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Part of 2015 International Zhautykov Olympiad

Problems(2)

triangle with area n

Source: IZHO2015.P1

Each point with integral coordinates in the plane is coloured white or blue. Prove that one can choose a colour so that for every positive integer

n

there exists a triangle of area

n

having its vertices of the chosen colour.

geometryanalytic geometrycombinatorics proposedcombinatorics

determine the maximum n

Source: IZHO2015 P4

Determine the maximum integer

n

such that for each positive integer

k \le \frac{n}{2}

there are two positive divisors of

n

with difference

k

floor functionnumber theoryleast common multiplenumber theory unsolved