$$\left. \begin{array} { l } 2 ^ { a } \cdot 3 ^ { b } \equiv 0 ( \bmod 12 ) \\ 2 ^ { b } \cdot 3 ^ { a } \equiv 0 ( \bmod 27 ) \end{array} \right\}$$
For positive integers a and b that satisfy both congruences simultaneously, what is the minimum value of the sum $a + b$?
A) 3
B) 4
C) 5
D) 6
E) 7

For $1 < n < 50$, how many integers n are there such that the number of positive divisors is 3?
A) 2
B) 3
C) 4
D) 5
E) 7

a, b are positive integers, p is a prime number and
$$a ^ { 3 } - b ^ { 3 } = p$$
Given this, which of the following is the equivalent of $a ^ { 2 } + b ^ { 2 }$ in terms of $p$?
A) $\frac { p + 1 } { 2 }$
B) $\frac { p + 3 } { 2 }$
C) $\frac { p + 2 } { 3 }$
D) $\frac { 2 p - 1 } { 2 }$
E) $\frac { 2 p + 1 } { 3 }$

Let n be a positive integer. If every prime number p that divides n also divides $p ^ { 2 }$ into n, then n is called a powerful number.
Which of the following is NOT a powerful number?
A) 27
B) 64
C) 72
D) 99
E) 108

An operation $\Theta$ is defined on the set of integers for every integers a and b as
$$a \ominus b = a - b + 1$$
Regarding the $\Theta$ operation, I. The identity element is 1. II. It has the commutative property. III. It has the associative property. Which of these statements are true?
A) Only I
B) I and II
C) I and III
D) II and III
E) I, II and III

n is an integer greater than 1 and
$$\begin{aligned} & 73 \equiv 3 ( \bmod n ) \\ & 107 \equiv 2 ( \bmod n ) \end{aligned}$$
Given this, what is the sum of the possible values of n?
A) 39
B) 41
C) 47
D) 51
E) 54

For a positive integer n, the greatest odd divisor of n is denoted by $\overline{n}$. The terms of the sequence $( a _ { n } )$ are defined for $n = 1,2 , \ldots$ as
$$a _ { n } = \begin{cases} n + 1 , & \text{if } n \equiv 1 ( \bmod 4 ) \\ n - 1 , & \text{if } n \equiv 3 ( \bmod 4 ) \end{cases}$$
Given this, what is the difference $a _ { 18 } - a _ { 12 }$?
A) 2
B) 4
C) 6
D) 8
E) 10

$$\lim _ { x \rightarrow \infty } \frac { e ^ { - 3 x } + e ^ { 2 x } } { \ln x + 3 e ^ { 2 x } }$$
What is the value of this limit?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 1 } { 3 }$
D) 0
E) 1

Let $n$ be a positive integer with $n \leq 20$. The sum
$$1 + 2 + 3 + \cdots + n$$
is divisible by 9. Accordingly, what is the sum of the possible values of n?
A) 50
B) 52
C) 56
D) 60
E) 64

Let $p , q , r$ be prime numbers with
$$2 < p < q < r < 15$$
Accordingly, how many different values can the product $p \cdot q \cdot r$ take?
A) 4
B) 6
C) 8
D) 10
E) 12

$\mathbf { a } < \mathbf { b } < \mathbf { c }$ are positive integers and
$$\begin{aligned} & \gcd ( a , b ) = 5 \\ & \gcd ( b , c ) = 4 \end{aligned}$$
Given this, what is the minimum value that the sum $\mathbf { a + b + c }$ can take?
A) 27
B) 35
C) 39
D) 45
E) 49

$$\frac { x + \frac { 1 } { x + 2 } } { 1 - \frac { 1 } { x + 2 } } = \frac { 1 } { 4 }$$
What is the value of $x$ that satisfies this equality?
A) $\frac { - 3 } { 2 }$
B) $\frac { - 3 } { 4 }$
C) $\frac { - 1 } { 4 }$
D) $\frac { - 5 } { 4 }$
E) $\frac { - 3 } { 8 }$

Let a, b and c be three consecutive even integers arranged from smallest to largest such that the geometric mean of b and c is $\sqrt { 2 }$ times the geometric mean of a and b.
Accordingly, what is the sum $\mathrm { a } + \mathrm { b } + \mathrm { c }$?
A) 12
B) 18
C) 24
D) 30
E) 36

Let n be an integer greater than 2, and let the largest prime divisor of n be denoted by $\tilde{n}$. The terms of the sequence $(a_n)$ are defined for $n \geq 2$ as
$$a _ { n } = \left\{ \begin{aligned} 1 & , \tilde{n} < 10 \\ - 1 & , \tilde{n} > 10 \end{aligned} \right.$$
Accordingly, what is the sum $\sum _ { n = 15 } ^ { 30 } a _ { n }$?
A) 2
B) 3
C) 4
D) 5
E) 6

The greatest common divisor of positive integers a and b is odd, and their least common multiple is even.
Accordingly, I. $a \cdot b$ II. $a + b$ III. $a ^ { b }$ Which of the following expressions always equals an odd number?
A) Only I
B) Only II
C) Only III
D) I and III
E) II and III

In the following table consisting of 100 unit squares numbered from 1 to 100, some squares will be painted.

1	2	3	$\ldots$	10
11	12	13	$\ldots$	20
.	.	.		.
.	.	.		.
.	.	.		.
91	92	93	..	100

Squares with even numbers are painted yellow, squares that are multiples of 3 are painted red, and squares that are multiples of 5 are painted blue.
For a square to be orange, it must be painted only yellow and red.
Accordingly, how many unit squares in the table are orange?
A) 8
B) 12
C) 13
D) 15
E) 18

$$1 ^ { 5 } + 2 ^ { 5 } + 3 ^ { 5 } + 4 ^ { 5 } + 5 ^ { 5 }$$
What is the remainder when this expression is divided by 7?
A) 4
B) 3
C) 2
D) 1
E) 0

Let $a$ and $b$ be distinct positive integers such that LCM(a,b) equals a prime number.
Accordingly,\ I. $a$ and $b$ are coprime numbers.\ II. The sum $a + b$ is an odd number.\ III. The product $\mathrm{a} \cdot \mathrm{b}$ is an odd number.
Which of the following statements are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and II\ E) II and III

Three-digit natural numbers $ADB$, $ADC$, $DAA$, $DAD$ $$\begin{aligned}& \mathrm{ADB} < \mathrm{DAA} \\& \mathrm{DAD} < \mathrm{ADC}\end{aligned}$$ satisfy the inequalities.\ Accordingly, which of the following orderings is correct?\ A) A $=$ D $<$ B $<$ C\ B) C $<$ A $=$ B $<$ D\ C) D $<$ A $=$ B $<$ C\ D) B $<$ A $=$ D $<$ C\ E) C $<$ A $=$ D $<$ B

Let $a$ and $b$ be natural numbers such that $$\begin{aligned}& 4 \cdot a \equiv 2 ( \bmod 11 ) \\& 4 \cdot b \equiv 5 ( \bmod 7 )\end{aligned}$$ the following congruences are given.\ Accordingly, what is the smallest value that the sum $\mathbf{a+b}$ can take?\ A) 7\ B) 9\ C) 11\ D) 13\ E) 15

Let $x$, $y$ and $z$ be distinct prime numbers,
$$\begin{aligned} & x ( z - y ) = 18 \\ & y ( z - x ) = 40 \end{aligned}$$
the equalities are given.
Accordingly, what is the sum $\mathbf { x } + \mathbf { y } + \mathbf { z }$?
A) 17 B) 19 C) 21 D) 23 E) 25

Let A, B, and C be different digits other than zero,
ABC CAB BCA The three-digit natural numbers are divisible by 4, 5, and 9 respectively. Accordingly, what is the product A · B · C?
A) 150
B) 180
C) 200
D) 210
E) 240

Let p, r, and t be different prime numbers;

• Integer multiples of p form set A,
• Integer multiples of r form set B,
• Integer multiples of t form set C.

It is known that two of the numbers 220, 245, 330, and 350 are elements of the blue-colored set, and the other two are elements of the yellow-colored set. Accordingly, what is the sum $\mathbf { p } + \mathbf { r } + \mathbf { t }$?
A) 10
B) 14
C) 15
D) 21
E) 23

When 6 of the integers from 1 to 9 are placed in the boxes below such that each box contains a different number, all equalities are satisfied.
$$\begin{aligned} & \square + \square = 5 \\ & \square - \square = 5 \\ & \square : \square = 5 \end{aligned}$$
Accordingly, what is the sum of the unused integers?
A) 23
B) 21
C) 19
D) 17
E) 15

Let $a$, $b$, and $c$ be prime numbers,
$$a ( a + b ) = c ( c - b ) = 143$$
Given the equalities, accordingly, what is the sum $a + b + c$?
A) 22
B) 26
C) 30
D) 32