MathDB

1

Part of 2023 Rioplatense Mathematical Olympiad

Problems(2)

Pythagorean cycle

Source: Rioplatense L2 2023 #1

12/6/2023
An integer n≥3n\geq 3 is poli-pythagorean if there exist nn positive integers pairwise distinct such that we can order these numbers in the vertices of a regular nn-gon such that the sum of the squares of consecutive vertices is also a perfect square. For instance, 33 is poli-pythagorean, because if we write 44,117,24044,117,240 in the vertices of a triangle we notice: 442+1172=1252,1172+2402=2672,2402+442=244244^2+117^2=125^2, 117^2+240^2=267^2, 240^2+44^2=244^2 Determine all poli-pythagorean integers.
number theorycombinatorics
Mixing squares with prime

Source: Rioplatense L3 2023 #1

12/6/2023
Determine all triples (x,y,p)(x,y,p) of positive integers such that pp is prime, p=x2+1p=x^2+1 and 2p2=y2+12p^2=y^2+1.
number theory