[/quote]
,
and 
Show that
Let 
Show that
be a triangle. 
is the midpoint of segment 
,
,
are selected on sides 
,
respectively such that 
and 
intersect at a point 
.
intersects line 
.
,
and the tangent to 
concur.
and 
,
.
past 
and point 
on segment 
such that 
.
be on the opposite side of 
from 
such that 
and 
.
intersect the circumcircle of 
again at 
,
intersect the circumcircle of 
again at 
.
,
are collinear.
people are playing a game at Cheesy's math casino, where 
be a positive integer. A subset of length 
from the set of integers from 
to 
,
for 
.
be the set of positive integers that do not exceed 
as the smallest positive integer that allows 
satisfying 
,
,
.
be a quadrilateral and let 
be a line. Let 
at points 
respectively. Given that these six points on 
,
are coaxal.
and let 
,
,
be a sequence of non-negative real numbers. Find the maximum value of 
in terms of 
positive real numbers such that 
Prove that![\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+(k+5)(a+b+c) \geqslant 3(k+6),\]](http://latex.artofproblemsolving.com/3/d/a/3da7c98ff14a70b4bbbb6504bfb2c397fd148e45.png)
for all ![$0 \leqslant k \leqslant k_0 = \frac{3\big(\sqrt[3]{2}+\sqrt[3]{4}\big)-7}{2}.$](http://latex.artofproblemsolving.com/7/7/5/77526925e38a958cb4167555e8fbc1056506c522.png)
or![$$a=b=\frac{\sqrt[3]{4}+2\sqrt[3]{2}-2}{3},\,c=\frac{2\big(1+\sqrt[3]{2}\big)}{3},$$](http://latex.artofproblemsolving.com/c/b/d/cbd0471249c6ac97348f9748ee55aa5e56af03ca.png)
or any cyclic permutation.
,
.
Y by
Problem #277 (Source: Mu Alpha Theta 1992)
Find![$\color[rgb]{0.35,0.35,0.35}\displaystyle\sum_{n=0}^\infty\frac{\sin (nx)}{3^n}$](//latex.artofproblemsolving.com/a/f/3/af33799c6924b61378db323645ea3e1b9c4e8c9d.png)
if ![$\color[rgb]{0.35,0.35,0.35}\sin x=1/3$](//latex.artofproblemsolving.com/c/e/a/cea5c46445caa512d94d655350170fb19518a2c8.png)
and ![$\color[rgb]{0.35,0.35,0.35} 0\le x\le \pi/2$](//latex.artofproblemsolving.com/9/8/b/98b7d75c944002b8b9c11dbe8cca0d402e6fbb67.png)
.
I know what cosine of x is also positive because of the value of x. I've also tried to see if the value of sin(nx) ever repeats, but it doesn't. Can anyone give me a hint (not the full solution) on how to start on solving this problem? Thank you.
Find
![$\color[rgb]{0.35,0.35,0.35}\displaystyle\sum_{n=0}^\infty\frac{\sin (nx)}{3^n}$](http://latex.artofproblemsolving.com/a/f/3/af33799c6924b61378db323645ea3e1b9c4e8c9d.png)
if ![$\color[rgb]{0.35,0.35,0.35}\sin x=1/3$](http://latex.artofproblemsolving.com/c/e/a/cea5c46445caa512d94d655350170fb19518a2c8.png)
and ![$\color[rgb]{0.35,0.35,0.35} 0\le x\le \pi/2$](http://latex.artofproblemsolving.com/9/8/b/98b7d75c944002b8b9c11dbe8cca0d402e6fbb67.png)
.I know what cosine of x is also positive because of the value of x. I've also tried to see if the value of sin(nx) ever repeats, but it doesn't. Can anyone give me a hint (not the full solution) on how to start on solving this problem? Thank you.
.
.
