OBM 2005

A natural number is a palindrome when one obtains the same number when writing its digits in reverse order. For example, $481184$ , $131$ and $2$ are palindromes.

Determine all pairs $(m,n)$ of positive integers such that $\underbrace{111\ldots 1}_{m\ {\rm ones}}\times\underbrace{111\ldots 1}_{n\ {\rm ones}}$ is a palindrome.

Determine the smallest real number $C$ such that the inequality $$C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16}$$ holds for all positive real numbers $x_1,x_2,x_3,x_4,x_5$ .

A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$ .

We have four charged batteries, four uncharged batteries and a radio which needs two charged batteries to work.

Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batteries in the radio and check if the radio works or not.

Given positive integers $a,c$ and integer $b$ , prove that there exists a positive integer $x$ such that $$a^x + x \equiv b \pmod c,$$ that is, there exists a positive integer $x$ such that $c$ is a divisor of $a^x + x - b$ .