be a triangle, 
its medial triangle, and 
)
).
be the circumcevian triangle of 
be the pedal triangle of 
be the reflection in 
of 
.
,
cyclically.
be the reflection in 
of 
,
cyclically.
be the reflection in 
of 
,
cyclically.
,
,
and the orthocentroidal circle of 
be an acute-angled scalene triangle with incircle 
and circumcircle 
.
and the tangent to 
meets line 
.
again at 
is a trapezium.
be an arrangement of 
.
cups labeled 
,
-
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 numbe
is integer for all natural 
.
has 
.
intersects 
.
intersects 
at 
is the midpoint of 
.
that, for all 
,![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](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)
.
.
-
numbers 
each equal to either 
or 
.
so that, for any 
and indices 
so that 
for all 
,
Y by Adventure10
Let 
be a square with sides length 
. Let 
be a path within which does not meet itself and which is composed of line segments 
with 
. Suppose that for every point 
on the boundary of there is a point of at a distance from no greater than 
. Prove that there are two points 
and 
of such that the distance between and is not greater than 
and the length of the part of which lies between and is not smaller than 
.
be a square with sides length 
. Let 
be a path within which does not meet itself and which is composed of line segments 
with 
. Suppose that for every point 
on the boundary of there is a point of at a distance from no greater than 
. Prove that there are two points 
and 
of such that the distance between and is not greater than 
and the length of the part of which lies between and is not smaller than 
.
is quite trivial in one direction, but if they ean the shortest distance, we just take points very close to the boundary and with angles close to 
degrees except in the corners, what is the mistake in those ideas?
.
to 
along 
from the current point. Obviously at any point the red points are a finite union of red segments (with possibly length zero). Suppose such points 
and 
are opposite sides of squares, then we must completely paint 
.
.
,
and 
was painted before 
