MathDB

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Part of 2006 Poland - Second Round

Problems(2)

system of inequalities and equations

Source: Polish second round 2006

2/24/2006
Positive integers a,b,c,x,y,za,b,c,x,y,z satisfy: a2+b2=c2a^2+b^2=c^2, x2+y2=z2x^2+y^2=z^2 and ∣x−a∣≤1|x-a| \leq 1 , ∣y−b∣≤1|y-b| \leq 1. Prove that sets {a,b}\{a,b\} and {x,y}\{x,y\} are equal.
inequalitiessymmetryalgebra proposedalgebra
bounded and periodic sequence

Source:

2/25/2006
Let cc be fixed natural number. Sequence (an)(a_n) is defined by: a1=1a_1=1, an+1=d(an)+ca_{n+1}=d(a_n)+c for n=1,2,...n=1,2,.... where d(m)d(m) is number of divisors of mm. Prove that there exist kk natural such that sequence ak,ak+1,...a_k,a_{k+1},... is periodic.
pigeonhole principleLaTeXnumber theory proposednumber theory