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.
and radius 
.
. and radius ..
exist that satisfy 
for all real 
and the domain of f is strictly within the set of 
, the range being 
?
consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
consecutive even numbers, such that the square of their product divides the sum of their squares.
prime and 
a positive integer. Determine all pairs 
satisfying the equation: 
be a positive integer and let 
and 
be sets of integers satisfying the following conditions:
, 
and is a subset of 
, 
iff 
.
be a prime such that: 
. Simplify:
be a given integer. The Mint issues coins of 
different values 
, where each 
is a positive integer (the number of coins of each value is unlimited). A set of values 
is called lucky, if the sum 
can be collected in a unique way (namely, by taking one coin of each value).
with 
(b) Prove that every lucky set of values satisfies 
with edges 
. three perpendiculars to the three lateral faces. from the apex such that these three points lie above the apex.
and 
.![$\sqrt[3]{ab-1}+\sqrt[3]{bc-1}+\sqrt[3]{ca-1} \le \sqrt[3]{(a+b+c)^2}$](http://latex.artofproblemsolving.com/9/6/3/9633b5849794285e44519c745c1f616eceb77711.png)
.
Let 
.
and 
.
,
.
is tangent to the circle at point 
.
represent the midpoint of 
and 
represent the intersection between 
.
