Problem 1 (yau-contest 2012)
Section: Analysis and Differential Equations

Compute the integral $$\int_{0}^{\infty} \frac{x^{p}}{1+x^{2}}\,dx,\quad -1<p<1.$$

Problem 2 (yau-contest 2012)
Section: Analysis and Differential Equations

Construct a one to one conformal mapping from the region
$$U=\{z\in C\mid |z-\tfrac{i}{2}|<\tfrac{1}{2}\}/\{z\mid |z-\tfrac{i}{4}|<\tfrac{1}{4}\}$$
onto the unit disk.

Problem 3 (yau-contest 2012)
Section: Analysis and Differential Equations

Let $f(x)$ be a nonlinear $C^{2}$ function on $\mathbb{R}$. Show that
$$\sup |f'(x)|^{2} \le 4\, \sup |f(x)|\, \sup |f''(x)|.$$

Problem 4 (yau-contest 2012)
Section: Analysis and Differential Equations

Let $f(x)$ be a real measurable function defined on $[a,b]$. Let $n(y)$ be the number of solutions of the equation $f(x)=y$. Prove that $n(y)$ is a measurable function on $\mathbb{R}$.

Problem 5 (yau-contest 2012)
Section: Analysis and Differential Equations

For $1<p,q<\infty$, $\tfrac{1}{p}+\tfrac{1}{q}=1$, let $g\in L^{q}$. Consider the linear functional $F$ on $L^{p}$ given by: $F(f)$ is equal to the integral of $fg$. Show that $$\|F\|=\|g\|_{q}.$$

Problem 6 (yau-contest 2012)
Section: Analysis and Differential Equations

Let $\mathbb{R}^{n}_{+}=\{x=(x_{1},x_{2},\ldots,x_{n})\in \mathbb{R}^{n}\mid x_{n}>0\}$. Show that the formula
$$u(x)=\frac{2x_{n}}{n\,\alpha_{n}}\int_{\partial \mathbb{R}^{n}_{+}} \frac{g(y)}{|x-y|^{n}}\,dy,\qquad x\in \mathbb{R}^{n}_{+},$$
is a solution of the problem
$$\Delta u=0\text{ in }\mathbb{R}^{n}_{+},\qquad u=g \text{ on } \partial\mathbb{R}^{n}_{+},$$
where $\alpha_{n}$ is the volume of the unit $n$ dimensional sphere.