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 real numbers such that 
. Find the value of 
be real numbers such that 
and 
. Find the value of 
+1 w
, let 
be a point on arc 
of circle which doesn't contain 
. 
and intersect at 
. Let 
and 
be the reflection of about to 
and 
, respectively. 
intersects at 
, and 
intersects at 
. Prove that circumcenter of triangle 
lies on .
Prove that
Let 
Prove that
Let 
Prove that![$$ a+b^2+c\geq \frac{135 \sqrt[3]{90}-2}{2}$$](http://latex.artofproblemsolving.com/8/6/6/866623c65f54d6147a43b1b241c2cba96863b624.png)
such that 
for all 
such that 
.
be a positive integer such that the sum of its positive divisors is at least 
. Prove that has at least 
distinct prime factors.
is known as an interesting number if satisfies![\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]](http://latex.artofproblemsolving.com/0/c/a/0ca11fa76313b65fd76a3cceee90955960e44ef6.png)
.
is given. Find all functions 
such that for all 
we have
be an acute triangle, 
its orthocenter and the center of its nine point circle. Let be a point on the parallel through 
to such that 
and and are on different sides of and a point on the parallel through 
to such that 
and and are on different sides of . If 
and 
are the reflections of over and respectively, and 
are the intersections of 
and 
respectively with the circumcircle of , prove that the intersection of lines 
and 
lies on 
. be an acute triangle and its orthocenter. Let be a point on the parallel through to such that 
. Define and as points on the parallels through to and through to similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?
is defined by 
. Find all terms which are perfect squares.
be an acute triangle with 
. Let be the centroid of triangle and let 
be the foot of the perpendicular from to side . The median 
intersects the circumcircle 
of triangle at a second point . Let 
be the point where intersects . The line 
intersects the circle at a point , such that lies between and . Prove that the points 
and lie on a circle.
is diameter of 
then 
is circumcenter of 
Further 
is cyclic 
is a median of 30-60-90 
, so 
. 
. Thus 
. 
, so 
. So is cyclic and 
.
Prove that
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
perpendiculars to 
Let 
and 
If 
find 
