Let $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ be positive integers such that
$$\sum _ { i = 1 } ^ { 4 } \frac { 1 } { a _ { i } } = \frac { 11 } { 6 }$$
Then, $a _ { 4 } - a _ { 2 }$ equals
(A) 11
(B) 10
(C) 9
(D) 8 .

An examination has 20 questions. For each question the marks that can be obtained are either $-1$ or $0$ or $4$. Let $S$ be the set of possible total marks that a student can score in the examination. Then, the number of elements in $S$ is
(A) 93
(B) 94
(C) 95
(D) 96 .

The number of integers $n \geq 10$ such that the product $\binom { n } { 10 } \cdot \binom { n + 1 } { 10 }$ is a perfect square is
(A) 0
(B) 1
(C) 2
(D) 3

Let $a \geq b \geq c \geq 0$ be integers such that $2 ^ { a } + 2 ^ { b } - 2 ^ { c } = 144$. Then, $a + b - c$ equals:
(A) 7
(B) 8
(C) 9
(D) 10 .

Consider a right-angled triangle with integer-valued sides $a < b < c$ where $a, b, c$ are pairwise co-prime. Let $d = c - b$. Suppose $d$ divides $a$. Then
(a) Prove that $d \leq 2$.
(b) Find all such triangles (i.e. all possible triplets $a, b, c$) with perimeter less than 100.

There are 128 numbers $1,2 , \ldots , 128$ which are arranged in a circular pattern in clockwise order. We start deleting numbers from this set in a clockwise fashion as follows. First delete the number 2, then skip the next available number (which is 3 ) and delete 4 . Continue in this manner, that is, after deleting a number, skip the next available number clockwise and delete the number available after that, till only one number remains. What is the last number left ?
(A) 1
(B) 63
(C) 127
(D) None of the above.

The number of ways one can express $2 ^ { 2 } 3 ^ { 3 } 5 ^ { 5 } 7 ^ { 7 }$ as a product of two numbers $a$ and $b$, where $\operatorname { gcd } ( a , b ) = 1$, and $1 < a < b$, is
(A) 5 .
(B) 6 .
(C) 7 .
(D) 8 .

Prove that every positive rational number can be expressed uniquely as a finite sum of the form
$$a _ { 1 } + \frac { a _ { 2 } } { 2 ! } + \frac { a _ { 3 } } { 3 ! } + \cdots + \frac { a _ { n } } { n ! } ,$$
where $a _ { n }$ are integers such that $0 \leq a _ { n } \leq n - 1$ for all $n > 1$.

Define $a = p ^ { 3 } + p ^ { 2 } + p + 11$ and $b = p ^ { 2 } + 1$, where $p$ is any prime number. Let $d = \operatorname { gcd } ( a , b )$. Then the set of possible values of $d$ is
(A) $\{ 1,2,5 \}$.
(B) $\{ 2,5,10 \}$.
(C) $\{ 1,5,10 \}$.
(D) $\{ 1,2,10 \}$.

The number of all integer solutions of the equation $x ^ { 2 } + y ^ { 2 } + x - y = 2021$ is
(A) 5 .
(B) 7 .
(C) 1 .
(D) 0 .

For a positive integer $n$, the equation $$x ^ { 2 } = n + y ^ { 2 } , \quad x , y \text { integers} ,$$ does not have a solution if and only if
(A) $n = 2$.
(B) $n$ is a prime number.
(C) $n$ is an odd number.
(D) $n$ is an even number not divisible by 4 .

Any positive real number $x$ can be expanded as $x = a _ { n } \cdot 2 ^ { n } + a _ { n - 1 } \cdot 2 ^ { n - 1 } + \cdots + a _ { 1 } \cdot 2 ^ { 1 } + a _ { 0 } \cdot 2 ^ { 0 } + a _ { - 1 } \cdot 2 ^ { - 1 } + a _ { - 2 } \cdot 2 ^ { - 2 } + \cdots$, for some $n \geq 0$, where each $a _ { i } \in \{ 0,1 \}$. In the above-described expansion of 21.1875, the smallest positive integer $k$ such that $a _ { - k } \neq 0$ is:
(A) 3
(B) 2
(C) 1
(D) 4

For any positive integer $n$, and $i = 1, 2$, let $f_i(n)$ denote the number of divisors of $n$ of the form $3k + i$ (including 1 and $n$). Define, for any positive integer $n$, $$f(n) = f_1(n) - f_2(n)$$ Find the values of $f\left(5^{2022}\right)$ and $f\left(21^{2022}\right)$.

Suppose the numbers 71, 104 and 159 leave the same remainder $r$ when divided by a certain number $N > 1$. Then, the value of $3 N + 4 r$ must equal:
(A) 53
(B) 48
(C) 37
(D) 23

The number of positive integers $n$ less than or equal to 22 such that 7 divides $n ^ { 5 } + 4 n ^ { 4 } + 3 n ^ { 3 } + 2022$ is
(A) 7
(B) 8
(C) 9
(D) 10

A $3 \times 3$ magic square is a $3 \times 3$ rectangular array of positive integers such that the sum of the three numbers in any row, any column or any of the two major diagonals, is the same. For the following incomplete magic square

27	36
31

the column sum is
(A) 90
(B) 96
(C) 94
(D) 99

The number of triples $( a , b , c )$ of positive integers satisfying the equation $$\frac { 1 } { a } + \frac { 1 } { b } + \frac { 1 } { c } = 1 + \frac { 2 } { a b c }$$ and such that $a < b < c$, equals:
(A) 3
(B) 2
(C) 1
(D) 0

Determine all integers $n > 1$ such that every power of $n$ has an odd number of digits.

Let $n _ { 1 } , n _ { 2 } , \cdots , n _ { 51 }$ be distinct natural numbers each of which has exactly 2023 positive integer factors. For instance, $2 ^ { 2022 }$ has exactly 2023 positive integer factors $1,2,2 ^ { 2 } , \cdots , 2 ^ { 2021 } , 2 ^ { 2022 }$. Assume that no prime larger than 11 divides any of the $n _ { i }$'s. Show that there must be some perfect cube among the $n _ { i }$'s. You may use the fact that $2023 = 7 \times 17 \times 17$.

The number of consecutive zeroes adjacent to the digit in the unit's place of $401 ^ { 50 }$ is
(A) 3.
(B) 4.
(C) 49.
(D) 50.

Suppose $x$ and $y$ are positive integers. If $4 x + 3 y$ and $2 x + 4 y$ are divided by 7, then the respective remainders are 2 and 5. If $11 x + 5 y$ is divided by 7, then the remainder equals
(A) 0.
(B) 1.
(C) 2.
(D) 3.

Let $n$ be a positive integer having 27 divisors including 1 and $n$, which are denoted by $d _ { 1 } , \ldots , d _ { 27 }$. Then the product of $d _ { 1 } , d _ { 2 } , \ldots , d _ { 27 }$ equals
(A) $n ^ { 13 }$.
(B) $n ^ { 14 }$.
(C) $n ^ { \frac { 27 } { 2 } }$.
(D) $27 n$.

If $[ x ]$ denotes the largest integer less than or equal to $x$, then $$\left[ ( 9 + \sqrt { 80 } ) ^ { 20 } \right]$$ equals
(A) $( 9 + \sqrt { 80 } ) ^ { 20 } - ( 9 - \sqrt { 80 } ) ^ { 20 }$.
(B) $( 9 + \sqrt { 80 } ) ^ { 20 } + ( 9 - \sqrt { 80 } ) ^ { 20 } - 20$.
(C) $( 9 + \sqrt { 80 } ) ^ { 20 } + ( 9 - \sqrt { 80 } ) ^ { 20 } - 1$.
(D) $( 9 - \sqrt { 80 } ) ^ { 20 }$.

Suppose $f : \mathbb { Z } \rightarrow \mathbb { Z }$ is a non-decreasing function. Consider the following two cases: $$\begin{aligned} & \text { Case 1. } f ( 0 ) = 2 , f ( 10 ) = 8 \\ & \text { Case 2. } f ( 0 ) = - 2 , f ( 10 ) = 12 \end{aligned}$$ In which of the above cases it is necessarily true that there exists an $n$ with $f ( n ) = n$?
(A) In both cases.
(B) In neither case.
(C) In Case 1. but not necessarily in Case 2.
(D) In Case 2. but not necessarily in Case 1.

Let $n \geqslant 1$. The maximum possible number of primes in the set $\{n+6, n+7, \ldots, n+34, n+35\}$ is
(A) 7
(B) 8
(C) 12
(D) 13