and 
Prove that 
be real numbers such that 
.
Prove that 
![\[
\sqrt[3]{\sqrt[3]{x - \frac{3}{8}} - \frac{3}{8}} = x^3 + \frac{3}{8}
\]](http://latex.artofproblemsolving.com/b/b/9/bb99cd588b94b5716738e6e9ea01d15377b6efef.png)
has exactly two real positive solutions 
and 
.
.
be a regular nonadecagon. Lines 
and 
meet at 
.
.
be the 19th-roots of unity with 
.![\[X = \frac{\omega^0 \omega^4 (\omega^2 + \omega^3) - \omega^2 \omega^3 (\omega^0 + \omega^4)}{\omega^0 \omega^4 - \omega^2 \omega^3}\]](http://latex.artofproblemsolving.com/3/b/5/3b535dd8a08df864eb062ebdb05d836ae1682cd0.png)
![\[= \frac{\omega^6 + \omega^7 - \omega^5 - \omega^9}{\omega^4 - \omega^5}\]](http://latex.artofproblemsolving.com/a/0/1/a015858628e275d220ebb89f4647fb2cdc00dfde.png)
![\[= \frac{(\omega^5 - \omega^3) - (\omega^2 - \omega)}{\omega - 1}\]](http://latex.artofproblemsolving.com/b/6/a/b6ae9d840aa9c9ed0e951a3b9a58368815192460.png)
![\[= \omega^4 + \omega^3 + \omega\]](http://latex.artofproblemsolving.com/1/4/4/144976ecce9cb7e1b2ef0b96eaf69e6d72dafc39.png)
Therefore, ![\[ \frac{A_7 - X}{A_5 - X} \]](http://latex.artofproblemsolving.com/d/e/7/de7218c41a3500b606805cebb420553e127d0c38.png)
![\[ = \frac{\omega^6 - \omega^4 - \omega^3 - \omega}{- \omega^3 - \omega} \]](http://latex.artofproblemsolving.com/a/0/4/a048a571a14118eac1da00fbf524685783ea5e3c.png)
![\[ = 1 - \omega^3 \]](http://latex.artofproblemsolving.com/8/a/4/8a49e13dd1b819e6d8e3776a5735012406636d50.png)
We plot this on the complex plane. We let 
be the origin, 
still be 
,
be 
.
is isosceles with 
.![\[\angle BOA_1 = \frac{\pi - \frac{6\pi}{19}}{2} = \boxed{\frac{13\pi}{38}}\]](http://latex.artofproblemsolving.com/8/0/9/809e94631304088982e4cf5e420e73f8cddbcb18.png)
Y by
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )
Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )
Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
