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and 
Prove that 
be real numbers such that 
. Find the minimum of 
and 
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 
. Compute 
.
be a regular nonadecagon. Lines 
and 
meet at 
. Compute 
.
be a function, two times derivable on 
for which there exist 
such that![\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\]](http://latex.artofproblemsolving.com/a/3/1/a31861ef909a6e17ed8b495b9b9b16d887ba573f.png)
for all 
.
.
be positive real numbers .
, whether or not the polynomial 
irreducible over 
?
such that the equation in the unknown 
:
admits 
roots: 
s.t. 
.
does there exist an 
matrix 
whose entries are all in 
, such that 
is the matrix of all ones?
be real numbers such that 
. Prove that
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)
![$\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{lcm(1,2,3,...,n)}=e$](http://latex.artofproblemsolving.com/a/1/3/a139e5cb990f3386de109c701f8edc71b4054712.png)
Prove that
Let 
Prove that
and 
Prove that
then 
or 
there is nothing to prove otherwise suppose 
then
and 
satisfying 
and any positive integer 
,
.
Prove that
Prove that
Prove that
![\[N=a^2+b^2+a^3+b^3+5ab\]](http://latex.artofproblemsolving.com/5/6/2/5626daf0552be826f9981b781f0bdc605ed66ac6.png)
then this is a continuous surface and assume there is a singular point on this surface that serves as maxima. now taking partial derivatives we must have
since 
solving quadratic gives 
which yields 
as max
and 
and 
.
Let 
.