[/quote]
,
![\[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\]](http://latex.artofproblemsolving.com/2/d/2/2d248ce9756f0f09265dd538de3b16b22a944c46.png)
are 
and 
,
is 
and 
,
denote the circumradius. Prove that if 
,
.
.
,
,
be the midpoints of the sides 
,
,
respectively. A circle 
passing through 
and tangent to 
and 
at 
and 
and 
are symmetric to 
meets 
and 
at 
and 
,
meets 
.
.
.
intersects 
.
be 
.
,
,
,
,
,
similarly. Prove that 
,
,
are concurrent.
be a parallelogram with 
A point 
The circumcircle of 
meets the segment 
again at 
The circumcircle of triangle 
meets the segment 
at 
Prove that lines 
are concurrent.
,
intersect at point 
,
intersects at point 
intersects 
. Prove that points 
is incenter of triangle 
and 
is a line tangent to the incircle. Let 
be another line such that intersects 
respectively at 
.
to the incircle other than 
,
.
are similarly defined. Prove that 
are concurrent.
and 
.
.
is right.
is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer 
on the board with 
,
.
points 
and 
and 
is a parallelogram. The circle with center the midpoint 
intersect for the second time at the point 
.
and 
intersect in a point.
,
,
,
.
and 
.
.
Y by Adventure10, Mango247
I'm happy to have (probably) qualified for USAMO, but I haven't been preparing at all for USAMO. I know I can't do too well, but I'd like to do ok, and in particular I may be able to get into freshman MOSP.
Yes, I do know what a proof is, and I indeed have proof-writing experience from USAMTS, but haven't yet tackled problems at the USAMO level.
So, what should my strategy be? How much time should I spend trying to solve problems vs. reading solutions, or even learning new theorems/tools? Which subjects should I focus on the most?
Regarding geometry, what are some important theorems I should know?
Regarding inequalities, should I try to learn anything about them? If so, could someone list the major theorems/inequalities/ideas/tools I should learn? (I've seen AM-GM, WLOG, and Cauchy mentioned, but I forget what else).
And are there any other theorems etc I should know? (ie in other topics)
Yes, I do know what a proof is, and I indeed have proof-writing experience from USAMTS, but haven't yet tackled problems at the USAMO level.
So, what should my strategy be? How much time should I spend trying to solve problems vs. reading solutions, or even learning new theorems/tools? Which subjects should I focus on the most?
Regarding geometry, what are some important theorems I should know?
Regarding inequalities, should I try to learn anything about them? If so, could someone list the major theorems/inequalities/ideas/tools I should learn? (I've seen AM-GM, WLOG, and Cauchy mentioned, but I forget what else).
And are there any other theorems etc I should know? (ie in other topics)
)
![$\frac{a_1 + a_2 + ... + a_n}{n} \ge \sqrt[n]{a_1 a_2 a_3 ... a_n}$](http://latex.artofproblemsolving.com/6/3/e/63efbefd0fcad4d66cbbbd574447e5edf0739c1a.png)
:
,
,
(since you can just pick them in that order).
.
and so on, which trivializes alot of ineq...also, you can do 
where the three angles sum to 
,
