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5

Part of 1996 Putnam

Problems(2)

Putnam 1996 A5

Source:

6/4/2014
Let pp be a prime greater than 33. Prove that p2∣∑i=1⌊2p3⌋(pi). p^2\Big| \sum_{i=1}^{\left\lfloor\frac{2p}{3}\right\rfloor}\dbinom{p}{i}.
Putnamfloor functioncollege contests
Putnam 1996 B5

Source:

6/6/2014
Given a finite binary string SS of symbols X,OX,O we define Δ(S)=n(X)−n(O)\Delta(S)=n(X)-n(O) where n(X),n(O)n(X),n(O) respectively denote number of XX's and OO's in a string. For example Δ(XOOXOOX)=3−4=−1\Delta(XOOXOOX)=3-4=-1. We call a string SS \emph{balanced} if every substring TT of SS has −2≤Δ(T)≤2-2\le \Delta(T)\le 2. Find number of balanced strings of length nn.
Putnamfloor functioncollege contests