$ is said to have property $\mathcal{P}$ if there exist $A,B \in \mathcal{F}$ so that $A\subset B$ and $|B\setminus A| = 1$. i) Determine the least value $m$, so that any family $\mathcal{F}$ with $|\mathcal{F}| > m$ has property $\mathcal{P}$. ii) Describe all families $\mathcal{F}$ with $|\mathcal{F}| = m$, and not having property $\mathcal{P}$. (Dan Schwarz)