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.
is a perfect square number.
the set of all positive real numbers.
which have the property:![\[f(x)f(y)=2f(x+yf(x))\]](http://latex.artofproblemsolving.com/4/f/f/4ffbde35ff082585a188c1a4368d39338d1b07a4.png)
and 
.
be a function that satisfies the following equation for all real numbers 
:![\[
f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17.
\]](http://latex.artofproblemsolving.com/f/6/2/f625b922656061221157fdb5a3b958cfa7d57108.png)
Find the value of 
.
and 
such that does not divide 
for all positive integers 
.
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 
be a tetrahedron and 
its incenter, and let the line 
be perpendicular to 
. Find the angle between the planes 
and 
denote the planes 
For conveniece, let's cut the trihedron with the spherical surface 
with center 
and radius 
Rays 
cut at 
thus 
become internal angle bisectors of the spherical triangle 
Great circle 
cuts 
at 
such that 
coincide with the projections of on 
If 
is the projection of on 
then 
and 
Thus 
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 
.
