MathDB

4

Part of 2014 IMC

Problems(2)

IMC 2014, Problem 4

Source: IMC 2014

7/27/2016
Let n>6n>6 be a perfect number, and let n=p1e1⋅⋅⋅pkekn=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k} be its prime factorisation with 1<p1<⋯<pk1<p_1<\dots <p_k. Prove that e1e_1 is an even number. A number nn is perfect if s(n)=2ns(n)=2n, where s(n)s(n) is the sum of the divisors of nn.
(Proposed by Javier Rodrigo, Universidad Pontificia Comillas)
IMCcollege contestsnumber theoryprime factorization
IMC 2014, Problem 9

Source: IMC 2014

7/27/2016
We say that a subset of Rn\mathbb{R}^n is kk-almost contained by a hyperplane if there are less than kk points in that set which do not belong to the hyperplane. We call a finite set of points kk-generic if there is no hyperplane that kk-almost contains the set. For each pair of positive integers (k,n)(k, n), find the minimal number of d(k,n)d(k, n) such that every finite kk-generic set in Rn\mathbb{R}^n contains a kk-generic subset with at most d(k,n)d(k, n) elements.
(Proposed by Shachar Carmeli, Weizmann Inst. and Lev Radzivilovsky, Tel Aviv Univ.)
IMCcollege contestsset theory