# Putnam Competition 2021

The William Lowell Putnam Mathematics Competition is an annual event for US and Canadian undergraduate students to test their mathematical mettle! … or just to have fun trying to solve hard math problems. 😊 There’s no time commitment required to participate besides joining us on Saturday December 4 to compete. And there’s no pressure! Score any points at all and you’re a hero! Register for the competition here:

Email me or find me in Wubben 134J if you’d like to join us in preparing for the competition.

• (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.)
• (1985 B6)   Let $$G$$ be a finite set of real $$n\times n$$ matrices $$\{M_i\}$$, $$1 \leq i \leq r$$, which form a group under matrix multiplication. Suppose that $$\sum_{i=1}^r \mathrm{tr}(M_i)=0$$, where $$\mathrm{tr}(A)$$ denotes the trace of the matrix $$A$$. Prove that $$\sum_{i=1}^r M_i$$ is the $$n \times n$$ zero matrix.
• (2019 A2)   In the triangle $$\triangle ABC$$, let $$G$$ be the centroid, and let $$I$$ be the center of the inscribed circle. Let $$\alpha$$ and $$\beta$$ be the angles at the vertices $$A$$ and $$B$$, respectively. Suppose that the segment $$IG$$ is parallel to $$AB$$ and that $$\beta = 2 \tan^{-1} (1/3)$$. Find $$\alpha$$.
• (2000 A4)   Show that this improper integral converges: $\lim_{B \to \infty} \int\limits_0^B \sin(x) \sin\left(x^2\right) \,\mathrm{d}x$