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 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 
players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least 
next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
be a fixed positive real number. Find the minimum possible value 
such that for any positive reals 
, 
, 
, 
, we have
and with no common divisor greater than 1 such that 
divides 
. (Professor Yongjin Song)
and 
. Prove that there always exists a positive integer 
such that 
. is a power of a prime number, but for arbitrary then well.....
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
, , with , . Prove that there exists such that for every (For and , the notation represents . )
lie on the arc of the circumcircle of 
(this arc does not contain 
) such that 
. Let denotes the midpoint of the chord 
. Show that 
.
be the circumcenter of the . Then 
. We draw a normal line from the point 
at 
and let it intersect the circumcircle in the point
. Since 
we have that is a line of symmetry of 
. According to this,
. Now, from the triangle inequality
.
, so
. That is why 
.
![\[
\frac{5^{2000} - 1}{4}
\]](http://latex.artofproblemsolving.com/8/b/a/8ba5d1ed8bafd6653f9e6f351558a9976c43a57a.png)
is divided by 
.
is equivalent to 
,
.
and the triangle inequality imply that the left hand side of the desired inequality is at least 
.