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.
Prove that
Where 
Prove that
Prove that
Where 
Prove that
Let 
Prove that
, and 
be positive real numbers. Prove that![\[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\]](http://latex.artofproblemsolving.com/1/f/d/1fdfd64c9aafa8741ad1594db46cc16a403b1a99.png)
When does equality hold?
decide whether there exist pairwise distinct positive integers 
such that for every 
, 
divides 
.
of four distinct positive integers, we denote the sum 
by 
. Let 
denote the number of pairs 
with 
for which 
divides . Find all sets 
of four distinct positive integers which achieve the largest possible value of .
is the center of the circle. If three angles 
and 
be equal, find 
square is filled with numbers 
.The numbers inside four 
squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?
consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into 
pairs and cover each pair of points with a circle containing no other points.
. Let the vertex on 
be 
, and Let the vertex 
. 
is a parallelogram. Let side of the 
rhombus be 
and let 
. Then we get 
Now draw a line passing through point 
, parallel to 
. Say that 
, (parallel line passing through ) intersects 
. Then it is easy to see 
. ===> 
We have to show it is also equal to 
. Let the line passing through also intersect at 
. Then 
. 
==>
.
, 
. Done!
be a prime such that: 
.
,
.
.
.
and 
are arranged so that 
lies on the segment 
and 
lies on the segment 
.
.