MathDB

4

Part of 1991 Bundeswettbewerb Mathematik

Problems(2)

tiling a strip of width 1 with rectangles

Source: 1991 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p4

11/20/2022
A strip of width 11 is to be divided by rectangular panels of common width 11 and denominations long a1a_1, a2a_2, a3a_3, .... . . be paved without gaps (a1≠1a_1 \ne 1). From the second panel on, each panel is similar but not congruent to the already paved part of the strip. When the first nn slabs are laid, the length of the paved part of the strip is snsn. Given a1a_1, is there a number that is not surpassed by any sns_n? The accuracy answer has to be proven.
combinatoricscombinatorial geometrygeometryrectangle
non-negative elements

Source: Bundeswettbewerb Mathematik 1991, round two, problem 4

6/19/2004
Given wo non-negative integers aa and bb, one of them is odd and the other one even. By the following rule we define two sequences (an),(bn)(a_n),(b_n): a_0 = a,   a_1 = b,   a_{n+1} = 2a_n - a_{n-1} + 2   (n = 1,2,3, \ldots) b_0 = b,   b_1 = a,   b_{n+1} = 2a_n - b_{n-1} + 2   (n = 1,2,3, \ldots) Prove that none of these two sequences contain a negative element if and only if we have ∣a−b∣≤1|\sqrt{a} - \sqrt{b}| \leq 1.
algebra unsolvedalgebra