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

and

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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

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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 ?