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

and

Skip to content
#
Month: February 2016

# ELMO 2012 SL A9

# Continued Fractions and Linear Fractional Transformations

# Putnam 2014 B6

# Putnam 2014 A5

# Romania 2013 p4 grade 12

# ELMO Shortlist 2012 C8

# ELMO Shortlist 2012 C9

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

and

This is an article written by Evan O’Dorney for Intel STS 2011. It deals with rational approximation to some square roots. The article is at here

Let be a function for which there exists a constant such that for all Suppose also that for each rational number there exist integers and such that Prove that there exist finitely many intervals such that is a linear function on each and

Let Prove that the polynomials and are relatively prime for all positive integers and with

Given a natural number, a body with commutative property that a polynomial of degree and a subgroup of the additive group , Show that there is so .

Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair denote the complex number for . We define an -chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form or , where and 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 -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 -chessboard polygon can be tiled by lozenges, then it can be done so tastefully.

b) Prove that such a tasteful tiling is unique.

For a set of integers, define . Is there a constant such that for all positive integers , there exists a set of size such that ?