# [Solutions] HKUST Undergraduate Math Competition 2017

### Junior

1. Find all positive integers $n$ for which $3 n-4,4 n-5$ and $5 n-3$ are all prime numbers.
2. Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function such that $$\int_{0}^{1} f(x) d x=\int_{0}^{1} x f(x) d x=1.$$ Prove that $\int_{0}^{1} f^{2}(x) d x \geq 4$
3. Determine all real numbers $a$ such that the series $\displaystyle \sum_{n=1}^{\infty}\left(\cos \frac{a}{n}\right)^{n^{3}}$ is convergent.
4. Let $x_{0}, x_{1}, \ldots, x_{n}$ be distinct real numbers. Prove that there exist unique real numbers $a_{0}, a_{1}, \ldots, a_{n}$ such that for all polynomials $P(x)$ of degree $n$ or less with real coefficients, we have $$\int_{0}^{1} P(t) d t=\sum_{j=0}^{n} a_{j} P\left(x_{j}\right)$$
5. Three husband-and-wife couples would like to sit around a table with six seats for dinner. If each of these people is randomly assigned one of the six seats, determine the probability that none of the couples would sit next to each other.
6. Let $A$ and $B$ be $n \times n$ real matrices satisfying $A^{2}+B^{2}=A B$. Prove that if $B A-A B$ is invertible, then $n$ is divisible by $3 .$

### Senior

1. Suppose $a_{1}, a_{2}, a_{3}, \ldots$ is an infinite sequence of positive integers. Define a new sequence $b_{1}, b_{2}, b_{3}, \ldots$ by $$b_{1}=a_{1}, b_{2}=a_{2} b_{1}+1 \text { and for } k \geq 3, b_{k}=a_{k} b_{k-1}+b_{k-2}.$$ Prove that no two consecutive $b_{k}$ 's are even.
2. Let $n$ be an integer greater than $2 .$ A deck of $n$ cards with 3 aces and $n-3$ kings is shuffled, with all permutations equally probable. The cards are then turned over one after the other until two aces have appeared with the second ace being the $k$ -th card. Show that the expected value of $k$ is $(n+1) / 2$
3. Determine with proof whether or not the sequence $$\sum_{n=1}^{\infty} \sin \left(\pi \sqrt{n^{2}+1}\right)$$ converges.
4. Let $M$ be an $n \times n$ matrix over the real numbers $R .$ Prove that $$\operatorname{rank} M^{2} \leq \frac{\operatorname{rank} M+\operatorname{rank} M^{3}}{2}$$
5. Suppose $p(z), q(z)$ and $r(z)$ are continuous functions defined on $\mathbb{C}$ such that whenever $|z|=1,$ we have $p(z), q(z), r(z) \in \mathbb{R}$ and $$4 e^{p(z)+r(z)} \leq q(z)^{2}.$$ Show that there does not exist any entire function $f(z)$ such that $$z^{2} f(z)^{2} e^{p(z)}+z f(z) q(z)+e^{r(z)}=0 \quad \text { on }\{z \in \mathbb{C}:|z|=1\}$$
6. Let $p$ be a prime number. For a group $G$ of order $p^{4},$ suppose the center of $G$ has order $p^{2}$. A conjugacy class of $G$ is a set $\left\{g^{-1} x g: g \in G\right\}$ for some $x \in G .$ Determine the number of distinct conjugacy classes in $G$.
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