MathDB

3

Part of 1984 Bundeswettbewerb Mathematik

Problems(2)

a^2 + b^2 + c^2 = d^2 if ab is even or odd

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

11/21/2022
Let aa and bb be positive integers. Show that if a⋅ba \cdot b is even, then there are positive integers cc and dd with a2+b2+c2=d2a^2 + b^2 + c^2 = d^2; if, on the other hand, a⋅ba\cdot b is odd, there are no such positive integers cc and dd.
number theoryPerfect Square
a_{n+1} = a_n - b_n and b_{n+1} = 2b_n if a_n >=b_n, or else ...

Source: 1984 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

11/21/2022
The sequences a1,a2,a3,...a_1, a_2, a_3,... and b1,b2,b3,...b_1, b_2, b_3,... suffices for all positive integers nn of the following recursion: an+1=an−bna_{n+1} = a_n - b_n and bn+1=2bnb_{n+1} = 2b_n, if an≥bna_n \ge b_n, an+1=2ana_{n+1} = 2a_n and bn+1=bn−anb_{n+1} = b_n - a_n, if an<bna_n < b_n. For which pairs (a1,b1)(a_1, b_1) of positive real initial terms is there an index kk with ak=0a_k = 0?
algebrarecurrence relationSequences