AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be an acute triangle with 
and let 
and 
be its orthocenter and circumcenter, respectively. Let 
be the circle 
. The line 
and the circle of radius centered at 
cross at 
and 
, respectively. Prove that , the circle on diameter 
and circle 
are concurrent.![\[
a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0
\]](http://latex.artofproblemsolving.com/0/c/e/0ce2c259010b678238f11d35fc1328ea76f945f6.png)
where 
for 
?
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)
and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let 
be the midpoint of 
. Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
be a triangle with circumcircle 
, incenter 
, and -excenter 
. Let the incircle and the -excircle hit 
at 
and 
, respectively, and let be the midpoint of arc without . Consider the circle tangent to at and arc 
at 
. If 
intersects again at 
, prove that 
and 
meet on ., points , , and lie on sides , 
, and , respectively, such that
, 
, 
, 
, 
, 
.
? be an acute triangle with 
. be the circumcircle circumscribed to the triangle and the midpoint of the smallest arc of this circle. Let and points of the segments and 
respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let 
and be the intersections of lines 
and 
with other than , respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of
. Prove that 
are real numbers, find the smallest possible value of the expression:
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and are primes and is a perfect square, prove that 
is a perfect square.
be a positive integer. Aslı and Zehra are playing a game on an 
chessboard. Initially, 
stones are placed on the squares of the board. In each move, Aslı chooses a row or a column; Zehra chooses a row or a column. The number of stones in each square of the chosen row or column must change such that the difference between the number of stones in a square with the most stones and a square with the fewest stones in that same row or column is at most 1. For which values of can Aslı guarantee that after a finite number of moves, all squares on the board will have an equal number of stones, regardless of the initial distribution?
such that 
.![\[
P = \frac{a}{a^2 + b^3} + \frac{b}{b^2 + c^2} + \frac{c}{c^2 + a^3}
\]](http://latex.artofproblemsolving.com/c/c/b/ccbc6fc8868abdf90b8e4cd97ea4f2d07f7bba28.png)
such that .
![\[
P = \frac{a}{a^2 + b^3} + \frac{b}{b^2 + c^3} + \frac{c}{c^2 + a^3}
\]](http://latex.artofproblemsolving.com/1/3/9/139529da9f360d82378a2fac5e756daba4183bbf.png)
