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.
be an acute-angled triangle with 
, and let 
be its circumcentre and orthocentre respectively. Points 
lie on segments 
respectively, such that ![\[\angle ZOB=\angle YOC = 90^{\circ}.\]](http://latex.artofproblemsolving.com/8/f/e/8fe34cc0f04ff9a48f1a7eee89fc50d57272d936.png)
The perpendicular line from 
to line 
meets lines 
and 
at 
respectively. Let the tangents to the circumcircle of 
at points 
and 
meet at point 
. Prove that 
are concyclic.
such that 
is also a prime number.
and the recursive relation
for all natural number 
. Find the general form of 
.
marbles in 
colors such that there are exactly 
marbles of each color. She wants to arrange them on 
shelves, 
marbles on each shelf such that for any 
colors there is a shelf that has marbles of those colors.
be an integer. The numbers 
are split into 
groups of . Prove that there are at least 
quadruples 
such that they are all in different groups, 
and 
.
, it holds that
be positive real numbers so that 
. Prove that![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
be an acute triangle with orthocenter . Prove that there is a line 
which is parallel to 
and tangent to the incircles of 
and 
.
are three positive real numbers, prove that 
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be a positive integer. A positive integer is called a -good number if it satisfies has exactly 
digits in decimal representation (it cannot have leading zeros). digits and the last digits of are considered as two separate -digit.
is a -good number because![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-good numbers.
be the set of all positive integers greater than or equal to 
and let 
be a function such that 
for all 
if 
, determine 
,
![[asy]
size(150);
draw((-1,0)--(3,0)); // ground line
draw(circle((0,1),1)); // left big circle
draw(circle((2,1),1)); // right big circle
draw(circle((1,0.25),0.25)); // small circle in between
[/asy]](http://latex.artofproblemsolving.com/d/0/d/d0d9cb2706e9b913e2051e224b8e64537c7b8332.png)
;