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 a triangle, 
its medial triangle, and 
a point on the Thomson cubic (= 
) of . (Suppose that 
).
be the circumcevian triangle of wrt .
be the pedal triangle of wrt .
be the reflection in 
of 
. Define 
, 
cyclically.
be the reflection in 
of . Define 
, 
cyclically.
be the reflection in 
of . Define 
, 
cyclically.
, 
, 
and the orthocentroidal circle of are coaxal.
+1 w
lines are drawn on the plane such that no three of them are concurrent and no two are parallel. At least 
of the bounded regions formed are colored black. A triangle formed by three lines is called a \textit{good triangle} if it lies entirely within a black region. Prove that there are at least 
good triangles. (A good triangle is a bounded region with finite area.)
cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city 
, an ordered list of cities 
is called an antitour from if) appears in the list exactly once, and
, it is impossible to go from to 
by a sequence of exactly 
(not necessarily distinct) flights. for any city . Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city.
be an acute-angled scalene triangle with incircle 
and circumcircle 
. Suppose touches line at 
and the tangent to at meets line at 
. Two circles passing through and tangent to meet line 
again at 
and 
.
is a trapezium.
be an arrangement of 
. Find the maximum and minimum values of 
+1 w
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
and 
for which there exists a natural number
such that the number 
is integer for all natural 
.
has as the midpoint of and orthocenter 
. Let be an arbitrary point on the nine point circle of . The line through perpendicular to 
intersects at 
. The line through perpendicular to 
intersects 
at . If 
is the midpoint of , show that 
.
that, for all and positive real numbers 
, there is![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](http://latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
that satisfy the condition![\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3.
\]](http://latex.artofproblemsolving.com/2/7/d/27d266570e3a105a4b416bb7c55904ffabf7a07c.png)
Find the maximum value of the expression![\[
P = \dfrac{yz}{\sqrt[3]{3y^2z^2+ 3x^2y^2z^2+ x^2z^2 + x^2y^2}}
+ \frac{xz}{\sqrt[3]{3x^2z^2 + 3x^2y^2z^2 + x^2y^2 + y^2z^2}}
+ \frac{xy}{\sqrt[3]{3x^2y^2 + 3x^2y^2z^2 +y^2z^2 + x^2z^2}}.
\]](http://latex.artofproblemsolving.com/b/a/f/bafc95e59d2f28dbb4e02771db38ce736e3fd45c.png)
-sequence is a sequence of 
numbers 
each equal to either 
or 
. Determine the largest 
so that, for any -sequence, there exists an integer and indices 
so that 
for all 
, and 
intersect at points 
and they are internally tangent to circle 
at points 
,respectively.
intersect with at points 
,respectively.Another circle is internally tangent to and 
at 
.Prove that 
.(C.Ilyasov)
intersect at points and they are internally tangent to circle at points ,respectively. intersect with at points ,respectively.Another circle is internally tangent to and at .Prove that .(C.Ilyasov)
and 
. By Monge 's theorem it lies on so the inversion with pole H interchanges these two circles and fixes the small circle internally tangent to them so 
so the circle with centre H and radius HY is Apollonian circle of 
and we are done. 
.
.
