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.
![\begin{align*}
\frac{a}{c} &= b-\sqrt{b}+c \\
\sqrt{\frac{a}{c}} &= \sqrt{b}+1 \\
\sqrt[4]{\frac{a}{c}} &=\sqrt[3]{b}-1
\end{align*}](http://latex.artofproblemsolving.com/0/c/d/0cd22d36acedbcb10d76d1b884aa35418273ed52.png)
be a convex pentagon such that 
, 
, and 
. Prove that the perpendicular line from 
to 
and the line segments 
and 
are concurrent.
+1 w
be positive integers. Prove that the expression![\[\frac{\gcd(a+b,ab)}{\gcd(a,b)}\]](http://latex.artofproblemsolving.com/1/b/7/1b78a2027c3b7156fbb566901d2cfd088cc09f00.png)
is always a positive integer, and determine all possible values it can take.
that satisfies: for all different 
, we have 
?
be a triangle with circumcenter 
and internal bisector 
. Let cuts 
again at 
and 
cuts again at 
. Point 
lie on such that 
. The line pass through and perpendicular to intersects 
at 
respectively. Let circumcircle of 
cuts at 
. Prove that 
.
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist. and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
be the number of spanning trees in a graph 
, where may have multiple edges and loops.
and 
of , let 
, 
, and 
denote the graphs obtained by contracting the edges , , and both and in 
, respectively.![\[
k(G/\{e,f\}) \cdot k(G) \leq k(G/e) \cdot k(G/f)
\]](http://latex.artofproblemsolving.com/7/8/2/78296088b860654fe0310b75c3aaeaf980789cef.png)
over the circle in some order so that among any 
successive numbers at least 
numbers are multiplies of 
or 
(or both and ) ?
where a and b are relatively prime integers, what is b − a?
be the set of real numbers. Determine all functions 
such that 
and for any real numbers 
and 
,
and 
on sides 
and 
respectively of triangle are such that triangles and 
are similar, the angles at and being equal to those at 
and 
respectively. Find and so that triangle has minimum area.
and on sides and respectively of triangle are such that triangles and are similar, the angles at and being equal to those at and respectively. Find and so that triangle has minimum area.
, 
and 
concur at a point . By trivial angle chasing is the orthocenter of 
and then it is also the circumcenter of 
. That means 
, 
and 
are shortest when 
, 
, and are the midpoints of the sides of , it is trivial to check that the medial triangle is similar to 
satisfying
Prove
the following equation :
