[/quote]
,
such that :
.
such that 
,
and 
to obtain 
and 
,
be the intersection of the altitude through 
and the median through 
.
be the intersection of the altitude through 
and line 
.
be the intersection of the median through 
.
be an even positive integer. A sequence of 
with 
either the sum of the first 
.
and 
.
and 
intersect at point 
.
be the reflection of 
and 
intersect at point 
,
.
.
of positive integers with 
is called good if we can colour each integer red, blue, green or purple, in such a way that
of each 
consecutive integers at least one is coloured red;
of each 
consecutive integers at least one is coloured blue;
of each 
consecutive integers at least one is coloured green;
of each 
consecutive integers at least one is coloured purple.
are positive real numbers satisfied 
.
is written on the blackboard. Every time, Carmela can erase the number 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
be a square with sides length 
.
be a path within 
with 
.
.
and the length of the part of 
.
and the sets 
If 
has exactly three elements, prove that 
and 
are divisible by 
is the sum of digits of 
.
other books each in one copy?
.
such that![\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\]](http://latex.artofproblemsolving.com/4/d/2/4d269e5b682be6778e89237aa91da5e0399042a7.png)
so that the sum of its edges' lengths is minumum.
Y by
Let 
be a point in the plane, distinct from the vertices of 
. Consider 
the reflections of with respect to lines 
and 
, in this order.
a) Prove that
are collinear if and only if lies on the circumcircle of .
b) If does not lie on the circumcircle of and the centroids of triangles and 
coincide, prove that is equilateral.
be a point in the plane, distinct from the vertices of 
. Consider 
the reflections of with respect to lines 
and 
, in this order.a) Prove that
are collinear if and only if lies on the circumcircle of .b) If
does not lie on the circumcircle of and the centroids of triangles and 
coincide, prove that is equilateral.
,
,
.
.
.
lies on unit circle, hence reflecting 
and we have an equilateral triangle well-known.
