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6

Part of 1999 Putnam

Problems(2)

Putnam 1999 A6

Source:

12/22/2012
The sequence (an)n≥1(a_n)_{n\geq 1} is defined by a1=1,a2=2,a3=24,a_1=1,a_2=2,a_3=24, and, for n≥4,n\geq 4, an=6an−12an−3−8an−1an−22an−2an−3.a_n=\dfrac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}. Show that, for all nn, ana_n is an integer multiple of nn.
Putnamcollege contests
Putnam 1999 B6

Source:

12/22/2012
Let SS be a finite set of integers, each greater than 11. Suppose that for each integer nn there is some s∈Ss\in S such that gcd⁡(s,n)=1\gcd(s,n)=1 or gcd⁡(s,n)=s\gcd(s,n)=s. Show that there exist s,t∈Ss,t\in S such that gcd⁡(s,t)\gcd(s,t) is prime.
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