[/quote]
,
be an acute-angled triangle with 
.
be the orthocenter of triangle 
,
be the midpoint of the side 
.
be a point on the side 
and 
a point on the side 
such that 
and the points 
is perpendicular to the common chord of the circumscribed circles of triangle 
.
balls to 
boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?
,
.
so that 
bisects 
.
be the intersection point of the lines 
and 
be the semicircle with diameter 
at an interior point. A line through 
.
and 
is a rational number.
and other permutations.
![\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]](http://latex.artofproblemsolving.com/6/9/b/69b1964be5f92eb44a36b0b8604bf473fe27e210.png)
and 
denote its semiperimeter and its inradius, respectively. Prove that triangle 
whose side lengths are all positive integers with no common divisors and determine these integers.
.
is the intersection of the parallel line from 
be the point where the external bisector of 
intersects with 
be the projection of 
.
have a common point.
and let 
be the circle 
.
and the circle of radius 
cross 
and 
,
and circle 
are concurrent.
for which the following statement holds: among any five positive real numbers 
(not necessarily distinct), one can always choose distinct subscripts 
such that![\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]](http://latex.artofproblemsolving.com/a/e/e/aeeab8e846634c7922e9312ef139739ad3fbe6eb.png)
,
.
at 
intersects 
,
and 
meet on 
Y by Adventure10, Mango247
Hey guys,
Consdering the fact that my last post got more than 2 comments, I am going to post again. However, this time I am looking ahead. I can comfortably get a 5 on the AIME. With all of the books I have mentioned, I hope to increase this average to about 12+. I am pretty sure that I can make the USAMO next year, if I work hard all through summer. I will have done all "school math" through AP Calculus BC. I will be in my second to last year of high school, but will still technically be a sophmore. I would really like to make MOP next year. What I am wondering is how in the world do you prepare for it, so that you can produce full solutions on at least three problems on the USAMO and get partial credit on the rest. Could someone reccomend books in olympiad-level geometry, number theory, combinatorics, algebra, inequalities, and general problem solving techniques. Also, could someone give me a study plan so that I can at least be able to get 15 points on the USAMO??? I would really like to get a 25+ on the USAMO, so can someone tell me how to improve proof-skills, like making connections faster and in general writing cleaner solutions? I really appreciate it.
Yours truly,
ktz2014 (discovered to be one among many math problem solvers in TN)
BTW: TN dudes and dudets(if it applies) PM, we need to get a strong ARML team, Duke Math Meet Team, Harvard MIT Math Team, and any other contest teams. ASAP!!!!
Consdering the fact that my last post got more than 2 comments, I am going to post again. However, this time I am looking ahead. I can comfortably get a 5 on the AIME. With all of the books I have mentioned, I hope to increase this average to about 12+. I am pretty sure that I can make the USAMO next year, if I work hard all through summer. I will have done all "school math" through AP Calculus BC. I will be in my second to last year of high school, but will still technically be a sophmore. I would really like to make MOP next year. What I am wondering is how in the world do you prepare for it, so that you can produce full solutions on at least three problems on the USAMO and get partial credit on the rest. Could someone reccomend books in olympiad-level geometry, number theory, combinatorics, algebra, inequalities, and general problem solving techniques. Also, could someone give me a study plan so that I can at least be able to get 15 points on the USAMO??? I would really like to get a 25+ on the USAMO, so can someone tell me how to improve proof-skills, like making connections faster and in general writing cleaner solutions? I really appreciate it.
Yours truly,
ktz2014 (discovered to be one among many math problem solvers in TN)
BTW: TN dudes and dudets(if it applies) PM, we need to get a strong ARML team, Duke Math Meet Team, Harvard MIT Math Team, and any other contest teams. ASAP!!!!
