[/quote]
,
,
.
intersects 
at 
,
at 
.
passes from circumcenter of 
.
,
be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of 
-
as many lattice points as does 
,
of the lattice points contained in 
is 27 times the fraction of lattice points in 
.
be an infinite sequence of real numbers in the interval 
.![\[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\]](http://latex.artofproblemsolving.com/3/c/e/3cef10f63e9c2dde90b29c742ece8fd31e0a8e94.png)
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
and 
,
and 
.
externally tangent to 
and internally tangent to 
.
and 
intersect at a point lying on 
be inscribed in the circle 
.
intersects 
and 
at points 
and 
,
be the reflection of 
be the reflection of 
of triangle 
lies on the circle 
and 
coincide.
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
.
on the board?
and 
be positive real numbers with 
.
be the maximum possible value of 
for which the system of equations![\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]](http://latex.artofproblemsolving.com/1/3/4/13432a7d6e5c1465e08d513ee64c6c7ce99d75f9.png)
has a solution in 
satisfying 
and 
.
can be expressed as a fraction 
,
and 
are relatively prime positive integers. Find 
.
satisfying
Prove
:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
and 
.
?
lies inside triangle 
and 
.
.
.
Y by Diehard, trigonometry456103, Cyclicduck, airplanes1, fermat007, mymathboy, AkshajK, tc1729, Binomial-theorem, dantx5, mathgenius64, AwesomeToad, Amir Hossein, teethpaste, parmenides51, Adventure10, Mango247, and 3 other users
All right! After a few long weeks of compiling, revising, deleting, compiling, deleting again…We finally have finished the MOCK AIME! Note that this is definitely one of the hardest AIME’s ever written, partly because of the sheer amount of geometry necessary and it is comprised of tedious work. Do not be disheartened if you cannot manage to finish all of them; I respect those who can even do one problem; it is an accomplishment. Also because of this, the USAMO process may be a bit flawed, and therefore, if the results of the mock AIME do not reflect on true results, a.k.a. being too hard, we will create a shorter, much easier, proof oriented competition, similar in many ways to USAMO.
You will have 2 weeks to complete this competition. If you are struggling to meet this deadline at the fact that you are on vacation, have math camps, or something else like that, please PM me, and we will discuss making your submission deadline longer. The deadline is from now- 6/17/12-7/1/12- July 1st, 2012. Any submissions after this date will be void, unless you have PMed me.
We are hoping that you will treat this like the real AIME, and spend 3 hours on it, but we have no way in enforcing that. Please just remember that you are competing against other AOPSers, and honor is important.
When you are finished, please PM your answers in the format below.
Ex. Subject: MOCK AIME ANSWERS USER: (USER)
Body:
1. 244(ANS)
2. 244(ANS)
3. …
4. …
All the way to 15. Note that 244 is NOT the answer, just a test charge. We recommend that even if you do not know how to do the problem, please submit an answer for every single one, even if you are just guessing. That way, you still have a 0.01% chance at getting it right. PM them to me!
Please additionally note that since this is a mock AIME, the answers will be in the format the positive integers between 1-999, inclusive. So if your answer turns out to be something different, please PM me if you think there is a mistake.
We wish you the best of luck, and hope you do your best!
This thread will be locked until the end of the testing period. Please PM me if you have any questions or concerns. I will be posting the solutions at the end of the testing period.
EDIT: This thread will NOT be locked.
A huge thanks to Diehard and 1=2 for everything. This could never have happened with them two. Even a greater thanks to Diehard for creating everything in the first place! Thanks to 1=2 especially because of (his) amazing problem skills.
Problem 1. The area of the circle can be expressed as
, for some positive integer 
. Compute .
Change to number 4: Find
.
Number 5 is in degrees
Problem 6. Instead of "Find the ratio of the areas of these two triangles", it should say "The ratio of the areas of these two triangles can be expressed as
, where 
and are positive integers and 
is a prime. Compute 
.
Problem 10. The area of region
can be expressed as 
, for some integer . Compute .
#12: Take the answer mod 1000
Number 15 is integral.
Apologies for any inconviences this has caused.
I have attached the NEW AIME. This is the edited version, and I have changed all the mistakes. If you STILL think there are mistakes, PM me immediately.
Also note that I will accept new answers if you have PMed me the answers already, due to the mistakes.
You will have 2 weeks to complete this competition. If you are struggling to meet this deadline at the fact that you are on vacation, have math camps, or something else like that, please PM me, and we will discuss making your submission deadline longer. The deadline is from now- 6/17/12-7/1/12- July 1st, 2012. Any submissions after this date will be void, unless you have PMed me.
We are hoping that you will treat this like the real AIME, and spend 3 hours on it, but we have no way in enforcing that. Please just remember that you are competing against other AOPSers, and honor is important.
When you are finished, please PM your answers in the format below.
Ex. Subject: MOCK AIME ANSWERS USER: (USER)
Body:
1. 244(ANS)
2. 244(ANS)
3. …
4. …
All the way to 15. Note that 244 is NOT the answer, just a test charge. We recommend that even if you do not know how to do the problem, please submit an answer for every single one, even if you are just guessing. That way, you still have a 0.01% chance at getting it right. PM them to me!
Please additionally note that since this is a mock AIME, the answers will be in the format the positive integers between 1-999, inclusive. So if your answer turns out to be something different, please PM me if you think there is a mistake.
We wish you the best of luck, and hope you do your best!
This thread will be locked until the end of the testing period. Please PM me if you have any questions or concerns. I will be posting the solutions at the end of the testing period.
EDIT: This thread will NOT be locked.
A huge thanks to Diehard and 1=2 for everything. This could never have happened with them two. Even a greater thanks to Diehard for creating everything in the first place! Thanks to 1=2 especially because of (his) amazing problem skills.
Problem 1. The area of the circle can be expressed as
, for some positive integer 
. Compute .Change to number 4: Find
.Number 5 is in degrees
Problem 6. Instead of "Find the ratio of the areas of these two triangles", it should say "The ratio of the areas of these two triangles can be expressed as
, where 
and are positive integers and 
is a prime. Compute 
.Problem 10. The area of region
can be expressed as 
, for some integer . Compute .#12: Take the answer mod 1000
Number 15 is integral.
Apologies for any inconviences this has caused.
I have attached the NEW AIME. This is the edited version, and I have changed all the mistakes. If you STILL think there are mistakes, PM me immediately.
Also note that I will accept new answers if you have PMed me the answers already, due to the mistakes.
This post has been edited 10 times. Last edited by ProblemSolver1026, Jul 4, 2012, 2:45 PM
and 6*12 + 6*3 = 90?
,
.
are extensions on sides 
,
(these are vector quantities), then 
(
are positive integers and 
.
.
.
.
and the inequality is flipped. Hence 
and 
.
.
?
,
to make each angle a multiple of 30, which worked.
.
part; but you can still find it out.
does not satisfy the condition that 
,
,
