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### BdMO National 2008: Secondary

Posted: Mon Dec 19, 2016 9:01 pm
The function $f (x)$ is a complicated nonlinear function. It satisfies, $f(x) + f(1-x) = 1$ . Evaluate $\int_{0}^{1} f(x)dx$.

### Re: BdMO National 2008: Secondary

Posted: Fri Jan 13, 2017 6:37 pm
Is that a deriviative sign?

### Re: BdMO National 2008: Secondary

Posted: Fri Jan 13, 2017 10:44 pm
ahmedittihad wrote:Is that a deriviative sign?
That is a integreation sign.

$$\displaystyle \int_0^1f(x)dx=\dfrac{1}{2}\left(\int_0^1f(x)dx+\int_0^1f(1-x)dx\right)$$

$$\displaystyle =\dfrac{1}{2}\left(\int_0^1(f(x)+f(1-x))dx\right)$$

$$\displaystyle =\dfrac{1}{2}\int_0^1 dx=\dfrac{1}{2}$$

### Re: BdMO National 2008: Secondary

Posted: Sat Jan 14, 2017 12:46 pm
Do we need calculus in Math Olympiads?!

### Re: BdMO National 2008: Secondary

Posted: Sat Jan 14, 2017 1:42 pm
I think not. This problem just used the very basics. I can't say anything about the nationals, but in IMO, APMO and other contests, you won't need it.

### Re: BdMO National 2008: Secondary

Posted: Wed Feb 21, 2018 6:59 pm
A particular formula kills the problem