DE-MO 2007

Determine all real numbers $x$ such that for all positive integers $n$ the inequality $(1+x)^n \leq 1+(2^n -1)x$ is true.

Let $A$ be the set of odd integers $\leq 2n-1.$ For a positive integer $m$ , let $B=\{a+m\,| a\in A \}.$ Determine for which positive integers $n$ there exists a positive integer $m$ such that the product of all elements in $A$ and $B$ is a square.

We say that two triangles are oriented similarly if they are similar and have the same orientation. Prove that if $ALT, ARM, ORT,$ and $ULM$ are four triangles which are oriented similarly, then $A$ is the midpoint of the line segment $OU.$

Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.

Determine all finite sets $M$ of real numbers such that $M$ contains at least $2$ numbers and any two elements of $M$ belong to an arithmetic progression of elements of $M$ with three terms.