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
(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and 
be two fivedigit integers with the same permutation sum.
.
be a triangle. Let 
be a points on the segment 
such that 
. Let 
be the mid-point of 
. Let 
intersect 
in 
and 
in 
respectively. Determine 
.
, Its incircle 
and 
escircle 
are tangent to at 
and 
respectively. 
intersects line at 
. Line intersects at , and line intersects at 
. Line 
intersects at 
. Prove that 
.
that satisfy the following condition for any real numbers 
and 
such that:
be a triangle with circumcenter 
. Let 
be the midpoints of 
and 
respectively and let 
be the projection of on 
. Let be the projection of 
on 
. Let 
intersect 
at points 
and 
. Let 
intersct at points 
. Let 
intersect 
at 
and 
intersect at 
. Then show that 
are concurrent.
+1 w
and 
respectively be the incentre, circumcentre and circumcircle of triangle . Points 
are chosen on , such that 
, and point 
is the foot of the altitude from 
to 
. If 
are similarly defined, prove that lines 
and 
concurr on 
.
such that, for all real numbers 
with 
, 
)
)
:
Volume of a right square pyramid with base length 
and height 
Prove that
and let 
and 
and 
respectively. Let the circle with centre 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
