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Number of solutions bounded from above

Source: IMO Longlist 1989, Problem 94

September 18, 2008

algebraDiophantine equationlinear equationcountingIMO Shortlist

Problem Statement

Let a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{} be given and let N

(a_1, a_2, a_3)

be the number of solutions

(x_1, x_2, x_3)

of the equation \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1. where

x_1, x_2,

and

x_3

are positive integers. Prove that N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).