[/quote]
,
and 
Show that
Let 
Show that
Let 
Show that
such that
x>y>0
2
,
and 
lie on the side 
,
.
is the midpoint of 
.
.
are such that 
,
with circumcircle 
and circumcentre 
.
and 
be the tangents to 
and 
respectively, which meet at 
.
be the foot of the perpendicular from 
.
parallel to line 
meets 
.
passes through the midpoint of the line segment 
.
,
.
.
differents subjects, each consist of 
students. Given that for any 
different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least 
different subjects.
and 
,
and 
denote the product and sum of the digits of 
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples 
are integers.
students and a teacher with 
marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of 
.
other students. The same goes on until the teacher is not able to choose anymore student.
Prove that
Let 
Prove that
satisfying the following condition: 
. Prove that
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.
.
.
