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Part of 2013 Putnam

Problems(2)

Source:

Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all

20

39.

Show that there are two faces that share a vertex and have the same integer written on them.

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Source:

For positive integers

n,

let the numbers

c(n)

be determined by the rules

c(1)=1,c(2n)=c(n),

c(2n+1)=(-1)^nc(n).

Find the value of

\sum_{n=1}^{2013}c(n)c(n+2).

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