SMO 2010 Senior Round 2

In the triangle $ABC$ with $AC\gt AB$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of the perpendicular from $D$ onto $AC$. Let $F$ be the point on the line $DE$ such that $EF\cdot DC=BD\cdot DE$. Prove that $AF$ is perpendicular to $BF$.

The numbers $\frac11,\frac12,\ldots,\frac1{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x,y$, erases them and then writes down $x+y+xy$. He continues to do this until only one number is left on the blackboard. What is this number?

Given $a_1\ge1$ and $a_{k+1}\ge a_k+1$ for $k=1,2,\ldots,n$, show that $$a_1^3+a_2^3+\cdots+a_n^3\ge(a_1+a_2+\cdots+a_n)^2.$$

An infinite sequence of integers, $a_0,a_1,a_2,\ldots$, with $a_0\gt 0$, has the property that for every $n\ge0$, $a_{n+1}=a_n-b_n$, where $b_n$ is the number having the same sign as $a_n$ but having the digits written in reverse order. For example, if $a_0=1210$, then $a_1=1089$ and $a_2=-8712$, etc. Find the smallest value of $a_0$ so that $a_n\ne0$ for all $n\ge1$.

Let $p$ be a prime number and let $a_1,a_2,\ldots,a_k$ be distinct integers chosen from $1,2,\ldots,p-1$. For $1\le i\le k$, let $r_i^{(n)}$ denote the remainder of $na_i$ upon division by $p$, so $0\le r_i^{(n)}\lt p$. Define

$$S=\{n:1\le n\le p-1,\ r_1^{(n)}\lt \cdots\lt r_k^{(n)}\}.$$

Show that $S$ has fewer than $\frac{2p}{k+1}$ elements.