3

Part of 2010 May Olympiad

Problems(2)

consecutive integers such that sum of their squares is a square

Source: XVI May Olympiad (Olimpiada de Mayo) 2010 L2 P3

Find the minimum

k>2

for which there are

k

consecutive integers such that the sum of their squares is a square.

number theoryconsecutivePerfect Square

coloring positive integers into 3 colors

Source: XVΙ May Olympiad (Olimpiada de Mayo) 2010 L1 P3

Is it possible to color positive integers with three colors so that whenever two numbers with different colors are added, the result of their addition is the third color? (All three colors must be used.) If the answer is yes, indicate a possible coloration; if not, explain why.

Coloringcombinatorics