SMO 2009 Open Round 2

Let $O$ be the centre of the circle inscribed in a rhombus $ABCD$. Points $E,F,G,H$ are chosen on sides $AB,BC,CD,DA$, respectively, so that $EF$ and $GH$ are tangent to the inscribed circle. Show that $EH$ and $FG$ are parallel.

Editorial qualification: the source proof uses interior side points and nondegenerate tangents cutting off the corners $B,D$. If endpoints and tangents coinciding with sides are allowed, the literal statement can fail. The original wording is preserved; the intended-domain proof and an endpoint counterexample are given below. This question requires review.

A palindromic number is a number unchanged when the order of its digits is reversed. Prove that the arithmetic progression $18,37,\ldots$ contains infinitely many palindromic numbers.

For a positive integer $k$, define $A_n$ for $n=1,2,\ldots$ by
$$A_{n+1}=\frac{nA_n+2(n+1)^{2k}}{n+2},\qquad A_1=1.$$
Prove that $A_n$ is an integer for all $n\ge1$, and that $A_n$ is odd if and only if $n\equiv1$ or $2\pmod4$.

Find the largest constant $C$ such that
$$\sum_{i=1}^4\left(x_i+\frac1{x_i}\right)^3\ge C$$
for all positive real numbers $x_1,\ldots,x_4$ satisfying
$$x_1^3+x_3^3+3x_1x_3=x_2+x_4=1.$$

Find all integers $x,y,z$ with $2\le x\le y\le z$ such that
$$xy\equiv1\pmod z,\qquad xz\equiv1\pmod y,\qquad yz\equiv1\pmod x.$$