## Problem

Given

*a*Without calculating^{2 }+ 1 + a^{-2}= a + a^{-1 }*a*, find the value of*a*= ?^{129 }- a^{128}+ a^{127}- a^{126}## Answer Submission Is Not Available

Form is valid through December 2015

## Solution

Dear Visitors,

thank you for sending the solutions for the December problem.

I received many solutions, but majority were not right - we were looking for a number and not for an expression.

The first two correct answers were submitted by MO Braga and by Hyun Woo Lee.

Congratulations!!!

This is the solution of the December problem: The equation given in the problem, after multiplication of both parts by a^{2}and putting everything to the LHS, can be rewritten as*a*. This is a factor of a^{4}– a^{3}+ a^{2}– a + 1 = 0^{5}+ 1. Hence*a*and^{5}+ 1 = 0*a*Hence, the requested expression can be rewritten as^{5 }= -1.*a*^{125}*(a*^{4}– a^{3}+ a^{2}– a) = 1. Dixi.Waiting for your answers to the New Year problems,

Serge Hazanov

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