CMO 2002

Let $S$ be a subset of $\{1,2, \ldots, 9\}$, such that the sums formed by adding each unordered pair of distinct numbers from $S$ are all different. For example, the subset $\{1,2,3,5\}$ has this property, but $\{1,2,3,4,5\}$ does not, since the pairs $\{1,4\}$ and $\{2,3\}$ have the same sum, namely 5 .

What is the maximum number of elements that $S$ can contain?

Call a positive integer $n$ practical if every positive integer less than or equal to $n$ can be written as the sum of distinct divisors of $n$.

For example, the divisors of 6 are $\mathbf{1 , 2}, \mathbf{3}$, and $\mathbf{6}$. Since

$$1=\mathbf{1}, 2=\mathbf{2}, 3=\mathbf{3}, 4=\mathbf{1}+\mathbf{3}, 5=\mathbf{2}+\mathbf{3}, 6=\mathbf{6},$$

we see that 6 is practical.

Prove that the product of two practical numbers is also practical.

Prove that for all positive real numbers $a, b$, and $c$,

$$\frac{a^{3}}{b c}+\frac{b^{3}}{c a}+\frac{c^{3}}{a b} \geq a+b+c$$

and determine when equality occurs.

Each of the inequalities used in the solutions below has the property that equality holds if and only if $a=b=c$. Thus equality holds for the given inequality if and only if $a=b=c$.

Let $\Gamma$ be a circle with radius $r$. Let $A$ and $B$ be distinct points on $\Gamma$ such that $A B<\sqrt{3} r$. Let the circle with centre $B$ and radius $A B$ meet $\Gamma$ again at $C$. Let $P$ be the point inside $\Gamma$ such that triangle $A B P$ is equilateral. Finally, let $C P$ meet $\Gamma$ again at $Q$. Prove that $P Q=r$.

Let $\mathbb{N}=\{0,1,2, \ldots\}$. Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that

$$x f(y)+y f(x)=(x+y) f\left(x^{2}+y^{2}\right)$$

for all $x$ and $y$ in $\mathbb{N}$.