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.
such that for all integer 
and a subset 
of ![$[n],$](http://latex.artofproblemsolving.com/d/2/7/d27d2715f4f95c24a28b7c6371a006d4c803b2cf.png)
satisfy 
for all 
then 
+1 w
be the set of positive integers. Find all functions 
such that for all 
, ![\[m^2f(m) + n^2f(n) + 3mn(m + n)\]](http://latex.artofproblemsolving.com/0/f/f/0ff783fa217b7be4bda5006e417f1471ea3a6fca.png)
is a perfect cube.
![$\mathbb{R}[x]$](http://latex.artofproblemsolving.com/a/8/8/a88b9f4858016b1771635c611d44b56161a08009.png)
be the set of all polynomials with real coefficients. Find all functions ![$f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$](http://latex.artofproblemsolving.com/0/3/3/033be3ab043bd47e537f01791c813739ad5c7da5.png)
satisfying the following conditions:
maps the zero polynomial to itself,![$P \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/f/7/c/f7c4c970908aa5570a9afcc45fc16c1d36cc3d5d.png)
, 
, and![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
, the polynomials 
and 
have the same set of real roots.
be a square of center 
and let 
be the symmetric of the point 
with respect to point 
. Let 
be the intersection of 
and 
, and let be the intersection of 
and 
. Show that 
is the angle bisector of 
.
be reals such that 
Show that
Let 
and Show that
be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers 
represent the number of unit squares that have exactly 
edges on the boundary of respectively. Find the largest real number 
such that the inequality 
holds for each polygon constructed with these conditions.
, and for a positive integer 
, prove that
.
and 
with real coefficients satisfying
![\[
a_1 = 1, \quad a_2 = 1 + \frac{1}{2}, \quad a_3 = 1 + \frac{1}{2} + \frac{1}{3}, \quad \dots, \quad a_{2024} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2024}
\]](http://latex.artofproblemsolving.com/0/7/d/07d972e3d517cb54a6822e2b4273e1fc5d07604d.png)
![\[
A = \frac{1}{a_1^2} + \frac{1}{2a_2^2} + \frac{1}{3a_3^2} + \dots + \frac{1}{2024 a_{2024}^2}.
\]](http://latex.artofproblemsolving.com/6/a/8/6a8f1300773d8753b775294370d54ee3693a5e03.png)
![\[
\frac{11}{9} < A < 2.
\]](http://latex.artofproblemsolving.com/7/d/8/7d87464c995366c8214e2f6699d26952e385d8e5.png)
and 
.
Prove that
and 
Find the value range of 
,
.
![$\frac{a^4+b^4}{a^3+b^3}+\frac{b^4+c^4}{b^3+c^3}+\frac{c^4+a^4}{c^3+a^3}\ge a+b+c \ge3\sqrt[3]{abc}=3$](http://latex.artofproblemsolving.com/2/7/6/2761477d952109a0bb7f6fe4a336a89d2b1b0ded.png)
Equality iff 
be positive real numbers.Prove that 
be positive real numbers.Prove that 
equality iff 
equality iff 
.
and 
.
Expand the left side, then use AM-GM to prove the inequality
which is true (all terms on left are square of right)
be positive real numbers such that 
and 
Prove that
Note by AM-GM that
as desired. (The inequality is strict because the equality case 
is not possible as given in the problem statement.)
and 
be positive numbers such that 
Prove that
,
and replace 
by a new 
and 
by a new 
.
.
which, due to both sides being positive, is equivalent to squaring both sides
which in turn is equivalent to its both sides squared again then the terms collected and factored
QED
be positive numbers such that 
Prove that
is convex on 
.
QED
as suggested by the attachment to the above post?
The same procedure applies to the term involving 
and 
Prove that
be positive reals. Prove that
and 
Prove that
be positive numbers such that 
Prove that 
be positive numbers such that ![\[ \overset{n}{\underset{i=1}{\sum}} \frac{1}{a_i+n-1} \le 1\]](http://latex.artofproblemsolving.com/d/6/b/d6b71e62d8728548b4dce109e9dac1a9b5e29b77.png)
Prove that