4

Part of 1992 Bundeswettbewerb Mathematik

Problems(2)

i+a for i = 1, 2, 3, ..., k$ is even for set {a_i}

Source: 1992 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p4

\{a_1, a_2, ... a_k\}

of positive integers with

a_1 < a_2 < a_3 < ... < a_k

is named alternating if

i+a

i = 1, 2, 3, ..., k

is even. The empty set is also considered to be alternating. The number of alternating subsets of

\{1, 2, 3,..., n\}

A(n)

. Develop a method to determine

A(n)

n \in N

and calculate hence

A(33)

connected sequences revisited

Source: Bundeswettbewerb Mathematik 1992, round two, problem 4

For three sequences

(x_n),(y_n),(z_n)

with positive starting elements

x_1,y_1,z_1

we have the following formulae: x_{n+1} = y_n + \frac{1}{z_n} y_{n+1} = z_n + \frac{1}{x_n} z_{n+1} = x_n + \frac{1}{y_n} (n = 1,2,3, \ldots) a.) Prove that none of the three sequences is bounded from above. b.) At least one of the numbers

x_{200},y_{200},z_{200}

is greater than 20.

algebra unsolvedalgebra