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Part of 2000 Putnam

Problems(2)

Putnam 2000 A2

Source:

9/6/2011
Prove that there exist infinitely many integers nn such that nn, n+1n+1, n+2n+2 are each the sum of the squares of two integers. [Example: 0=02+020=0^2+0^2, 1=02+121=0^2+1^2, 2=12+122=1^2+1^2.]
PutnamquadraticsalgebrapolynomialinductionDiophantine equationcollege contests
Putnam 2000 B2

Source:

9/6/2011
Prove that the expression gcd(m,n)n(nm) \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} is an integer for all pairs of integers n≥m≥1 n \ge m \ge 1 .
Putnamnumber theorygreatest common divisorcollege contests