ELMO 2012 SL A9

Let a,b,c be distinct positive real numbers, and let k be a positive integer greater than 3. Show that

\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)


\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2)


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.

ELMO Shortlist 2012 C8

Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair (x,y) denote the complex number x+y\omega for \omega=e^{2\pi i/3}. We define an \omega-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form x=a or y=b, where a and b are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an \omega-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a tasteful tiling is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order).

a) Prove that if an \omega-chessboard polygon can be tiled by lozenges, then it can be done so tastefully.

b) Prove that such a tasteful tiling is unique.