Problem Set 3
For the meeting Thursday, October 15.
Arnold sent in this week’s problems.
Problem 1: Putnam 2025, A1
Let \(m_0\) and \(n_0\) be distinct positive integers. For every positive integer \(k\), define \(m_k\) and \(n_k\) to be the relatively prime positive integers such that
\[\frac{m_k}{n_k} = \frac{2m_{k-1}+1}{2n_{k-1}+1}.\]Prove that \(2m_k+1\) and \(2n_k+1\) are relatively prime for all but finitely many positive integers \(k\).
Problem 2: Putnam 2015, A1
Let \(A\) and \(B\) be distinct points on one branch of the hyperbola \(xy=1\). Choose \(P\) on the hyperbola between \(A\) and \(B\) so that the area of triangle \(APB\) is as large as possible. Prove that the region enclosed by the hyperbola and the chord \(AP\) has the same area as the region enclosed by the hyperbola and the chord \(PB\).
Problem 3: Putnam 1987, B1
Compute
\[\int_2^4 \frac{\sqrt{\ln(9-x)}}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}\,dx.\]Problem 4: Putnam 2002, B1
Shanille O’Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots?
Problem 5: Putnam 1992, A6
Four points are chosen independently and uniformly at random on the surface of a sphere. What is the probability that the tetrahedron they determine contains the center of the sphere?
Problem 6: IMO 1978, Problem 6
An international society has its members from six different countries. The list of members contains 1978 names, numbered \(1, 2, \ldots, 1978\). Prove that there is at least one member whose number is the sum of the numbers of two members from their own country, or twice the number of one member from their own country.