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
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
Let , for and let be a positive integer. Prove that , where denotes the -th derivative of .
IMC 2014 Day 2 Problem 3
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
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
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