APMO 1989

Let $x_{1}, x_{2}, \ldots, x_{n}$ be positive real numbers, and let

$$S=x_{1}+x_{2}+\cdots+x_{n}$$

Prove that

$$\left(1+x_{1}\right)\left(1+x_{2}\right) \cdots\left(1+x_{n}\right) \leq 1+S+\frac{S^{2}}{2!}+\frac{S^{3}}{3!}+\cdots+\frac{S^{n}}{n!}$$

Prove that the equation

$$6\left(6 a^{2}+3 b^{2}+c^{2}\right)=5 n^{2}$$

has no solutions in integers except $a=b=c=n=0$.

Let $A_{1}, A_{2}, A_{3}$ be three points in the plane, and for convenience,let $A_{4}=A_{1}, A_{5}=A_{2}$. For $n=1,2$, and 3 , suppose that $B_{n}$ is the midpoint of $A_{n} A_{n+1}$, and suppose that $C_{n}$ is the midpoint of $A_{n} B_{n}$. Suppose that $A_{n} C_{n+1}$ and $B_{n} A_{n+2}$ meet at $D_{n}$, and that $A_{n} B_{n+1}$ and $C_{n} A_{n+2}$ meet at $E_{n}$. Calculate the ratio of the area of triangle $D_{1} D_{2} D_{3}$ to the area of triangle $E_{1} E_{2} E_{3}$.

Let $S$ be a set consisting of $m$ pairs $(a, b)$ of positive integers with the property that $1 \leq a<$ $b \leq n$. Show that there are at least

$$4 m \frac{\left(m-\frac{n^{2}}{4}\right)}{3 n}$$

triples $(a, b, c)$ such that $(a, b),(a, c)$, and $(b, c)$ belong to $S$.

Determine all functions $f$ from the reals to the reals for which (1) $f(x)$ is strictly increasing, (2) $f(x)+g(x)=2 x$ for all real $x$, where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x))=x$ and $g(f(x))=x$ for all real x.)