There should be a list of generic titles for sourceless prob

For discussing Olympiad Level Algebra (and Inequality) problems
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Zawadx
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There should be a list of generic titles for sourceless prob

Unread post by Zawadx » Wed Feb 15, 2017 9:13 pm

Find ALL solutions of:

$$ f(x+y) + f(x-y) = 2f(x)cos y $$

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Thanic Nur Samin
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Re: There should be a list of generic titles for sourceless

Unread post by Thanic Nur Samin » Wed Feb 15, 2017 9:25 pm

Let $P(x,y)$ be the FE.

$$P(0,x)\Rightarrow f(x)+f(-x)=2f(0)\cos x=2A\cos x$$
$$P\left (x+\dfrac{\pi}{2}\right )\Rightarrow f(x)+f(x+\pi)=0$$
$$P\left (\dfrac{\pi}{2},\dfrac{\pi}{2}+x\right )\Rightarrow f(-x)+f(x+\pi)=-2f\left (\dfrac{\pi}{2}\right )\sin x=-2B\sin x$$

So, $f(x)=A\cos x+B\sin x$.

Substituting in, the FE works. So we have found the solution.
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.

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