# [Solutions] HKUST Undergraduate Math Competition 2014

### Junior

1. Prove that every real-coefficient polynomial $f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}$ on the interval $(-\infty,+\infty)$ is the difference of two polynomials $g$ and $h,$ each of which is increasing, where a function $p$ is increasing means for all real numbers $a$ and $b, a<b$ implies $p(a) \leq p(b)$
2. Let $A, B$ be invertible $n \times n$ complex matrices. Suppose that $A B=\lambda B A$ for some complex number $\lambda$. a) Prove that if $v$ is an eigenvector of $A,$ then so is $B v$. b) Prove that $\lambda$ is a root of unity.
3. Prove that if $y=f(x)$ satisfies the differential equation $$\frac{d y}{d x}=\frac{1}{3+2 \sin y+x^{2}}$$ on the interval $(-\infty,+\infty),$ then $f(x)$ must be bounded on the interval $(-\infty,+\infty)$
4. Evaluate the following sums showing all details $$\sum_{i=0}^{\infty} \sum_{j=0}^{\infty} \frac{(-1)^{i+j}(i-j)}{(i-j)^{2}-\frac{1}{4}} \quad \text { and } \quad \sum_{j=0}^{\infty} \sum_{i=0}^{\infty} \frac{(-1)^{i+j}(i-j)}{(i-j)^{2}-\frac{1}{4}}$$
5. For what values of $a>1$ is $$\int_{a}^{a^{2}} \frac{1}{x} \ln \frac{x-1}{32} d x$$ minimum? Here $\ln x$ is the natural logarithmic function.
6. A $13 \times 13$ grid of lights can be controlled according to a series of switches. For any $9 \times 9$ or $2 \times 2$ square of lights there is a switch that reverses each of those $9^{2}$ or $2^{2}$ lights. Initially, all $13^{2}$ lights are off. Determine whether or not it is possible to achieve every lighting configuration using some combination of switches.

### Senior

1. Prove that every real-coefficient polynomial $f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}$ on the interval $(-\infty,+\infty)$ is the difference of two polynomials $g$ and $h,$ each of which is increasing, where a function $p$ is increasing means for all real numbers $a$ and $b, a<b$ implies $p(a) \leq p(b)$
2. Let $\mathbb{N}$ denote the set of all positive integers. Prove that there exists an uncountable set $\mathcal{S}$ of subsets of $\mathbb{N}$ such that if $A, B \in \mathcal{S},$ then either $A \subseteq B$ or $B \subseteq A$.
3. Suppose $R$ is a ring (possibly commutative or non-commutative). If $x, y \in R$ and the element $1+x y$ is invertible, then prove that $1+y x$ is invertible, too.
4. Alec tosses a fair coin until he gets a tails. If the first tails is on the $k$ -th toss, he then rolls a fair die $k$ times. The probability that his $k$ rolls form a non-decreasing sequence can be written in the form $a^{b}-c,$ where $a, b, c$ are rationals. Compute this probabilty.
5. Let $b_{1}, b_{2}, b_{3}, \ldots$ be a strictly decreasing sequence and the series $\displaystyle \sum_{n=1}^{\infty} b_{n}$ converges. For $x>0,$ define $f(x)$ to be the number of $n$ 's satisfying $b_{n} \geq 1 / x .$ Prove that $\displaystyle \lim_{x \rightarrow+\infty} \frac{f(x)}{x}=0$
6. Let $M_{n}(\mathbb{R})$ be the vector space of all $n \times n$ matrices over the real numbers $\mathbb{R}$. For $A \in M_{n}(\mathbb{R}),$ let $A^{t}$ denote the transpose of $A .$ Define a linear operators $T_{A}: M_{n}(\mathbb{R}) \rightarrow$ $M_{n}(\mathbb{R})$ by $$T_{A}(X)=A X A^{t}.$$ Let $\operatorname{Tr}(A)$ and $\operatorname{Det}(A)$ denote the trace and determinant of $A$ respectively. Prove that the trace of $T_{A}$ is $\operatorname{Tr}(A)^{2}$ and the determinant of $T_{A}$ is $\operatorname{Det}(A)^{2 n}$
 MOlympiad.NET rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc LIÊN HỆ[email protected]
You can use $\LaTeX$ in comment