Problem Set 1

Handed out: September 1, 2026. Due: Tuesday, September 8.

Write your solutions up in LaTeX and email the PDF to enaig@tulane.edu.

Here is the LaTeX template: problem-set-1.tex.

Nobody is expected to get all five. These are low stakes: if you solve one, your name goes on this site, and we would love to have you walk the room through it at the September 8 meeting.

Problem 1

Can a \(10 \times 10\) board be tiled by \(1 \times 4\) pieces, placed horizontally or vertically? Prove your answer.

Problem 2

Prove that every sequence of \(n^2 + 1\) distinct real numbers contains a monotone subsequence of length \(n + 1\), increasing or decreasing. Then show \(n^2 + 1\) cannot be lowered.

Problem 3

An island holds \(13\) red, \(15\) green, and \(17\) blue chameleons. When two of different colors meet, both turn the third color. Can they all end up the same color?

Problem 4: Putnam 2016, B1

Let \(x_0 = 1\) and \(x_{n+1} = \ln(e^{x_n} - x_n)\) for \(n \geq 0\). Show that

\[x_0 + x_1 + x_2 + \cdots\]

converges, and find its sum.

Problem 5: Putnam 2012, A1

Let \(d_1, d_2, \ldots, d_{12}\) be real numbers in the interval \((1, 12)\). Show that there exist distinct indices \(i, j, k\) such that \(d_i\), \(d_j\), \(d_k\) are the side lengths of an acute triangle.

Solutions

Posted after September 8.