Solving A Polynomial Equation | Math Olympiads

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Комментарии
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Hocam çok basit gibi görünen ama muhteşem bir çözüm 🤩

turkishwagnerian
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Saying P has to be a polynomial is a bit heavy handed. All you really need is for P to be continuous at 0.
[P(4x) + P(2x)] - [P(2x) + P(x)] = P(4x) - P(x) = 18x+2 - (9x+2) = 9x.

P(0) + P(2*0) = 9*0+2
2P(0) = 2
P(0)=1.

(P(x) - P(x/4)) + (P(x/4) - P(x/16) ) + (P(x/16)- P(x/64)) + .... = 9/4 x + 9/16 x + 9/64x + ...
P(x) - P(0) = 9x ( 1/4 + 1/16 + 1/64 ... ) = 9x (1/3) = 3x
P(x) = 3x + P(0) = 3x+1.

Other solutions are possible if P doesn't have to be continuous at 0.
E.g. P(x) = 3x if floor(log2(x)) is even, and P(x) = 3x+2 if it's odd.

eilonlavi
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"Polynomial" is any number of terms with any finite integer power... It need not be a line.

byronwatkins
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I want to know which software you use for your presentations. I would love to learn how to do presentations like this for work.

Roq-stone
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Зможети вирішити таке:
f(x) + f(x²) = x² + x⁴ ?

Germankacyhay
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Calling this a polynomial equation is a travesty. Some explanation of why you assume the solution is linear is needed.

Gotta hand it to you. You get a ton of page views working simple home made problems.

What keeps people coming back? Maybe your soothing unctuous voice?!?

MyOneFiftiethOfADollar