Problem 1 (yau-contest 2010)
Section: Analysis and Differential Equations

a) Let $x_k$, $k=1,\ldots,n$ be real numbers from the interval $(0,\pi)$, and define
$$x=\frac{1}{n}\sum_{i=1}^n x_i.$$
Show that
$$\prod_{k=1}^n \frac{\sin x_k}{x_k} \le \left(\frac{\sin x}{x}\right)^n.$$

b) From
$$\int_0^{\infty} e^{-x^2}\,dx=\frac{\sqrt{\pi}}{2},$$
calculate the integral
$$\int_0^{\infty} \sin(x^2)\,dx.$$

Problem 2 (yau-contest 2010)
Section: Analysis and Differential Equations

Let $f:\mathbb{R}\to\mathbb{R}$ be any function. Prove that the set of points $x$ in $\mathbb{R}$ where $f$ is continuous is a countable intersection of open sets.

Problem 3 (yau-contest 2010)
Section: Analysis and Differential Equations

Let $f(z)$ be holomorphic in $D:\ |z|<1$ and $|f(z)|\le 1$ ($z\in D$). If $z_0$ is a point in $D$ such that both $z_0$ and $-z_0$ are zeros of order $m$ of $f(z)$ and $0<|z_0|\le \frac{m-1}{m}$, then $|f(0)|<e^{-2}.$

Problem 4 (yau-contest 2010)
Section: Analysis and Differential Equations

Find a harmonic function $f$ on the right half-plane such that, when approaching any point in the positive half of the $y$-axis, the function has limit $1$, while, when approaching any point in the negative half of the $y$-axis, the function has limit $-1.$

Problem 5 (yau-contest 2010)
Section: Analysis and Differential Equations

Let $K(x,y)\in L^1([0,1]\times[0,1])$. For all $f\in C^0[0,1]$, the space of continuous functions on $[0,1]$, define a function
$$Tf(x)=\int_0^1 K(x,y)f(y)\,dy.$$
Prove that $Tf\in C^0([0,1])$. Moreover $\Omega=\{Tf\mid \|f\|_{\sup}\le 1\}$ is pre-compact in $C^0([0,1])$, i.e. every sequence in $\Omega$ has a converging subsequence, where $\|f\|_{\sup}=\sup\{|f(x)|\mid x\in[0,1]\}$. (Hint: Every Lebesgue integrable function over the square can be approximated by polynomial functions in the $L^1$ norm.)

Problem 6 (yau-contest 2010)
Section: Analysis and Differential Equations

Consider the equation $\dot{x}=-x+f(t,x)$, where $|f(t,x)|\le \phi(t)|x|$ for all $(t,x)\in\mathbb{R}\times\mathbb{R}$, $\int_0^{\infty}\phi(t)\,dt<\infty$. Prove that every solution approaches zero as $t\to\infty$.