# Russell's Fibonacci coin bags

Algebra Level 5

Russell has $$2013$$ bags of coins to give away to two friends, Andrew and Brian. Bag $$i$$ has $$F_i$$ coins, where $$F_i$$ is the $$i$$th term of the Fibonacci sequence, and each bag must be given to either Andrew or Brian. If there are $$N$$ ways to distribute all of the bags so that Andrew and Brian get the same number of coins, what is $$\log_2 N$$?

This problem is shared by Russell F. from NIMO Summer Contest 2012.

Details and assumptions

The Fibonacci sequence is defined by $$F_1 = 1, F_2 = 1$$ and $$F_{n+2} = F_{n+1} + F_{n}$$ for $$n \geq 1$$.

Disregard physical impossibilities for the number of coins.

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