France | Math Olympiad Exponent Simplification Problem | Can You Solve this? Find "X"

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In this Math Olympiad Algebra Problem, you'll learn tips and tricks of solving International Math Olympiad exams quickly. #IMO #matholympiad #algebra #radicalequations #simplify #exponential

[@user-vx1nq3xq7p]

Thanks

[@superacademy247]

Welcome

[@SidneiMV]

Great! Congratulations sir.

[@superacademy247]

Many thanks!

[@ichdu6710]

I just discovered the fact, that the original equation with m and the roots is only defined when m > =0 and the solution must be between 0 and 1. This can be verified by a simply inspection. Therefore the two received negative solutions of the y-equation can be cancelled immediately the 6th root of a number m must be positive. Only the positive solution remains. m = [(root(3)-1) /3]^6 can be easily calculated by [(root(3)-1)]^2 = 4 -2*root(3) using the binomial formula then multiply this with [(root(3)-1) /3]^2 and so on [(root(3)-1)]^6 = [(root(3)-1)]^2 * [(root(3)-1)]^2 * [(root(3)-1)]^2 = [4 - 2*root(3)]^2* [4 -2*root(3)] = (28 - 16root(3) ) * (4 - 2*root(3)) = 208 - 120root(3)

[@superacademy247]

Nice 👍💯 discovery. Thanks

[@user-gs3fk2ig4r]

Let a=m⅓, b=m½. Thus a+b=2/27, a³=b² => a³=(2/27-a)² => (27)²a³=(2-27a)² =>(9a)³=(2-3•(9a))². Put z=9a. We have z³=(2-3z)² => z³-9z²+12z-4=0 (z-1)(z²-8z+4)=0. Therefore, z=1, 4+2•3½, 4-2•3½ => a=1/9, (4+2•3½)/9=((3½+1)/3)², (4-2•3½)/9 =((3½-1)/3)². In case a=1/9, 0

[@superacademy247]

And you simpliify your result

[@moemirza1865]

Actually, -1/3 is on of the answers. Substitute directly into the equation not separating -1 from 1/3 !?

[@superacademy247]

You've to find m and then you substitute m into the equation

[@sathyaNarrayanan1992]

how is this olympiad question

[@superacademy247]

It's. It is not a walk in the park to producing that beautiful result.

[@sathyaNarrayanan1992]

hmm,