# [Solutions] HKUST Undergraduate Math Competition 2015

### Junior

1. Given the value of $\displaystyle I=\int_{0}^{+\infty} \frac{\ln x}{1+x^{2}} d x$ is a number. Find that number.
2. Let $P_{n}$ be the number of permutations $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ of the integers $1,2,3, \ldots, n$ such that $$1 x_{1} \leq 2 x_{2} \leq 3 x_{3} \leq \cdots \leq n x_{n}.$$ a) Find the values of $P_{1}, P_{2}, P_{3}$ and $P_{4}$. b) Find the value of $P_{20}$
3. Let $n$ be a positive integer and $A$ be an $n \times n$ matrix over complex numbers. If $A^{j}=0$ for some positive integer $j,$ then prove that $A^{n}=0$
4. For every positive integer $n,$ let $\langle n\rangle$ be the closest integer to $\sqrt{n}$. Determine the value of $$\lim _{n \rightarrow \infty} \sum_{j=1}^{n} \frac{2^{(j)}+2^{-\langle j\rangle}}{2^{j}}$$
5. For all real numbers $x$ and $y$ satisfying $y>x>0,$ prove that $y^{x^{y}}>x^{y^{x}}$.
6. Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function such that $f, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$ and $\frac{\partial^{2} f}{\partial x \partial y}$ are continuous and each of them at every point of $\mathbb{R}^{2}$ is not zero. If $f \frac{\partial^{2} f}{\partial x \partial y}=\frac{\partial f}{\partial x} \frac{\partial f}{\partial y}$ at all points in $\mathbb{R}^{2}$, then prove that there exist functions $g, h: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x, y)=g(x) h(y)$ at every $(x, y) \in \mathbb{R}^{2}$

### Senior

1. A point $(x, y)$ in the coordinate plane is called a lattice point if both coordinates $x$ and $y$ are integers. Prove that no three lattice points in the plane are the vertices of an equilateral triangle.
2. For every real number $a,$ define $x_{1}=a$ and $x_{n+1}=f\left(x_{n}\right)$ for $n=1,2,3, \ldots,$ where $f(x)=1+\dfrac{x^{2}}{1+x^{2}} .$ Determine the set of all real numbers $a$ such that the sequence $x_{1}, x_{2}, x_{3}, \ldots$ defined above is convergent.
3. Let $A$ be a $n \times n$ positive definite matrix over $\mathbb{R} .$ Prove that there exists a $n \times n$ symmetric matrix $B$ such that $A=B^{2} .$ Let $C$ and $D$ be $n \times n$ positive definite matrices. Prove that all the eigenvalues of $C D$ are real and positive.
4. Let $a$ and $b$ be positive integers such that for every positive integer $n, b^{n}+n$ is divisible by $a^{n}+n .$ Prove that $a=b$.
5. Let $f: \mathbb{R} \rightarrow(0,+\infty)$ be continuous and $\int_{-\infty}^{+\infty} f(x) d x=1 .$ Let $0<\alpha<1 .$ If $[a, b]$ is an interval of minimal length such that $\int_{a}^{b} f(x) d x=\alpha,$ then prove that $f(a)=f(b)$
6. Let $D=\{z \in \mathbb{C}:|z|<1\} .$ Let $f: D \rightarrow D$ be analytic (or holomorphic) and satisfy $f(0)=0 .$ If there exists a real number $r \in(0,1)$ such that $f(r)=f(-r)=0,$ then prove that for all $z \in D$ $$\begin{array}{c} |f(z)| \leq|z|\left|\frac{z^{2}-r^{2}}{1-r^{2} z^{2}}\right| \end{array}$$
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