A walk with 2026 vertices in a simple graph is called "shaking" if the first vertex has odd degree and last vertex has even degree. Prove that the number of "shaking" walks in any finite simple graph is even.
Let M be a positive integer, and let a,b,c be positive integers in the range [M,M+sqrt(M/2)) such that a^3 b + b^3 c + c^3 a is divisible by abc. Prove that a=b=c.
Consider a perfect rooted binary tree of depth n. (That is, every node has either 0 or 2 children, and all leaves have the same depth). Every node is given a weight drawn independently from some fixed distribution D. For any path starting from the root and ending at a leaf, the average weight of the path is the arithmetic mean of the weights assigned to the nodes on the path. Once our weighting is fixed, we look at the largest average weight of any path from the root to a leaf. Let Eₙ denote the expected value of this largest average weight of path over all weightings of the tree. Then find the limit as n →infinity of Eₙ, in the cases of:
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D=U({0,1}) is a Bernoulli distribution.
D=U([0,1]) is a continuous uniform distribution.