2

Part of 2012 China National Olympiad

Problems(2)

Source: 2012cmo,problem5

Consider a square-free even integer

n

p

(n,p)=1

p\le 2\sqrt{n}

; 3) There exists an integer

k

p|n+k^2

. Prove that there exists pairwise distinct positive integers

a,b,c

n=ab+bc+ca

.
Proposed by Hongbing Yu

modular arithmeticinequalitiesnumber theory proposednumber theory

Matrices that can be converted to 0 by certain operations

Source: 2012 China Mathematical Olympaid P2

p

be a prime. We arrange the numbers in

{\{1,2,\ldots ,p^2} \}

p \times p

A = ( a_{ij} )

. Next we can select any row or column and add

1

to every number in it, or subtract

1

from every number in it. We call the arrangement good if we can change every number of the matrix to

0

in a finite number of such moves. How many good arrangements are there?

linear algebramatrixgeometrygeometric transformationsymmetryalgebrapolynomial