MathDB

3

Part of 2001 Polish MO Finals

Problems(2)

Fibonaccis Sequence

Source: Poland 2001

11/9/2004
A sequence x0=Ax_0=A and x1=Bx_1=B and xn+2=xn+1+xnx_{n+2}=x_{n+1} +x_n is called a Fibonacci type sequence. Call a number CC a repeated value if xt=xs=cx_t=x_s=c for tt different from ss. Prove one can choose AA and BB to have as many repeated value as one likes but never infinite.
algebra unsolvedalgebra
partition of a set

Source:

11/25/2005
Given positive integers n1<n2<...<n2000<10100n_1<n_2<...<n_{2000}<10^{100}. Prove that we can choose from the set {n1,...,n2000}\{n_1,...,n_{2000}\} nonempty, disjont sets AA and BB which have the same number of elements, the same sum and the same sum of squares.
quadraticspigeonhole principlecombinatorics proposedcombinatorics