Putnam preparation (Table of contents)

Number Theory

Problem 1

Find all positive integers \(n\) such that \(n(n+1)\) is a perfect square.

Problem 2

Assume that \(x\) and \(y\) are non-negative integers such that \(15x+11y\) is divisible by \(37\). Prove that \(7x+15y\) is divisible by \(37\).

Problem 3

There are \(n\) books in a library. If books are to be arranged in boxes with \(7\) books in each box, then \(5\) books remain. If they are arranged with \(9\) books in each box, then \(3\) books remain, and if they are arranged with \(11\) books in each box, then \(7\) books remain. What is the smallest possible value for \(n\).

Problem 4

Assume that \(n_1\), \(n_2\), \(\dots\), \(n_k\) are positive integers whose greatest common divisor is equal to \(d\). Prove that there exist integers \(\alpha_1\), \(\alpha_2\), \(\dots\), \(\alpha_k\) such that \[\alpha_1n_1+\alpha_2n_2+\cdots+\alpha_kn_k=d.\]

Problem 5

Let \(n\geq 3\) be an odd integer. Prove that every integer \(l\) satisfying \(1\leq l\leq n\) can be represented as a sum or difference of two integers each of which is less than \(n\) and relatively prime to \(n\).

Problem 6

Prove that there is no positive integer \(n\) for which \(n^5\) can be written as a product of six consecutive positive integers.

Problem 7

Let \(n\geq 5\) be a natural number. Prove that the following two statements are equivalent:

  • (a) Neither of the numbers \(n\) and \(n+1\) is prime.

  • (b) The closest integer to \(\displaystyle\frac{(n-1)!}{n^2+n}\) is even.

Remark. If \(k\in\mathbb Z\), the closest integer to \(k+\frac12\) is \(k+1\).

Problem 8

Does there exist \(k,n\in\mathbb N\) such that \(k\geq 2\) and for which the set \(\{n,n+1,n+2, \dots, n+101\}\) can be partitioned into \(k\) disjoint subsets all of which have equal products of elements?