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Part of 1995 Putnam

Problems(2)

Putnam 1995 A4

Source:

7/1/2014
Suppose we have a necklace of nn beads. Each bead is labelled with an integer and the sum of all these labels is n−1n-1. Prove that we can cut the necklace to form a string whose consecutive labels x1,x2,⋯ ,xnx_1, x_2,\cdots , x_n satisfy \sum_{i=1}^{k}x_i\le k-1  \forall \;\;1\le k\le n
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Putnam 1995 B4

Source:

7/1/2014
Evaluate : 2207−12207−12207−⋯8 \sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-\cdots}}} Express your expression in the form a+bcd\frac{a+b\sqrt{c}}{d} where a,b,c,d∈Za,b,c,d\in \Bbb{Z}.
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