Writeup: Putnam 2007 A2

19 Jul, 2025

Problem Statement

Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy=1$ and both branches of the hyperbola $xy=-1$.

My Solution

Note that the convexity requirement makes this problem a lot easier. Let our convex set be $S$. Then note that $S$ must contain at least 4 points: 2 on the branches of $xy=1$, and $2$ on the branches of $xy=-1$. In any clockwise order, let these points be $A,B,C,D$.

Let $|A|$ denote the area of a set $A$

Then note that the convex hull of $ABCD=H$ satisfies $|H|\le |A|$ since $H$ must be contained within $A$.

Let the points $A,B,C,D$ be $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$, and let $x_1{y}_1=1$, $x_3{y}_3=1$, $x_2{y}_2=-1$, $x_4{y}_4=-1$. Then by the shoelace theorem, we must have

By AM-GM this quantity is at least $4$.

We can achieve this by using the corners $(1,1),(1,-1),(-1,1),(-1,-1)$.