MathDB

2

Part of 2016 IMC

Problems(2)

IMC 2016, Problem 2

Source: IMC 2016

7/27/2016
Let kk and nn be positive integers. A sequence (A1,…,Ak)\left( A_1, \dots , A_k \right) of n×nn\times n real matrices is preferred by Ivan the Confessor if Ai2≠0A_i^2\neq 0 for 1≤i≤k1\le i\le k, but AiAj=0A_iA_j=0 for 1≤i1\le i, j≤kj\le k with i≠ji\neq j. Show that k≤nk\le n in all preferred sequences, and give an example of a preferred sequence with k=nk=n for each nn.
(Proposed by Fedor Petrov, St. Petersburg State University)
IMCIMC 2016Matriceslinear algebracollege contests
IMC 2016, Problem 7

Source: IMC 2016

7/28/2016
Today, Ivan the Confessor prefers continuous functions f:[0,1]→Rf:[0,1]\to\mathbb{R} satisfying f(x)+f(y)≥∣x−y∣f(x)+f(y)\geq |x-y| for all pairs x,y∈[0,1]x,y\in [0,1]. Find the minimum of ∫01f\int_0^1 f over all preferred functions.
(Proposed by Fedor Petrov, St. Petersburg State University)
IMCIMC 2016college contestsfunctionminimum value