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.
is centered at the apex of a right square pyramid with base length 
and height 
The volume of the space where the sphere overlaps with the pyramid can be expressed in the form 
Find 
+1 w
has how many digits?
on the diagonal 
of a given rectangle 
, such that 
. The point 
is the intersection point between 
and the parallel line to 
that contains 
. Prove that the triangle 
is equilateral if and only if is a square.
satisfies 
. Find the value of 
.
be a convex quadrilateral, such that
and 
. is a kite.
be a triangle with orthocentre 
. The tangent lines from 
to the circle with diameter 
touch this circle at 
and 
. Prove that 
and are collinear.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are distinct positive integers and 
for every positive integer 
. The terms of the sequence are not known and the goal is to find the period . To do this, at each move it possible to reveal the value of a term of the sequence at your choice. is one of the first 
prime numbers, find for which values of there exist a strategy that allows to find revealing at most 
terms of the sequence.
:
Volume of a right square pyramid with base length 
and height 
prove that 
Prove that
is cut along the planes 
,
.
.