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4

Part of 1986 Bundeswettbewerb Mathematik

Problems(2)

a_{n+1} = 1/16 (1 + 4a_n +\sqrt{1 + 24a_n})

Source: 1986 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p4

11/21/2022
The sequence a1,a2,a3,...a_1, a_2, a_3,... is defined by a1=1   ,   an+1=116(1+4an+1+24an)   (n∈N∗).a_1 = 1\,\,\,, \,\,\,a_{n+1} =\frac{1}{16}(1 + 4a_n +\sqrt{1 + 24a_n}) \,\,\,(n \in N^* ). Determine and prove a formula with which for every natural number nn the term ana_n can be computed directly without having to determine preceding terms of the sequence.
algebrarecurrence relation
subsets of a set with m elements, related

Source: 1986 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p4

11/21/2022
Given the finite set MM with mm elements and 19861986 further sets M1,M2,M3,...,M1986M_1,M_2,M_3,...,M_{1986}, each of which contains more than m2\frac{m}{2} elements from MM . Show that no more than ten elements need to be marked in order for any set MiM_i (i=1,2,3,...,1986i =1, 2, 3,..., 1986) contains at least one marked element.
combinatoricsSubsets