Problem 1 (yau-contest 2013)
Section: Analysis and Differential Equations

Suppose that $f$ is an integrable function on $\mathbb{R}^d$. For each $\alpha>0$, let $E_\alpha=\{x\mid |f(x)|>\alpha\}$. Prove that:
$$\int_{\mathbb{R}^d} |f(x)|\,dx=\int_0^\infty m(E_\alpha)\,d\alpha.$$

Problem 2 (yau-contest 2013)
Section: Analysis and Differential Equations

Let $p(z)$ be a polynomial of degree $d\ge 2$, with distinct roots $a_1,a_2,\cdots,a_d$. Show that
$$\sum_{i=1}^d \frac{1}{p'(a_i)}=0.$$

Problem 3 (yau-contest 2013)
Section: Analysis and Differential Equations

Let $\alpha$ be a number such that $\alpha/\pi$ is not a rational number. Show that:
1) $\displaystyle \lim_{N\to\infty}\frac1N\sum_{n=1}^N e^{ik(x+n\alpha)}=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{ikt}\,dt.$
2) For every continuous periodic function $f: \mathbb{R} \to \mathbb{C}$ of period $2\pi$, we have
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^N f(x+n\alpha)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)\,dt.$$

Problem 4 (yau-contest 2013)
Section: Analysis and Differential Equations

Let $u$ be a positive harmonic function over the punctured complex plane $\mathbb{C}/\{0\}$. Show that $u$ must be a constant function.

Problem 5 (yau-contest 2013)
Section: Analysis and Differential Equations

Suppose $H=L^2(B)$, $B$ is the unit ball in $\mathbb{R}^d$. Let $K(x,y)$ be a measurable function on $B\times B$ that satisfies
$$|K(x,y)|\le A|x-y|^{-d+\alpha}$$
for some $\alpha>0$, whenever $x,y\in B$. Define
$$Tf(x)=\int_B K(x,y)f(y)\,dy.$$
(a) Prove that $T$ is a bounded operator on $H$.
(b) Prove that $T$ is compact.

Problem 6 (yau-contest 2013)
Section: Analysis and Differential Equations

Let $A$ be a $n\times n$ real non-degenerate symmetric matrix. For $\lambda\in\mathbb{R}^+$, we define: $\displaystyle \int_{\mathbb{R}} \exp(i\lambda x^2)\,dx = \lim_{\varepsilon\to 0^+}\int_{-\infty}^{\infty}\exp(i\lambda x^2-\tfrac12\varepsilon x^2)\,dx$. Show that:
$$\int_{\mathbb{R}^n} \exp\big(i\tfrac{\lambda}{2}\langle A x, x\rangle - i\langle x,\xi\rangle\big)\,dx
= \left(\frac{2\pi}{\lambda}\right)^{n/2} |\det(A)|^{-1/2} \exp\left(-\frac{i}{2\lambda}\langle A^{-1}\xi,\xi\rangle\right)\exp\left(i\frac{\pi}{4}\operatorname{sgn}A\right).$$
Here $\lambda\in\mathbb{R}^+$, $\xi\in\mathbb{R}^n$, $\operatorname{sgn}(A)=\nu_+(A)-\nu_-(A)$, $\nu_+(A)$ ($\nu_-(A)$) is the number of positive (negative) eigenvalues of $A$.