## IMC 2014 Day 2 Problem 5

For every positive integer , denote by the number of permutations of such that for every . For , denote by the number of permutations of such that for every and for every . Prove that

**IMC 2014 Day 2 Problem 5**

## IMC 2014 Day 2 Problem 4

We say that a subset of is -*almost contained* by a hyperplane if there are less than points in that set which do not belong to the hyperplane. We call a finite set of points -*generic* if there is no hyperplane that -almost contains the set. For each pair of positive integers and , find the minimal number such that every finite -generic set in contains a -generic subset with at most elements.

**IMC 2014 Day 2 Problem 4**

## IMC 2014 Day 2 Problem 3

Let , for and let be a positive integer. Prove that , where denotes the -th derivative of .

**IMC 2014 Day 2 Problem 3**

## IMC 2014 Day 2 Problem 2

Let be a symmetric matrix with real entries, and let denote its eigenvalues. Show that

and determine all matrices for which equality holds.

**IMC 2014 Day 2 Problem 2**

## IMC 2014 Day 2 Problem 1

For a positive integer denote its -th decimal digit by , i.e. and . Suppose that for some sequence there are only finitely many zeros in the sequence . Prove that there are infinitely many positive integers that do not occur in the sequence .

**IMC 2014 Day 2 Problem 1**

## IMC 2014 Day 1 Problem 5

Let be a close broken line consisting of line segments in the Euclidean plane. Suppose that no three of its vertices are collinear and for each index , the triangle has counterclockwise orientation and , using the notation modulo . Prove that the number of self-intersections of the broken line is at most .

**IMC 2014 Day 1 Problem 5**

## IMC 2014 Day 1 Problem 4

Let be a perfect number and be its prime factorization with . Prove that is an even number.

A number is *perfect* if , where is the sum of the divisors of .

**IMC 2014 Day 1 Problem 4**