MathDB

5

Part of 2012 Czech-Polish-Slovak Junior Match

Problems(2)

1/2 (c - a) (c - b) is perfect square for positive integers such a^2 + b^2 = c^2

Source: Czech-Polish-Slovak Junior Match 2012, Individual p5 CPSJ

3/11/2020
Positive integers a,b,ca, b, c satisfying the equality a2+b2=c2a^2 + b^2 = c^2. Show that the number 12(c−a)(c−b)\frac12(c - a) (c - b) is square of an integer.
number theoryPythagorean TriplePerfect Square
k + a^k = m + 2a^m, diophantine in Z+

Source: Czech-Polish-Slovak Junior Match 2012, Team p5 CPSJ

3/12/2020
Find all triplets (a,k,m)(a, k, m) of positive integers that satisfy the equation k+ak=m+2amk + a^k = m + 2a^m.
Diophantine equationdiophantinenumber theory