Problem 1 (yau-contest 2014)
Section: Analysis

Calculate the integral:

$$\int_0^{\infty} \frac{\log x}{1 + x^2}\, dx.$$

Problem 2 (yau-contest 2014)
Section: Analysis

Construct an increasing function on $\mathbb{R}$ whose set of discontinuities is precisely $\mathbb{Q}$.

Problem 3 (yau-contest 2014)
Section: Analysis

Prove that any bounded analytic function $F$ over $\{ z \mid r < |z| < R \}$ can be written as $F(z) = z^{\alpha} f(z)$, where $f$ is an analytic function over the disk $\{ z \mid |z| < R \}$ and $\alpha$ is a constant.

Problem 4 (yau-contest 2014)
Section: Analysis

Let $D \subset \mathbb{R}^n$ be a bounded open set, $f : \bar{D} \to \bar{D}$ is a smooth map such that its Jacobian $\left| \frac{\partial f}{\partial x} \right| \equiv 1$, where $\bar{D}$ denotes the closure of $D$.

Prove

(a) for each small ball $B_{\varepsilon}(x)$, there exists a positive integer $k$ such that $f^{k}(B_{\varepsilon}(x)) \cap B_{\varepsilon}(x) \neq \varnothing$, where $B_{\varepsilon}(x)$ denotes the ball centered at $x$ with radius $\varepsilon$;

(b) there exists $x \in \bar{D}$ and a sequence $k_1, k_2, \cdots, k_j, \cdots$ such that $f^{k_j}(x) \to x$ as $k_j \to \infty$.

Problem 5 (yau-contest 2014)
Section: Analysis

Let $u$ be a subharmonic function over a domain $\Omega \subset \mathbb{C}$, i.e., it is twice differentiable and $\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \ge 0$. Prove that $u$ achieves its maximum in the interior of $\Omega$ only when $u$ is a constant.

Problem 6 (yau-contest 2014)
Section: Analysis

Suppose that $\phi \in C_0^{\infty}(\mathbb{R}^n)$, $\int_{\mathbb{R}^n} \phi\, dx = 1$. Let $\phi_{\varepsilon}(x) = \varepsilon^{-n} \phi(x/\varepsilon)$, $x \in \mathbb{R}^n$, $\varepsilon > 0$. Prove that if $f \in L^p(\mathbb{R}^n)$, $1 \le p < \infty$, then $f * \phi_{\varepsilon} \to f$ in $L^p(\mathbb{R}^n)$, as $\varepsilon \to 0$. It is not true for $p = \infty$.