### 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 $.

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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 $.

Posted: **Fri Jan 13, 2017 6:37 pm**

Is that a deriviative sign?

Posted: **Fri Jan 13, 2017 10:44 pm**

That is a integreation sign.ahmedittihad wrote:Is that a deriviative sign?

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

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

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

Posted: **Sat Jan 14, 2017 12:46 pm**

Do we need calculus in Math Olympiads?!

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.

Posted: **Wed Feb 21, 2018 6:59 pm**

A particular formula kills the problem