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Minimum Perimeter of a Triangle

Source: 2003 China Second Round Olympiad

August 30, 2014
geometryperimeterfloor functioncalculusintegrationmodular arithmeticalgebra

Problem Statement

Let the three sides of a triangle be ℓ,m,n\ell, m, n, respectively, satisfying ℓ>m>n\ell>m>n and {3ℓ104}={3m104}={3n104}\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}, where {x}=x−⌊x⌋\{x\}=x-\lfloor{x}\rfloor and ⌊x⌋\lfloor{x}\rfloor denotes the integral part of the number xx. Find the minimum perimeter of such a triangle.
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