# Putnam 2014 B6

Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$

For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$?
Ben Green and I have just uploaded to the arXiv our new paper “On sets defining few ordinary lines“, submitted to Discrete and Computational Geometry. This paper asymptotically solves two old questions concerning finite configurations of points $latex {P}&fg=000000$ in the plane $latex {{mathbb R}^2}&fg=000000$. Given a set $latex {P}&fg=000000$ of $latex {n}&fg=000000$ points in the plane, define an ordinary line to be a line containing exactly two points of $latex {P}&fg=000000$. The classical Sylvester-Gallai theorem, first posed as a problem by Sylvester in 1893, asserts that as long as the points of $latex {P}&fg=000000$ are not all collinear, $latex {P}&fg=000000$ defines at least one ordinary line:
It is then natural to pose the question of what is the minimal number of ordinary lines that a set of $latex {n}&fg=000000$ non-collinear points can generate. In 1940, Melchior gave an elegant proof of the Sylvester-Gallai theorem based…