MathDB

2

Part of 1990 Bundeswettbewerb Mathematik

Problems(2)

perfect squares sequence, a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)

Source: Germany Federal - Bundeswettbewerb Mathematik 1990, round 1, p2

2/21/2020
The sequence a0,a1,a2,...a_0,a_1,a_2,... is defined by a0=0,a1=a2=1a_0 = 0, a_1 = a_2 = 1 and an+2+an−1=2(an+1+an)a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n) for all n∈Nn \in N. Show that all ana_n are perfect squares .
SequencePerfect Squarerecurrence relationalgebra
A(n) =[(n+1)/2] [(n+2)/2} , distinct points in the plane

Source: Germany Federal - Bundeswettbewerb Mathematik 1990, round 2, p2

2/21/2020
Let A(n)A(n) be the least possible number of distinct points in the plane with the following property: For every k=1,2,...,nk = 1,2,...,n there is a line containing precisely kk of these points. Show that A(n)=[n+12][n+22]A(n) =\left[\frac{n+1}{2}\right] \left[\frac{n+2}{2}\right]
combinatorial geometrycombinatorics