Olympiad problems

2019 ISI Entrance Examination, 5

A subset $\bf{S}$ of the plane is called [i]convex[/i] if given any two points $x$ and $y$ in $\bf{S}$, the line segment joining $x$ and $y$ is contained in $\bf{S}$. A quadrilateral is called [i]convex[/i] if the region enclosed by the edges of the quadrilateral is a convex set. Show that given a convex quadrilateral $Q$ of area $1$, there is a rectangle $R$ of area $2$ such that $Q$ can be drawn inside $R$.

1980 Czech And Slovak Olympiad IIIA, 3

Tags: geometry , combinatorial geometry , convex set , combinatorics

The set $M$ was formed from the plane by removing three points $A, B, C$, which are vertices of the triangle. What is the smallest number of convex sets whose union is $M$? [hide=original wording] Množina M Vznikla z roviny vyjmutím tří bodů A, B, C, které jsou vrcholy trojúhelníka. Jaký je nejmenší počet konvexních množin, jejichž sjednocením je M?[/hide]

1962 Putnam, B3

Tags: real analysis , bounded , convex set , geometry

Let $S$ be a convex region in the euclidean plane containing the origin. Assume that every ray from the origin has at least one point outside $S$. Prove that $S$ is bounded.