Assume that a type of eucalyptus has an expected exponential growth rate in the first years after planting, modeled by the function $y(t) = a^{t-1}$, in which $y$ represents the height of the plant in meters, $t$ is considered in years, and $a$ is a constant greater than 1. The graph represents the function $y$.
Also assume that $y(0)$ gives the height of the seedling when planted, and it is desired to cut the eucalyptus when the seedlings grow 7.5 m after planting.
The time between planting and cutting, in years, is equal to
(A) 3.
(B) 4.
(C) 6.
(D) $\log_{2} 7$.
(E) $\log_{2} 15$.

The government of a city is concerned about a possible epidemic of an infectious disease caused by bacteria. To decide what measures to take, it must calculate the reproduction rate of the bacteria. In laboratory experiments of a bacterial culture, initially with 40 thousand units, the formula for the population was obtained:
$$p(t) = 40 \cdot 2^{3t}$$
where $t$ is the time, in hours, and $p(t)$ is the population, in thousands of bacteria.
In relation to the initial quantity of bacteria, after 20 min, the population will be
(A) reduced to one third.
(B) reduced to half.
(C) reduced to two thirds.
(D) doubled.
(E) tripled.

A loan was made at a monthly rate of $i\%$, using compound interest, in eight fixed and equal installments of $P$.
The debtor has the possibility of paying off the debt early at any time, paying for this the present value of the remaining installments. After paying the $5^{\text{th}}$ installment, he decides to pay off the debt when paying the $6^{\text{th}}$ installment.
The expression that corresponds to the total amount paid for the loan settlement is
(A) $P \left[ 1 + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } \right]$
(B) $P \left[ 1 + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { 2i } { 100 } \right) } \right]$
(C) $P \left[ 1 + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } \right]$
(D) $P \left[ \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { 2i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { 3i } { 100 } \right) } \right]$
(E) $P \left[ \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 3 } } \right]$

Organochlorine pesticides were widely used in agriculture; however, due to their high toxicity and persistence in the environment, they were banned. Consider the application of 500 g of an organochlorine pesticide to a crop and that, under certain conditions, the half-life of the pesticide in the soil is 5 years.
The mass of pesticide over 35 years will be closest to
(A) $3.9 \mathrm{~g}$.
(B) $31.2 \mathrm{~g}$.
(C) $62.5 \mathrm{~g}$.
(D) $125.0 \mathrm{~g}$.
(E) $250.0 \mathrm{~g}$.

A researcher analyzed the data on the number of new cases of a disease in a city over a period of 5 consecutive years and organized them in the table below.

Year	New cases
1	200
2	400
3	800
4	1600
5	3200

Based on this data, the researcher modeled the number of new cases as a function of the year $x$ by the expression $f(x) = 100 \cdot 2^x$.
If this trend continues, in which year will the number of new cases first exceed 100,000?
(A) 9
(B) 10
(C) 11
(D) 12
(E) 13

A person invests R\$\,2{,}000.00 in a savings account that yields 0.5\% per month in simple interest. After 12 months, this person withdraws all the money.
What is the total amount withdrawn, in reais?
(A) R\$\,2{,}100.00
(B) R\$\,2{,}110.00
(C) R\$\,2{,}120.00
(D) R\$\,2{,}130.00
(E) R\$\,2{,}140.00

The compound interest on R\$\,1{,}000.00 at a rate of 10\% per year for 2 years is:
(A) R\$\,100.00
(B) R\$\,200.00
(C) R\$\,210.00
(D) R\$\,220.00
(E) R\$\,230.00

Consider the following function defined for all real numbers $x$ $$f(x) = \frac{2018}{100 + e^{x}}$$ How many integers are there in the range of $f$?

Let $f_0(x) = x$. For $x > 0$, define functions inductively by $f_{n+1}(x) = x^{f_n(x)}$. So $f_1(x) = x^x$, $f_2(x) = x^{x^x}$, etc. Note that $f_0(1) = f_0'(1) = 1$.
Statements
(25) As $x \rightarrow 0^+$, $f_1(x) \rightarrow 1$. (26) As $x \rightarrow 0^+$, $f_2(x) \rightarrow 1$. (27) $\int_0^1 f_3(x)\, dx = 1$. (28) The derivative of $f_{123}$ at $x = 1$ is 1.

[14 points] Let $\mathbb { R } _ { + }$ denote the set of positive real numbers. A one-to-one and onto function $f : \mathbb { R } _ { + } \rightarrow \mathbb { R } _ { + }$ is called golden if $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R } _ { + }$.
(i) Find all golden functions (if any) of the form $f ( x ) = a x ^ { b }$. Find all golden functions (if any) of the form $f ( x ) = a b ^ { x }$. In both cases $a$ and $b$ are suitable real numbers.
(ii) Show that there is no one-to-one and onto function $f : \mathbb { R } \rightarrow \mathbb { R }$ such that $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R }$.

For the function $f ( x ) = \frac { 4 ^ { x } } { 4 ^ { x } + 2 }$, select all correct statements from . [4 points]
ㄱ. $f \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 }$ ㄴ. $f ( x ) + f ( 1 - x ) = 1$ ㄷ. $\sum _ { k = 1 } ^ { 100 } f \left( \frac { k } { 101 } \right) = 50$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄱ, ㄴ, ㄷ

For positive numbers $a , b$ and natural numbers $m , n$ satisfying the inequality $a ^ { m } < a ^ { n } < b ^ { n } < b ^ { m }$, which of the following is correct? [3 points]
(1) $a < 1 < b , m > n$
(2) $a < 1 < b , m < n$
(3) $a < b < 1 , m < n$
(4) $1 < a < b , m > n$
(5) $1 < a < b , m < n$

When the two roots of the equation $4 ^ { x } - 7 \cdot 2 ^ { x } + 12 = 0$ are $\alpha , \beta$, find the value of $2 ^ { 2 \alpha } + 2 ^ { 2 \beta }$. [3 points]

For two exponential functions $f ( x ) = 4 ^ { x }$, $g ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$ with domain $\{ x \mid - 1 \leqq x \leqq 3 \}$, let $M$ be the maximum value of $f ( x )$ and $m$ be the minimum value of $g ( x )$. What is the value of $M m$? [3 points]
(1) 8
(2) 6
(3) 4
(4) 2
(5) 1

Even when the surroundings suddenly become dark, the human eye perceives the change gradually. After the light intensity suddenly changes from 1000 to 10, and $t$ seconds have elapsed, the light intensity $I ( t )$ perceived by a person is $$I ( t ) = 10 + 990 \times a ^ { - 5 t } \text{ (where } a \text{ is a constant with } a > 1 \text{)}$$ After the light intensity suddenly changes from 1000 to 10, let $s$ seconds elapse until the person perceives the light intensity as 21. What is the value of $s$? (Here, the unit of light intensity is Td (troland).) [3 points]
(1) $\frac { 1 + 2 \log 3 } { 5 \log a }$
(2) $\frac { 1 + 3 \log 3 } { 5 \log a }$
(3) $\frac { 2 + \log 3 } { 5 \log a }$
(4) $\frac { 2 + 2 \log 3 } { 5 \log a }$
(5) $\frac { 2 + 3 \log 3 } { 5 \log a }$

The graph of the function $y = k \cdot 3 ^ { x } ( 0 < k < 1 )$ intersects the graphs of the two functions $y = 3 ^ { - x }$ and $y = - 4 \cdot 3 ^ { x } + 8$ at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When the ratio of the $x$-coordinates of points P and Q is $1 : 2$, find the value of $35 k$. [4 points]

The graph of the function $y = k \cdot 3 ^ { x }$ ($0 < k < 1$) meets the graphs of the two functions $y = 3 ^ { - x }$ and $y = - 4 \cdot 3 ^ { x } + 8$ at points P and Q, respectively. When the ratio of the $x$-coordinates of points P and Q is $1 : 2$, find the value of $35k$. [4 points]

When the graph of the function $f ( x ) = 2 ^ { x }$ is translated by $m$ in the $x$-direction and by $n$ in the $y$-direction, the graph of the function $y = g ( x )$ is obtained. By this translation, point $\mathrm { A } ( 1 , f ( 1 ) )$ moves to point $\mathrm { A } ^ { \prime } ( 3 , g ( 3 ) )$. When the graph of the function $y = g ( x )$ passes through the point $( 0,1 )$, what is the value of $m + n$? [3 points]
(1) $\frac { 11 } { 4 }$
(2) 3
(3) $\frac { 13 } { 4 }$
(4) $\frac { 7 } { 2 }$
(5) $\frac { 15 } { 4 }$

What is the minimum value of the function $y = 3 + \log _ { 3 } \left( x ^ { 2 } - 4 x + 31 \right)$? [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8

Two exponential functions $f ( x ) = a ^ { b x - 1 } , g ( x ) = a ^ { 1 - b x }$ satisfy the following conditions.
(a) The graphs of the function $y = f ( x )$ and the function $y = g ( x )$ are symmetric with respect to the line $x = 2$.
(b) $f ( 4 ) + g ( 4 ) = \frac { 5 } { 2 }$
What is the value of $a + b$, the sum of the two constants $a , b$? (where $0 < a < 1$) [3 points]
(1) 1
(2) $\frac { 9 } { 8 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 11 } { 8 }$
(5) $\frac { 3 } { 2 }$

Two exponential functions $f ( x ) = a ^ { b x - 1 } , g ( x ) = a ^ { 1 - b x }$ satisfy the following conditions.
(a) The graphs of $y = f ( x )$ and $y = g ( x )$ are symmetric with respect to the line $x = 2$.
(b) $f ( 4 ) + g ( 4 ) = \frac { 5 } { 2 }$
What is the value of the sum of the two constants $a + b$? (where $0 < a < 1$) [3 points]
(1) 1
(2) $\frac { 9 } { 8 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 11 } { 8 }$
(5) $\frac { 3 } { 2 }$

When the graph of the exponential function $y = 5 ^ { x - 1 }$ passes through the two points $( a , 5 ) , ( 3 , b )$, find the value of $a + b$. [3 points]

Shellfish filter suspended matter. When the water temperature is $t \left( { } ^ { \circ } \mathrm { C } \right)$ and the individual weight is $w ( \mathrm {~g} )$, the amounts filtered in 1 hour by shellfish A and B (in L) are $Q _ { \mathrm { A } }$ and $Q _ { \mathrm { B } }$ respectively, and the following relational equations hold. $$\begin{aligned} Q _ { \mathrm { A } } & = 0.01 t ^ { 1.25 } w ^ { 0.25 } \\ Q _ { \mathrm { B } } & = 0.05 t ^ { 0.75 } w ^ { 0.30 } \end{aligned}$$ When the water temperature is $20 { } ^ { \circ } \mathrm { C }$ and the individual weights of shellfish A and B are each 8 g, the value of $\frac { Q _ { \mathrm { A } } } { Q _ { \mathrm { B } } }$ is $2 ^ { a } \times 5 ^ { b }$. What is the value of $a + b$? (Here, $a$ and $b$ are rational numbers.) [3 points]
(1) 0.15
(2) 0.35
(3) 0.55
(4) 0.75
(5) 0.95

On the coordinate plane, the graph of the exponential function $y = a ^ { x }$ is reflected about the $y$-axis, then translated 3 units in the $x$-direction and 2 units in the $y$-direction. The resulting graph passes through the point $( 1,4 )$. What is the value of the positive number $a$? [3 points]
(1) $\sqrt { 2 }$
(2) 2
(3) $2 \sqrt { 2 }$
(4) 4
(5) $4 \sqrt { 2 }$

For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$, respectively. Find the number of all ordered pairs $( a , b )$ of $a , b$ satisfying the following condition. For example, $a = 4 , b = 5$ satisfies the following condition. [4 points]
(A) $2 \leq a \leq 10, 2 \leq b \leq 10$
(B) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.