Problem 1 (yau-contest 2022)
Section: Analysis and Differential Equations

For $n \ge 1$, consider the integral
$$I_n = \int_{[0,1]^n} \frac{n}{x_1 + x_2 + \cdots + x_n}\, dx_1 \cdots dx_n.$$
Prove that $\lim_{n\to\infty} I_n$ exists.

Problem 2 (yau-contest 2022)
Section: Analysis and Differential Equations

Let $U \subset \mathbb{C}$ be a non-empty open set and $f : U \to U$ be a non-constant holomorphic function. Prove that, if $f \circ f = f$, then $f(z) \equiv z$ for all $z \in U$.

Problem 3 (yau-contest 2022)
Section: Analysis and Differential Equations

Let $X \subset \mathbb{R}$ be a set with positive (Lebesgue) measure. Show that we can find an arithmetic progression of $2022$ terms in $X$, i.e., there exist $x_1, \ldots, x_{2022} \in X$ so that the $x_{i+1}-x_i$'s are all equal and positive, $i=1, \ldots, 2021$.

Problem 4 (yau-contest 2022)
Section: Analysis and Differential Equations

Let $C([0,1])$ be the space of all continuous $\mathbb{C}$-valued functions equipped with $L^\infty$-norm. Let $P \subset C([0,1])$ be a closed linear subspace. Assume that the elements of $P$ are polynomials. Prove that $\dim P < \infty$.

Problem 5 (yau-contest 2022)
Section: Analysis and Differential Equations

Let $\Omega \subset \mathbb{R}^3$ be a bounded domain with smooth boundary. Assume that $u \in C(\mathbb{R}^3 \setminus \Omega)$ is a harmonic function on $\mathbb{R}^3 \setminus \Omega$ so that $u|_{\partial \Omega} = 1$ and $\lim_{|x|\to\infty} |u(x)| = 0$.

Prove that for such $u$, $\lim_{|x|\to\infty} |x|\,u(x)$ exists.

Problem 6 (yau-contest 2022)
Section: Analysis and Differential Equations

Let $f(x,y) \in C^1(\mathbb{R}^2)$. We assume that there exists $C>0$ so that for all $(x,y) \in \mathbb{R}^2$, $\left|\frac{\partial f}{\partial y}(x,y)\right| \le C$. Prove that the following ODE has a globally defined solution for all $y(0) = y_0 \in \mathbb{R}$:
$$
\begin{cases}
\frac{d}{dx} y(x) = f(x, y(x)), \\
 y(0) = y_0.
\end{cases}
\tag{1}
$$
In addition, we assume that $f$ is $1$-periodic in $x$, i.e., for all $(x,y) \in \mathbb{R}^2$, we have $f(x+1, y) = f(x, y)$. Prove that if (1) admits a globally defined bounded solution, then (1) admits a periodic solution.