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.
of positive integers satisfying 
and 
for all positive integers 
and 
For a given positive integer 
find the maximum possible value of 
such for any 
:
.
and 
be nonzero real numbers satisfying![\[
a_1^2 b_1^2 (a_1 + b_1) + a_2^2 b_2^2 (a_2 + b_2) + \cdots + a_n^2 b_n^2 (a_n + b_n) \leq 7,
\]](http://latex.artofproblemsolving.com/a/7/e/a7ea7af24ac529090f3b2e305be43e2728b79eb4.png)
![\[
\frac{1}{a_1} + \cdots + \frac{1}{a_n} = \frac{1}{4}, \quad \frac{1}{b_1} + \cdots + \frac{1}{b_n} = \frac{1}{3}.
\]](http://latex.artofproblemsolving.com/e/f/3/ef341b850979a9b780f5daaa49f46f1d1324cdbd.png)
Find the maximum value of![\[
a_1 b_1 + a_2 b_2 + \cdots + a_n b_n.
\]](http://latex.artofproblemsolving.com/d/3/3/d338bf5be4f1a218e150bfd84173e988ee92ddad.png)
![\[
P(x) = a_0 + a_1 x^2 + a_2 x^4 + \cdots + a_{10} x^{20}
\]](http://latex.artofproblemsolving.com/0/d/c/0dcd08456f253de08f3b880c7a1fddc07b335932.png)
be a polynomial of degree 20 with only even powers of 
.
be 
.![\[
(x_1^2 + 1)(x_2^2 + 1) \cdots (x_{20}^2 + 1) = 2025,
\]](http://latex.artofproblemsolving.com/b/a/4/ba4bc5f526565e56a9a685fe01572c1366241084.png)
find the **minimum value** of 
.
?
by the recurrence![\[
a(n) = \left| a(n-1) - a(n-2) \right|
\]](http://latex.artofproblemsolving.com/e/7/9/e7981725c3dfd9fd1a0e22c17eded1a4cde84e9f.png)
for all 
, with initial values 
, 
, where 
.
, there exists a positive integer 
such that![\[
a(i) = a(i+3)
\]](http://latex.artofproblemsolving.com/d/c/7/dc75144ca27873b2420e79b5e8f045be864a4c2e.png)
for all integers 
.
be positive real numbers satisfying![\[
a^2 + c^2 = b(a + c).
\]](http://latex.artofproblemsolving.com/9/c/0/9c0730d15b1177ee23d3f65bc2fe8b2cc85d0c62.png)
Let![\[
m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).
\]](http://latex.artofproblemsolving.com/e/b/6/eb6f340defa4b38e35292062b202d138b9de843b.png)
Find the value of 
. with real coefficients such that for all reals 
such that 
we have the following relations![\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]](http://latex.artofproblemsolving.com/4/4/9/4491a53f9c6f96c18e42eb984b8e83566140bfdb.png)
. Let 
and 
be the points on sides 
and 
respectively. Circumcircles of 
and 
intersect at the points 
and 
. Prove that is a point on side 
if and only if 
?
satisfy that: 
and 
Prove that
,![\[
A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\},
\]](http://latex.artofproblemsolving.com/e/9/2/e92becb6264b8e5a9753f2288436a48523a25f29.png)
and![\[
B = \left[2\sqrt{ab}, a + b\right].
\]](http://latex.artofproblemsolving.com/2/d/2/2d25aa634389b768dc26408480bb71b316c0bf17.png)
.
,
