MathDB

2

Part of 2013 Pan African

Problems(2)

Functional: f(x)f(y)+f(x+y)=xy

Source: Pan African MO 2013 Q2

6/30/2013
Find all functions f:R→Rf:\mathbb{R}\to\mathbb{R} such that f(x)f(y)+f(x+y)=xyf(x)f(y)+f(x+y)=xy for all real numbers xx and yy.
functionalgebra unsolvedalgebrafunctional equation
Every 3x3 square of an nxn board has such a colouring

Source: Pan African MO 2013 Q5

6/30/2013
The cells of an n×nn\times n board with n≥5n\ge 5 are coloured black or white so that no three adjacent squares in a row, column or diagonal are the same colour. Show that for any 3×33\times 3 square within the board, two of its corner squares are coloured black and two are coloured white.
symmetrycombinatorics unsolvedcombinatorics