MathDB

1991 Bundeswettbewerb Mathematik

Part of Bundeswettbewerb Mathematik

Subcontests

(4)
1
2
4
2

tiling a strip of width 1 with rectangles

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.

non-negative elements

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.
2
2

f(n) = g^n + 1 where g is even positive integer

Let gg be an even positive integer and f(n)=gn+1f(n) = g^n + 1 , (n∈N∗)(n \in N^* ). Prove that for every positive integer nn we have: a) f(n)f(n) divides each of the numbers f(3n),f(5n),f(7n)f(3n), f(5n), f(7n) b) f(n)f(n) is relative prime to each of the numbers f(2n),f(4n),f(6n),...f(2n), f(4n),f(6n),...

17 blue connecting segments

In the space there are 8 points that no four of them are in the plane. 17 of the connecting segments are coloured blue and the other segments are to be coloured red. Prove that this colouring will create at least four triangles. Prove also that four cannot be subsituted by five. Remark: Blue triangles are those triangles whose three edges are coloured blue.
3
2