Show that there is no infinite arithmetic progression consisting of distinct integers all of which are squares.

Find the remainder given by $3 ^ { 89 } \times 7 ^ { 86 }$ when divided by 17.

Let $a _ { 1 } , a _ { 2 } , \ldots , a _ { 100 }$ be 100 positive integers. Show that for some $m , n$ with $1 \leq m \leq n \leq 100 , \sum _ { i = m } ^ { n } a _ { i }$ is divisible by 100.

(a) A computer program prints out all integers from 0 to ten thousand in base 6 using the numerals $0,1,2,3,4$ and 5. How many numerals it would have printed?
(b) A 3-digit number $abc$ in base 6 is equal to the 3-digit number $cba$ in base 9. Find the digits.

A sequence of integers $c _ { n }$ starts with $c _ { 0 } = 0$ and satisfies $c _ { n + 2 } = a c _ { n + 1 } + b c _ { n }$ for $n \geq 0$, where $a$ and $b$ are integers. For any positive integer $k$ with $\operatorname { gcd } ( k , b ) = 1$, show that $c _ { n }$ is divisible by $k$ for infinitely many $n$.

Let $f ( x )$ be a polynomial with integer coefficients such that for each nonnegative integer $n , f ( n ) = \mathrm { a }$ perfect power of a prime number, i.e., of the form $p ^ { k }$, where $p$ is prime and $k$ a positive integer. ($p$ and $k$ can vary with $n$.) Show that $f$ must be a constant polynomial using the following steps or otherwise. a) If such a polynomial $f ( x )$ exists, then there is a polynomial $g ( x )$ with integer coefficients such that for each nonnegative integer $n , g ( n ) =$ a perfect power of a fixed prime number. b) Show that a polynomial $g ( x )$ as in part a must be constant.

(a) Let $f \in \mathbb { Z } [ x ]$ be a non-constant polynomial with integer coefficients. Show that as $a$ varies over the integers, the set of divisors of $f ( a )$ includes infinitely many different primes.
(b) Assume known the following result: If $G$ is a finite group of order $n$ such that for integer $d > 0$, $d \mid n$, there is no more than one subgroup of $G$ of order $d$, then $G$ is cyclic. Using this (or otherwise) prove that the multiplicative group of units in any finite field is cyclic.

A positive integer $N$ has its first, third and fifth digits equal and its second, fourth and sixth digits equal. In other words, when written in the usual decimal system it has the form $xyxyxy$, where $x$ and $y$ are the digits. Show that $N$ cannot be a perfect power, i.e., $N$ cannot equal $a ^ { b }$, where $a$ and $b$ are positive integers with $b > 1$.

What is the smallest positive integer $n$ for which $\frac { 50 ! } { 24 ^ { n } }$ is not an integer?

(a) Show that there are exactly 2 numbers $a$ in $\{2, 3, \ldots, 9999\}$ for which $a^2 - a$ is divisible by 10000. Find these values of $a$.
(b) Let $n$ be a positive integer. For how many numbers $a$ in $\{2, 3, \ldots, n^2 - 1\}$ is $a^2 - a$ divisible by $n^2$? State your answer suitably in terms of $n$ and justify.

For an arbitrary integer $n$, let $g(n)$ be the GCD of $2n + 9$ and $6n^2 + 11n - 2$. What is the largest positive integer that can be obtained as the value of $g(n)$? If $g(n)$ can be arbitrarily large, state so explicitly and prove it.

A positive integer $n$ is called a magic number if it has the following property: if $a$ and $b$ are two positive numbers that are not coprime to $n$ then $a + b$ is also not coprime to $n$. For example, 2 is a magic number, because sum of any two even numbers is also even. Which of the following are magic numbers? Write your answers as a sequence of four letters (Y for Yes and N for No) in correct order.
(i) 129
(ii) 128
(iii) 127
(iv) 100.

Determine the cardinality of set of subrings of $\mathbb{Q}$. (Hint: For a set $P$ of positive prime numbers, consider the smallest subring of $\mathbb{Q}$ that contains $\left\{\left.\frac{1}{p}\right\rvert\, p \in P\right\}$.)

Let $$f(x) = \sum_{n \geq 1} \frac{\sin\left(\frac{x}{n}\right)}{n}$$ Show that $f$ is continuous. Determine (with justification) whether $f$ is differentiable.

Let $A$ be a non-empty finite sequence of $n$ distinct integers $a_{1} < a_{2} < \cdots < a_{n}$. Define
$$A + A = \left\{ a_{i} + a_{j} \mid 1 \leq i, j \leq n \right\}$$
i.e, the set of all pairwise sums of numbers from $A$. E.g., for $A = \{1,4\}$, $A + A = \{2,5,8\}$.
(a) Show that $|A + A| \geq 2n - 1$. Here $|A + A|$ means the number of elements in $A + A$.
(b) Prove that $|A + A| = 2n - 1$ if and only if the sequence $A$ is an arithmetic progression.
(c) Find a sequence $A$ of the form $0 < 1 < a_{3} < \cdots < a_{10}$ such that $|A + A| = 20$.

Find all pairs $(p, n)$ of positive integers where $p$ is a prime number and $p^{3} - p = n^{7} - n^{3}$.

We say that two subsets $X$ and $Y$ of $\mathbb{R}$ are order-isomorphic if there is a bijective map $\phi : X \longrightarrow Y$ such that for every $x_1 \leq x_2 \in X$, $\phi(x_1) \leq \phi(x_2)$, where '$\leq$' denotes the usual order on $\mathbb{R}$. Choose the correct statement(s) from below:
(A) $\mathbb{N}$ and $\mathbb{Z}$ are not order-isomorphic;
(B) $\mathbb{N}$ and $\mathbb{Q}$ are order-isomorphic;
(C) $\mathbb{Z}$ and $\mathbb{Q}$ are order-isomorphic;
(D) The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ are pairwise non-order-isomorphic.

Four children $\mathrm{K}$, $\mathrm{L}$, $\mathrm{M}$ and R are about to run a race. They make some predictions as follows.
K says: M will win. Myself will come second. R says: M will come second. L will be third. M says: L will be last. R will be second. After the race, it turns out that each person has made exactly one correct and one incorrect prediction. Write the result of the race in the order from first to the last.

A country's GDP grew by $7.8\%$ within a period. During the same period the country's per-capita-GDP ($=$ ratio of GDP to the total population) increased by $10\%$. During this period, the total population of the country increased/decreased by $\_\_\_\_$ \%. (Choose the correct option and fill in the blank if possible.)

You are told that $n = 110179$ is the product of two primes $p$ and $q$. The number of positive integers less than $n$ that are relatively prime to $n$ (i.e. those $m$ such that $\operatorname{gcd}(m,n) = 1$) is 109480. Write the value of $p + q$. Then write the values of $p$ and $q$.

Let $f : \mathbb{C} \longrightarrow \mathbb{C}$ be an entire function such that $f(z+1) = f(z+\imath) = f(z)$ for every $z \in \mathbb{C}$. Choose the correct statement(s) from below:
(A) $f$ is constant;
(B) $f(z) = 0$ for every $z \in \mathbb{C}$;
(C) There exist complex numbers $a, b$ such that for every $x, y \in \mathbb{R}$, $f(x + \imath y) = a\sin(x) + \imath b\cos(y)$;
(D) $f$ is not necessarily constant but $|f(z)|$ is constant.

A function $f(x)$ is defined by the following formulas
$$f(x) = \begin{cases} x^{2} + 1 & \text{when } x \text{ is irrational} \\ \tan(x) & \text{when } x \text{ is rational} \end{cases}$$
At how many $x$ in the interval $[0, 4\pi]$ is $f(x)$ continuous?

Let $U \subseteq \mathbb{R}$ be a non-empty open subset. Choose the correct statement(s) from below:
(A) $U$ is uncountable;
(B) $U$ contains a closed interval as a proper subset;
(C) $U$ is a countable union of disjoint open intervals;
(D) $U$ contains a convergent sequence of real numbers.

We want to construct a nonempty and proper subset $S$ of the set of non-negative integers. This set must have the following properties. For any $m$ and any $n$,
if $m \in S$ and $n \in S$ then $m + n \in S$ and if $m \in S$ and $m + n \in S$ then $n \in S$.
For each statement below, state if it is true or false.
(i) 0 must be in $S$.
(ii) 1 cannot be in $S$.
(iii) There are only finitely many ways to construct such a subset $S$.
(iv) There is such a subset $S$ that contains both $2015^{2016}$ and $2016^{2015}$.

Let $R$ be a commutative ring. The characteristic of $R$ is the smallest positive integer $n$ such that $a + a + \cdots + a$ ($n$ times) is zero for every $a \in R$, if such an integer exists, and zero, otherwise. Choose the correct statement(s) from below:
(A) For every $n \in \mathbb{N}$, there exists a commutative ring whose characteristic is $n$;
(B) There exists an integral domain with characteristic 57;
(C) The characteristic of a field is either 0 or a prime number;
(D) For every prime number $p$, every commutative ring of characteristic $p$ contains $\mathbb{F}_p$ as a subring.