MathDB

2

Part of 2007 IMC

Problems(2)

Minimal and maximal possible rank of a matrix

Source: IMC 2007, Day 1, Problem 2

8/6/2007
Let n≥2 n\ge 2 be an integer. What is the minimal and maximal possible rank of an n×n n\times n matrix whose n2 n^{2} entries are precisely the numbers 1,2,…,n2 1, 2, \ldots, n^{2}?
linear algebramatrixinductionIMCcollege contests
Sum of fourth powers divisible by 29

Source: IMC 2007 Day 2 Problem 2

8/7/2007
Let x x, y y and z z be integers such that S=x4+y4+z4 S = x^{4}+y^{4}+z^{4} is divisible by 29. Show that S S is divisible by 294 29^{4}.
modular arithmeticalgebrapolynomialIMCcollege contests