PUTNAM PRACTICE PROBLEMS
INTEGERS
1) Let S be the set of all integers that can be written as a sum of 5s and 7s. What is the largest integer not in S?
2) (A-1-1955) Let a be the square root of 2 and b the square root of 3. Prove that there is no set of integers m,n,p except 0,0,0 for which m+ns+pb=0.
3) (A-1-1989) How many primes among the positive integers, written as usual in base 10, are alternating 1s and 0s, beginning and ending with 1?
4) (A-1-1940) Prove that if f(x) is a polynomial with integral coefficients, and there exists a positive integer $k$ such that none of the integers f(1),f(2),...,f(k) is divisible by k, then f(x) has no integral root.
5) (A-2-2000) Prove that there exists infinitely many integers n, such that n,n+1, and n+2 are each the sum of two squares of integers. [Example: 0=0x0+0x0,1=0x0+1x1,2=1x1+1x1.]