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Prove that there exist integers st....

Source: Indian IMOTC 2005 Day 4 Problem 2

September 23, 2005
number theory unsolvednumber theory

Problem Statement

Given real numbers a,α,β,σ and ϱa,\alpha,\beta, \sigma \ and \ \varrho s.t. σ,ϱ>0\sigma, \varrho > 0 and σϱ=116\sigma \varrho = \frac{1}{16}, prove that there exist integers xx and yy s.t. −σ≤(x+α(ax+y+β)≤ϱ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho
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