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 $a_{2} = 0$.

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) $( 9 - \sqrt { 80 } ) ^ { 20 }$.

$25 ^ { 190 } - 19 ^ { 190 } - 8 ^ { 190 } + 2 ^ { 190 }$ is divisible by
(1) neither 14 nor 34
(2) 14 but not by 34
(3) 34 but not by 14
(4) both 14 and 34

(1) Answer the following questions.
(i) Consider an integer $a$. When $a$ is divided by 5, the remainder is 4. So, $a$ can be represented as
$$a = \mathbf { A } \, k + \mathbf { B } \quad ( k \text{ : an integer}).$$
Hence, when $a ^ { 2 }$ is divided by 5, the remainder is $\mathbf { C }$.
(ii) The number written as the three-digit number $120_{(3)}$ in the base-3 system is $\mathbf{DE}$ in the decimal system.
The greatest natural number that can be expressed in three digits using the base-3 system is $\mathbf { F G }$ in the decimal system, and the smallest is $\mathbf { H }$ in the decimal system.
(2) For each of $\mathbf { I }$, $\mathbf { J }$ in the following statements, choose the correct answer from among (0) $\sim$ (3) below.
In the following, let $a$ be an integer and $b$ be a natural number.
(i) "When $a$ is divided by 5, the remainder is 4" is $\mathbf { I }$ for "when $a ^ { 2 }$ is divided by 5, the remainder is $\mathbf { C }$".
(ii) "$b$ satisfies $6 \leqq b \leqq 30$" is $\mathbf { J }$ for "$b$ is a three-digit number in the base-3 system".
(0) a necessary condition but not a sufficient condition
(1) a sufficient condition but not a necessary condition
(2) a necessary and sufficient condition
(3) neither a necessary condition nor a sufficient condition

If a number of the form $7k + 4$ is divisible by 3 without remainder, how many positive integers k less than 21 are there?
A) 8 B) 9 C) 7 D) 6 E) 5

$$\left. \begin{array} { l } 2 ^ { a } \cdot 3 ^ { b } \equiv 0 ( \bmod 12 ) \\ 2 ^ { b } \cdot 3 ^ { a } \equiv 0 ( \bmod 27 ) \end{array} \right\}$$
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

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

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

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

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

The three-digit natural number ABA divided by the two-digit natural number A1 gives a quotient of 13 and a remainder of 19.
Accordingly, what is the sum $A + B$?
A) 8
B) 9
C) 10
D) 11
E) 12

Where $a$ and $b$ are integers, $$a + 5b, \quad 2a + 3b \quad \text{and} \quad 3a + b$$ It is known that two of these numbers are odd and one is even.
Accordingly, I. (expression from figure) II. $2a + b$ III. $a \cdot b$ which of these expressions is an even number?
A) Only II
B) Only III
C) I and II
D) I and III
E) II and III