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.
of the pizza, where 
is a positive integer. Find .
is called stylish if the set 
can be partitioned into 
pairs such that the sum of the elements in each pair is a power of 
. For example, 
is stylish because the set 
can be partitioned as 
, with sums , 
, and respectively. Determine the number of stylish numbers less than 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
that verify the following equation for all 
:
is a prime number
to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
and 
, but am confused on how to prove this statement for general vectors 
and 
although its supposed to be a quick comparison.
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
such that 
.
be a quadrilateral inscribed in a circle 
, and let 
be a point on the extension of 
such that 
and 
are tangent to . The tangent at 
intersects at 
and the line 
at 
. Let 
be the second point of intersection between 
and . Prove that 
, , are collinear.
be a triangle with incenter 
, and suppose the incircle is tangent to 
and 
at and 
. Denote by 
and 
the reflections of and over . Let be the intersection of 
with 
, and let 
be the midpoint of . Prove that 
and 
are perpendicular.
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of . Let 
be the circle with diameter . Let 
be the intersection of 
and . Let be the orthocenter of . Let 
be the intersection of and . Let 
be the lines through tangent to 
respectively. Let be the intersection of 
and 
. Let be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter , centroid and orthocenter . Prove that 
is constant.
opposite the vertex 
be tangent to the side at the point 
. Define the points 
on and 
on analogously, using the excircles opposite and , respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled.
centimetres. Prove that a circular coin of radius 
centimetre can be placed to cover at least twelve of the points.
. We see that any vertex along with the two closest midpoints and form a cyclic quadrilateral of two right angle triangles. Since the hypotenuse for each of these triangles is the distance from to a vertex, or , this quadrilateral's circumcircle has radius . Thus, all six of these regions can be covered entirely by a circle of radius . By pigeonhole principle, one of these regions must contain 
points, and we are done. 
and 
Prove that
