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.
with incentre 
and excentres 
and 
, opposite the vertices 
and 
respectively. The incircle touches 
and 
at 
and 
respectively. Prove that the circles 
and 
have a common point other than .
and 
, then find the maximum value of ![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
. Points 
and 
lie on segments and 
, respectively. Lines 
and 
intersect at point 
. The circumcircles of triangles 
and 
intersect at a second point 
. The circumcircles of triangles and 
intersect at a second point 
. Point 
lies on segment 
such that 
. Prove that triangles 
and 
are similar.
be an acute triangle with orthocenter 
and circumcenter 
. Let and be points on the line segments and respectively such that 
is a parallelogram. Prove that 
.
, let 
be the midpoint of side 
. Let the line perpendicular to at point intersect 
at 
. If 
is tangent to at 
, find 
.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line. be an acute triangle with incentre and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
be a scalene triangle with circumcenter , incenter , and incircle 
. Let touch the sides 
, 
, and 
at points , , and respectively. Let be the projection of to 
. The line 
intersects the circumcircle of again at point 
. The circumcircles of 
and 
intersect again at points 
and 
respectively. Prove that the lines 
, 
, and 
are concurrent.
, 
, 
, , 
, represent different digits. Suppose 
is the largest number that satisfies the equation 
What is the value of 
?
, so 
. However, 
results in 
, which is not allowed, thus 
. In fact, 
results in 
which is allowed. Our final answer is 
.
)
can be is 
so let equal to since can't be greater than 
otherwise it would become a 
digit number, which wouldn't satisfy the expression to the right of the equation. digit number we note that can either be 
or as the digits must be distinct since 
. hence .
, , , , , represent different digits. Suppose is the largest number that satisfies the equation What is the value of ?
which is C
works. Then, we get C.
always equals 
then there will be no carryovers before that when doing the multiplication, because if there were carryovers, then 
which is not equal to 
So the problem just condenses into 
because if it were 
then 
then 
and if 
and if 
because 
or 
because 
We can multiply 124 by 8 to get 992, which isn't what we want, because 
in this case. So 
and when we do 
will become 
which satisfies the fact that all of the letters must be distinct.
results in 
,
Then 
.
,
,
gives 
,
and we are done.
