[/quote]
,
,
and 
,
and 
denote the product and sum of the digits of 
![\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]](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 
with the perpendicular line to 
be the point where the external bisector of 
intersects with 
be the projection of 
.
have a common point.
and let 
and 
be the circle 
.
and the circle of radius 
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)
,
,
.
be the midpoint of arc 
at 
intersects 
,
and 
meet on 
.
.
be the second intersection point of the circumcircle circumscribed to 
with 
and 
and 
with 
,
and 
and 
with lines 
line passes through the midpoint of 
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.
is a perfect square.
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 
reals, and 
.
?
Y by Mango247
Consider acute triangles 
and 
, with 
, such that 
and 
are on opposite sides of line 
. Denote by 
the foot of the perpendicular line to 
through 
and by 
the foot of the perpendicular line to 
through 
. Let 
be the orthocenter of triangle and 
be the orthocenter of . Show that lines 
and 
are concurrent.
and 
, with 
, such that 
and 
are on opposite sides of line 
. Denote by 
the foot of the perpendicular line to 
through 
and by 
the foot of the perpendicular line to 
through 
. Let 
be the orthocenter of triangle and 
be the orthocenter of . Show that lines 
and 
are concurrent.
