The last digit of $( 2004 ) ^ { 5 }$ is
(A) 4
(B) 8
(C) 6
(D) 2

The last digit of $( 2004 ) ^ { 5 }$ is
(A) 4
(B) 8
(C) 6
(D) 2

Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a^n + b^n$ is prime, what can you say about $n$?
(A) The integer $n$ must be 2
(B) The integer $n$ need not be 2, but must be a power of 2
(C) The integer $n$ need not be a power of 2, but must be even
(D) None of the above is necessarily true

Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a ^ { n } + b ^ { n }$ is prime, what can you say about $n$?
(A) The integer $n$ must be 2
(B) The integer $n$ need not be 2, but must be a power of 2
(C) The integer $n$ need not be a power of 2, but must be even
(D) None of the above is necessarily true

The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is
(A) 7
(B) 8
(C) 11
(D) 12

The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is
(A) 7
(B) 8
(C) 11
(D) 12

$N$ is a 50 digit number. All the digits except the 26th from the right are 1. If $N$ is divisible by 13, then the unknown digit is
(A) 1
(B) 3
(C) 7
(D) 9

$N$ is a 50 digit number. All the digits except the 26th from the right are 1. If $N$ is divisible by 13, then the unknown digit is
(A) 1
(B) 3
(C) 7
(D) 9

If $n$ is a positive integer such that $8n + 1$ is a perfect square, then
(A) $n$ must be odd
(B) $n$ cannot be a perfect square
(C) $2n$ cannot be a perfect square
(D) none of the above

If $n$ is a positive integer such that $8 n + 1$ is a perfect square, then
(A) $n$ must be odd
(B) $n$ cannot be a perfect square
(C) $2 n$ cannot be a perfect square
(D) none of the above

The digit in the unit's place of the number $1! + 2! + 3! + \ldots + 99!$ is
(A) 3
(B) 0
(C) 1
(D) 7

The digit in the unit's place of the number $1 ! + 2 ! + 3 ! + \ldots + 99 !$ is
(A) 3
(B) 0
(C) 1
(D) 7

Let $d_1, d_2, \ldots, d_k$ be all the factors of a positive integer $n$ including 1 and $n$. If $d_1 + d_2 + \ldots + d_k = 72$, then $\frac{1}{d_1} + \frac{1}{d_2} + \cdots + \frac{1}{d_k}$ is:
(A) $\frac{k^2}{72}$
(B) $\frac{72}{k}$
(C) $\frac{72}{n}$
(D) none of the above

Let $d _ { 1 } , d _ { 2 } , \ldots , d _ { k }$ be all the factors of a positive integer $n$ including 1 and $n$. If $d _ { 1 } + d _ { 2 } + \ldots + d _ { k } = 72$, then $\frac { 1 } { d _ { 1 } } + \frac { 1 } { d _ { 2 } } + \cdots + \frac { 1 } { d _ { k } }$ is:
(A) $\frac { k ^ { 2 } } { 72 }$
(B) $\frac { 72 } { k }$
(C) $\frac { 72 } { n }$
(D) none of the above

Let the sequence $\left\{ a _ { n } \right\} _ { n \geq 1 }$ be defined by
$$a _ { n } = \tan ( n \theta )$$
where $\tan ( \theta ) = 2$. Show that for all $n , a _ { n }$ is a rational number which can be written with an odd denominator.

Let $g : \mathbb { N } \rightarrow \mathbb { N }$ with $g ( n )$ being the product of the digits of $n$.
(a) Prove that $g ( n ) \leq n$ for all $n \in \mathbb { N }$.
(b) Find all $n \in \mathbb { N }$, for which $n ^ { 2 } - 12 n + 36 = g ( n )$.

In the Mathematics department of a college, there are 60 first year students, 84 second year students, and 108 third year students. All of these students are to be divided into project groups such that each group has the same number of first year students, the same number of second year students, and the same number of third year students. What is the smallest possible size of each group?
(A) 9
(B) 12
(C) 19
(D) 21.

Let $p _ { 1 } , p _ { 2 } , p _ { 3 }$ be primes with $p _ { 2 } \neq p _ { 3 }$, such that $4 + p _ { 1 } p _ { 2 }$ and $4 + p _ { 1 } p _ { 3 }$ are perfect squares. Find all possible values of $p _ { 1 } , p _ { 2 } , p _ { 3 }$.

A number is called a palindrome if it reads the same backward or forward. For example, 112211 is a palindrome. How many 6-digit palindromes are divisible by 495?
(A) 10
(B) 11
(C) 30
(D) 45

Let $a , b , c \in \mathbb { N }$ be such that $$a ^ { 2 } + b ^ { 2 } = c ^ { 2 } \text { and } c - b = 1$$ Prove that ( i ) $a$ is odd, ( ii ) $b$ is divisible by 4 ,
(iii) $a ^ { b } + b ^ { a }$ is divisible by $c$.

The number of pairs of integers $( x , y )$ satisfying the equation $x y ( x + y + 1 ) = 5 ^ { 2018 } + 1$ is:
(A) 0
(B) 2
(C) 1009
(D) 2018.

The sum of all natural numbers $a$ such that $a ^ { 2 } - 16 a + 67$ is a perfect square is:
(A) 10
(B) 12
(C) 16
(D) 22.

Prove that the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r > 1$, are of the form $n = 2^{l}$ for some $l \geq 0$.

Let $f$ be a polynomial with integer coefficients. Define $$a_{1} = f(0),\quad a_{2} = f(a_{1}) = f(f(0)),$$ and $$a_{n} = f(a_{n-1}) \quad \text{for } n \geq 3$$ If there exists a natural number $k \geq 3$ such that $a_{k} = 0$, then prove that either $a_{1} = 0$ or $a_{2} = 0$.

A particle is allowed to move in the $XY$-plane by choosing any one of the two jumps:

1. move two units to right and one unit up, i.e., $( a , b ) \mapsto ( a + 2 , b + 1 )$ or
2. move two units up and one unit to right, i.e., $( a , b ) \mapsto ( a + 1 , b + 2 )$.

Let $P = ( 30,63 )$ and $Q = ( 100,100 )$. If the particle starts at the origin, then
(A) $P$ is reachable but not $Q$.
(B) $Q$ is reachable but not $P$.
(C) both $P$ and $Q$ are reachable.
(D) neither $P$ nor $Q$ is reachable.