[/quote]
,
such that ![\[\left(g(m)+n\right)\left(g(n)+m\right)\]](http://latex.artofproblemsolving.com/5/a/c/5ac9fce315e8a524a3a01b345cfe76bc76056d68.png)
is a perfect square for all 
be an acute triangle inscribed in a circle of center 
.
intersect at 
and the circumcenter of 
is 
,
is a parallelogram.
is a perfect square number.
and 
are the midpoints of sides 
and 
respectively. Point 
lies inside the triangle. Let 
.
.
is good if there exists a permutation 
of 
such that 
is perfect square for all 
.
.
of positive integers such that 
and 
are both powers of 
.
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 
for all positive integers 
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)
consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
.
Let 
.
Y by PikaPika999
For any fixed integer 
, prove that there exists a positive integer 
, such that for any integer 
, 
can be expressed by a sum of positive integers 
's as
![\[n=a_1+a_2+\dots+a_m,\]](//latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.
, prove that there exists a positive integer 
, such that for any integer 
, 
can be expressed by a sum of positive integers 
's as![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.This post has been edited 1 time. Last edited by DinDean, Apr 23, 2025, 12:34 AM
Reason: I forgot one condition for a_i's.
Reason: I forgot one condition for a_i's.
and 
?
"
