China | A nice Algebra Problem | Math Olympiad Question | Solve a+3125^a=0 | Lambert w Function

Chinese | A nice Algebra Problem | Math Olympiad Question | Solve a+3125^a=0 | Lambert w Function
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[@walterwen2975]

Math Olympiad Question: a + 3125ᵃ = 0; a = ? First method: a = - 3125ᵃ, a² = (- 3125ᵃ)² = (5⁵ᵃ)² = 5¹⁰ᵃ, (a²)¹⸍²ᵃ = (5¹⁰ᵃ)¹⸍²ᵃ, a¹⸍ᵃ = 5⁵ a¹⸍ᵃ = (a⁻¹)⁻⁽¹⸍ᵃ⁾ = (- 1/a)⁽⁻¹⸍ᵃ⁾ = 5⁵, - 1/a = 5; a = - 1/5 Second method: - a = 3125ᵃ, (- a)¹⸍ᵃ = (3125ᵃ)¹⸍ᵃ, (- a)¹⸍ᵃ = (- a)⁻ᵃ = 3125 = 5⁵ (- a)⁻ᵃ = 5⁵ = 5⁽⁻¹⁾⁽⁻⁵⁾ = (5⁻¹)⁻⁵ = (1/5)¹⸍⁽¹⸍⁵⁾; - a = 1/5; a = - 1/5 Answer check: a + 3125ᵃ = - 1/5 + (5⁵)⁽⁻¹⸍⁵⁾ = - 1/5 + 5⁻¹ = - 1/5 + 1/5 = 0 Final answer: a = - 1/5

[@JossWainwright]

In competition-type problem solving, the first thing to do is look for a trick, or symmetry or similar that can simplify things. Fore example, check the properties of 3125. Seeing it is 5^5, if you just do the obvious rewrite as a + 5^(5a) = 0 to make it simpler to think about the LHS function, then it's not hard to guess the answer. Then show the function is increasing, so there can be no other answer. That is much faster, which matters in a competition.

[@superacademy247]

Thanks for your kind advice.

[@icheichigo]

This question make sense only when a1/e, and for y>1,it has only one solution for x, so b=1/5 is the only solution.

[@nikolayguzman331]

More Faster: a

[@manasishastry345]

very nice.

[@prime423]

the number 3125 gives the methodology obvious.its 5 with exponent 5!!Thats the clue and a teaching moment.

[@josecaffomarruffo2186]

VER EL VIDEO ENTERO

[@superacademy247]

Yes. They should watch over and over again to master the two most powerful 💪 methods presented.