# [Solutions] HKUST Undergraduate Math Competition 2016

### Junior

1. For every positive integer $n,$ let $f(n)$ be the number of ordered pairs $(x, y)$ of positive integers satisfying $n(x+y)=x y$.
a) Prove that for every positive integer $n, f(n)$ is odd.
b) Find, with proof, a two-term formula for $f\left(2^{n}\right)$
2. Determine all continuous functions $f:(0,+\infty) \rightarrow \mathbb{R}$ such that $f(1)=3$ and for all $x, y>0$ $$\int_{1}^{x y} f(t) d t=x \int_{1}^{y} f(t) d t+y \int_{1}^{x} f(t) d t$$
3. Let $\displaystyle H_{n}=\sum_{k=1}^{n} 1 / k .$ Show that the infinite series $$\sum_{n=1}^{\infty} \frac{H_{n+1}}{n(n+1)}$$ converges and find its value.
4. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be twice differentiable and $f^{\prime \prime}(x) \leq 0$ for all $x \in \mathbb{R} .$ Prove that $$\int_{0}^{1} f\left(t^{2}\right) d t \leq f\left(\frac{1}{3}\right)$$
5. Let $A$ and $B$ be $n \times n$ matrices over $\mathbb{R}$ such that $A+B$ is invertible. Prove that $A(A+B)^{-1} B=B(A+B)^{-1} A$
6. Let $p(z)$ be a nonconstant polynomial with real coefficients and only real roots. Prove that for every real number $r,$ the polynomial $p(z)-r p^{\prime}(z)$ has only real roots.

### Senior

1. Let $V$ be a real vector space and $T: V \rightarrow V$ be a linear transformation. If $v_{1}, v_{2}, \ldots, v_{n}$ are non-zero eigenvectors of $T$ with eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ respectively such that $\lambda_{i} \neq \lambda_{j}$ for $i \neq j,$ then prove that $v_{1}, v_{2}, \ldots, v_{n}$ are linearly independent. Let $c_{1}, c_{2}, \ldots, c_{n}$ be positive numbers such that $c_{1}<c_{2}<\cdots<c_{n} .$ Show the functions $f_{i}(x)=\sin \left(c_{i} x\right)(i=1,2, \ldots, n)$ are linearly independent real-valued functions on the real line.
2. Given a set $S=\left\{a_{1}, a_{2}, \ldots, a_{k}\right\}$ with $a_{1}>a_{2}>\ldots>a_{k},$ define its alternating sum by $$A(S)=a_{1}-a_{2}+a_{3}-\cdots+(-1)^{k+1} a_{k} .$$ For example, $A(\{4\})=4$, $A(\{7,3,1\})=7-3+1=5 .$ Find, with proof, a one-term formula for the sum of $A(S)$ over all non-empty subsets $S$ of $\{1,2, \ldots, n\}$.
3. Let $i$ be the square root of -1 in $H=\{z \in \mathbb{C} \mid \operatorname{Im} z>0\} .$ Let $\mathrm{SL}_{2}(\mathbb{R})$ be the group of $2 \times 2$ real matrices with determinant $1 .$ For $g=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in \mathrm{SL}_{2}(\mathbb{R})$ and $z \in H$ define action $g \cdot z=\dfrac{a z+b}{c z+d}$.
a) Prove that $\left\{g \in \mathrm{SL}_{2}(\mathbb{R}) \mid g \cdot i=i\right\}=\left\{\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right) \mid \theta \in \mathbb{R}\right\}$ and for every $z \in H,$ there exists $g \in \mathrm{SL}_{2}(\mathbb{R})$ such that $g \cdot i=z$.
b) Prove that for any $z, w \in H,$ there exists $g \in \mathrm{SL}_{2}(\mathbb{R})$ such that $g \cdot z$ and $g \cdot w$ are both on the vertical line passing through $i$
4. Let $X_{n}=\{1,2, \ldots, n\}$ and $Y_{n}$ be the set of all ordered pairs $(A, B),$ where $A$ and $B$ are non-empty disjoint subsets of $X_{n} .$ Prove the number of elements in $Y_{n}$ is $3^{n}-2^{n+1}+1$
5. Show that the power series representation of $\displaystyle f(z)=\sum_{n=0}^{\infty} \frac{z^{n}(z-1)^{2 n}}{n !}$ with center at 0 cannot have three consecutive zero coefficients.
6. Calculate the line integral $\displaystyle \oint_{C} \frac{y d x-x d y}{4 x^{2}+y^{2}}$ dot represents the origin.
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