3

Part of 2009 ITAMO

Problems(2)

find all nice ones

Source: ItaMO 2009, p3

A natural number

n

is called nice if it enjoys the following properties: • The expression is made up of

4

decimal digits; • the first and third digits of

n

are equal; • the second and fourth digits of

n

are equal; • the product of the digits of

n

n^2

. Determine all nice numbers.

number theory proposednumber theory

number of k which are n-squared

Source: ItaMO 2009, p6

A natural number

k

n

-squared if by colouring the squares of a

2n \times k

chessboard, in any manner, with

n

different colours, we can find

4

separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of

n

, the smallest natural

k

n

geometryrectanglefunctioncombinatorics unsolvedcombinatorics