[/quote]
,
be an acute triangle. Points 
,
lie on a line in this order and satisfy 
.
and 
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
with diameter 
is given. The point 
intersects 
and 
lies on 
is tangent to 
.
,
colours (so that adjacent vertices have different colours).
edges can be removed from the graph so that it remains connected.
be the incenter and the circumcenter of triangle 
be the circumcenters of triangle 
.
be the midpoints of segments 
.
,
.
Prove that
E
matrix equals either 
or 
Let 
denote the sum of eight products of entries in each row. Also, let 
denote the sum of eight products of entries in each column. Find the maximum possible value of 
In other words, find![$$ {\rm max}\ \left[ \sum_{i=1}^8\ \prod_{j=1}^8\ a_{ij} -
\sum_{j=1}^8\ \prod_{i=1}^8\ a_{ij} \right]
$$](http://latex.artofproblemsolving.com/2/d/e/2de7fd40366073445d289356d0f4e66b542cb60c.png)
matrix.
Show that there exists a constant 
such that for all positive integer 
![\[\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.\]](http://latex.artofproblemsolving.com/f/e/9/fe933e886e28c2ecf1fa880d0e650ac7a2998aaa.png)
Proposed by Cheng Jiang
is written on the board. A move consists of adding either 
or 
to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is 
,
of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
such that 
is the cube of a positive integer. Find 
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!!!!
