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-angled triangle with 
. Let 
be the orthocenter of triangle 
, and let 
be the midpoint of the side 
. Let 
be a point on the side 
and 
a point on the side 
such that 
and the points , , are on the same line. Prove that the line 
is perpendicular to the common chord of the circumscribed circles of triangle and 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?
, define 
and 
, where 
and 
denote the product and sum of the digits of , respectively. Find all solutions to 
be a triangle with 
, and let be the foot of the altitude from 
. A point is chosen inside the triangle 
so that 
bisects 
. Let 
be the intersection point of the lines 
and . Let 
be the semicircle with diameter that meets the segment 
at an interior point. A line through is tangent to at 
. Prove that the lines 
and meet on .
is a rational number.
and other permutations.
be a triangle such that![\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]](http://latex.artofproblemsolving.com/6/9/b/69b1964be5f92eb44a36b0b8604bf473fe27e210.png)
and 
denote its semiperimeter and its inradius, respectively. Prove that triangle is similar to a triangle 
whose side lengths are all positive integers with no common divisors and determine these integers.
be a triangle with circumcenter 
. Point 
is the intersection of the parallel line from to with the perpendicular line to from . Let 
be the point where the external bisector of 
intersects with . Let 
be the projection of onto 
. Prove that the lines 
have a common point.
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.
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 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 . 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
are real numbers, find the smallest possible value of the expression:
be a natural number . Define 
be cut number at 
if
Prove that if 
be cut number at 
prove that 
.
