1. MIT Which positive integers can be expressed as the sum of two or more consecutive positive integers? E.g. the first three examples are: \[ 3 = 1+2 \qquad 5=2+3 \qquad 6=1+2+3 \]
2. MIT In how many ways can the positive integer \(n\) be written as a sum of positive integers, taking order into account? E.g. for \(n=3\), there are the four ways: \[ 1+1+1 \qquad 1+2 \qquad 2+1 \qquad 3\]
3. MIT Find an integer \(n\) whose first digit is three, such that the integer \(3n/2\) is the result of removing the three at the beginning of \(n\) and putting it at the end.
4. Putnam & Beyond Find all functions \(f\colon \mathbf{N} \to \mathbf{N}\) such that for all \(n \in \mathbf{N},\) \[f(n) + 2f\big(f(n)\big) = 3n+5\,.\]
5. Austrian-Polish Mathematics Competition Three infinite arithmetic progressions are given, whose terms are positive integers. Assuming that each of the numbers 1, 2, 3, 4, 5, 6, 7, 8 occurs in at least one of these progressions, show that \(1980\) must occur in one of them.
6. Putnam 2003 A1 Let \(n\) be a fixed positive integer. How many ways are there to write \(n\) as a sum of positive integers, \( n = a_1 + a_2 + \dotsb + a_k, \) with \(k\) an arbitrary positive integer and \(a_1 \leq a_2 \leq \dotsb \leq a_k \leq a_1 + 1?\) E.g. for \(n=4\) there are four ways: \[4\qquad 2+2\qquad 1+1+2\qquad 1+1+1+1\]
7. Putnam 2000 A2 Prove that there exist infinitely many integers \(n\) such that \(n,\) \(n+1,\) and \(n+2\) are each the sum of the squares of two integers. E.g.: \[0=0^2+0^2 \qquad 1=0^2+1^2 \qquad 2=1^2+1^2\]
8. Putnam 2010 A1 Given a positive integer \(n,\) what is the largest \(k\) such that the numbers \(1,2,\dotsc,n\) can be put into \(k\) boxes so that the sum of the numbers in each box is the same? [When \(n=8,\) the example \(\{1,2,3,6\},\{4,8\},\{5,7\}\) shows that the largest \(k\) is at least \(3.\)]
9. MIT What positive integers can be expressed as the difference of two squares? E.g. the first five are: \[ 1 = 1^2-0^2 \qquad 3 = 2^2-1^2 \qquad 4 = 2^2-0^2 \qquad 5= 3^2-2^2 \qquad 7=4^2-3^2 \]
10. Putnam 2017 A1 Let \(S\) be the smallest set of positive integers such that \(2\) is in \(S\), \(n\) is in \(S\) whenever \(n^2\) is in \(S\), and \((n+5)^2\) is in \(S\) whenever \(n\) is in \(S\). Which positive integers are not in \(S\)? (The set \(S\) is “smallest” in the sense that \(S\) is contained in any other such set.)