This is from a Brazilian math contest for college students (OBMU):
Given a positive integer $n$, find the maximum value of
$$x_1^3+x_2^3+ \cdots + x_n^3$$
where $x_j$ is a real number for all $j \in \{1,2,\cdots, n\}$ such that $x_1 + x_2 + \cdots + x_n = 0$ and $x_1^2 + x_2^2 + \cdots + x_n^2 = 1$ .
On
Hint: Try to use the technique of Fourier Analysis. We can reasonably guess that the maximum occurs when $x_i = -\frac{1}{\sqrt{n(n-1)}}, \forall i<n$ and $x_n = \sqrt{1-\frac{1}{n}}.$
View $x_{\cdot}$ as a function defined on the group $\mathbf{Z}/n\mathbf{Z}$. Let $\zeta = \exp(\frac{2\pi i}{n})$ be $n$-th root of unity and define its $r$-th Fourier coefficient as $s^{}(r) = \frac{1}{n}\sum_{k=1}^{n}x_k\zeta^{-rk}.$ We note that $$x_j = \sum_{r=0}^{n-1}s(r)\zeta^{rj}, \quad \forall j=1,\ldots,n,$$ and $$ \sum_{1\leq j\leq n} |x_j|^2 = n\sum_{1\leq r\leq n}|s(r)|^2, $$ since $\frac{1}{n}\sum_{1\leq r\leq n} \zeta^{rm} =1_{\{m=0\}}$. Express the constraints in the language of $s(r)$ (including $x$ is real-valued.) Then we get $$s(r) =\overline{s(n-r)},\; s(0) = 0,\; \sum_{1\leq r\leq n}|s(r)|^2 = \frac{1}{n}. $$ What is left is to express $Q = x_1^3 + \cdots + x_n^3$ as a function involving $s(r)$'s. From the first identity above, we have $$Q = n\sum_{1\leq r,r'\leq n}s(r)\overline{s(r')}s(r-r'). $$ Note that the support is $$R = \{(r,r')\;| \;1\leq r,r'\leq n-1, r\neq r'\}.$$ By applying Cauchy-Schwarz, we have $$\begin{eqnarray}|\sum_{1\leq r,r'\leq n}s(r)\overline{s(r')}s(r-r')|^2&\leq& \sum_{(r,r')\in R}|s(r)|^2|s(r-r')|^2\sum_{(r,r')\in R}|s(r')|^2 \\ &=&\sum_{1 \leq r\leq n-1}|s(r)|^2\left(\frac{1}{n} - |s(r)|^2\right)\cdot\sum_{1 \leq r\leq n-1}\frac{1}{n} - |s(r)|^2 \\ &=&\left(\frac{1}{n^2} - \sum_{1 \leq r\leq n-1} |s(r)|^4\right)\frac{n-2}{n}\\ &\leq& \frac{(n-2)^2}{n^3(n-1)}, \end{eqnarray}$$ since $\frac{1}{n^2} \leq (n-1)\sum_{1 \leq r\leq n-1} |s(r)|^4$. Thus, we have $$Q \leq \frac{n-2}{\sqrt{n}\sqrt{n-1}},$$ as desired.
On
I come up with a solution using Lagrange Multipliers (one of the comments and one of answers suggests to use this). First, we need to realize that the domain
$$ S =\{ (x_1,x_2,\cdots,x_n) \in {\mathbb{R}}^n | x_1 + x_2 + \cdots + x_n = 0 \text{ and } x_1^2 + x_2^2 + \cdots + x_n^2 = 1\} $$
is closed (since the functions $(x_1,x_2,\cdots,x_n ) \mapsto x_1 + x_2 + \cdots + x_n$ and $(x_1,x_2,\cdots,x_n ) \mapsto x_1^2 + x_2^2 + \cdots + x_n^2$ are continuous) and bounded (since $x_1^2 + x_2^2 + \cdots + x_n^2 = 1$). Then S is compact.
Define $$f(x_1,x_2,\cdots,x_n ) = x_1^3 + x_2^3 + \cdots+ x_n^3 $$ $$g(x_1,x_2,\cdots,x_n ) = x_1^2 + x_2^2 + \cdots+ x_n^2 -1 $$ $$h(x_1,x_2,\cdots,x_n ) = x_1 + x_2 + \cdots+ x_n $$
Since $S$ is compact and $f$ is continuous, then $f$ reaches a maximum value in $S$. We can find the maximum using Lagrange Multipliers:
$$\nabla f = \lambda_1 \nabla g + \lambda_2 \nabla h$$
Thus,
$$ 3(x_1^2,x_2^2,\cdots,x_n^2) = 2\lambda_1(x_1,x_2,\cdots,x_n ) + \lambda_2(1,1,\cdots,1)$$
Then,
$$3x_i^2 = 2\lambda_1 x_i+ \lambda_2 \text{ } \forall i \in \{1,2,\cdots,n\} \label{1}\tag{1}$$
If we add all the equations and use the constraints equations, we obtain $\lambda_2= \frac{3}{n}$. Then,
$$3x_i^2 = 2\lambda_1 x_i+ \frac{3}{n} $$
Solving for $x_i$, we obtain
$$ x_i = \frac{\lambda_1 \pm \sqrt{\lambda_1^2 + \frac{9}{n}}}{3} $$
Let $k$ be a natural number less or equal to $n$. Therefore we can suppose
$$ x_i = \frac{\lambda_1 + \sqrt{\lambda_1^2 + \frac{9}{n}}}{3} \forall i \in \{1,2,\cdots,k\}$$
$$ x_i = \frac{\lambda_1 - \sqrt{\lambda_1^2 + \frac{9}{n}}}{3} \forall i \in \{k+1,k+2,\cdots,n\}$$
Firtly, notice that $k$ is different of $0$ and $n$, because of the constraint $x_1+ x_2 + \cdots + x_n =0$. Using the constraint $x_1^2 + x_2^2 + \cdots + x_n^2 = 1$ and after some simplifications, we obtain
$$\lambda_1 \left(n\lambda_1 + (2k-n) \sqrt{\lambda_1^2 + \frac{9}{n}} \right) = 0\label{2}\tag{2}$$
And using $x_1+ x_2 + \cdots + x_n =0$, we get
$$ n\lambda_1 + (2k-n) \sqrt{\lambda_1^2 + \frac{9}{n}} = 0 \label{3}\tag{3}$$
So, we just need to satisfy \ref{3}, because with that \ref{2} is automatically satisfied. Then, we get
$$\lambda_1 = \frac{3(n-2k)}{2\sqrt{(n-k)nk}} $$
Notice that we should consider the other solution with the other sign, but in the end we obtain the same maximum value for both cases. Multiplying \ref{1} by $x_i$:
$$ 3x_i^3 = 2\lambda_1x_i^2 + \frac{3x_i}{n} $$
Adding in all i's:
$$3f = 2\lambda_1 $$
Thus,
$$ f(k) = \frac{n-2k}{\sqrt{(n-k)nk}} = \frac{1}{\sqrt{n}}\left(\sqrt{\frac{n-k}{k}}-\sqrt{\frac{k}{n-k}}\right)$$
Remember that $k \in {2,3,\cdots, n-1}$. Making $x = \sqrt{\frac{k}{n-k}}$ and analyzing the derivative of
$$y: (0,1) \rightarrow \mathbb{R}$$ $$ x \mapsto\frac{1}{x}-x $$
We see that the maximum value of $f$ is when $k=1$, that is,
$$ \frac{n-2}{\sqrt{(n-1)n}} $$
One could indeed use Lagrange multipliers, but I thought I'd try to get geometric intuition in three dimensions.
This shows the constraint sphere $x_1^2 + x_2^2 + x_3^2=1$ and the constraint plane $x_1 + x_2 + x_3 = 0$, and the locus of their intersection, a (black) ring. The plane is perpendicular to the normalized vector ${\bf a} = (1/\sqrt{3}, 1/\sqrt{3}, 1/\sqrt{3})$. One (normalized) vector perpendicular to ${\bf a}$ is ${\bf b} = \left( \frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},-\frac{2}{\sqrt{6}} \right)$. Another (normalized) vector perpendicular to both is ${\bf c} = \left( -\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\right)$, determined by ${\bf c} = {\bf a} \times {\bf b}$.
Thus any potential solution point can be described as $(x_1, x_2, x_3) = \cos (\theta) {\bf b} + \sin (\theta) {\bf c}$.
Direct substitution, cubing the components and simplification yields $x_1^3 + x_2^3 + x_3^3 = -\frac{\cos (3 \theta )}{\sqrt{6}}$.
It is a simple matter to maximize this function of a single variable $\theta$ and find that the solution is $1/\sqrt{6}$.
Not surprisingly there are three equivalent solutions, corresponding to the permutation of the three variables.
As a check, I find the solution vector with $\theta = 1$ (from the graph) to be ${\bf s} = (-0.374432, 0.815587, -0.441155)$ indeed obeys the constraints and leads to the criterion sum-of-cubes to be $0.404163$, as visible on the graph.