$\lim _ { h \rightarrow 0 } \frac { \ln ( 4 + h ) - \ln ( 4 ) } { h }$ is
(A) 0
(B) $\frac { 1 } { 4 }$
(C) 1
(D) $e$
(E) nonexistent

If $\arcsin x = \ln y$, then $\frac { d y } { d x } =$
(A) $\frac { y } { \sqrt { 1 - x ^ { 2 } } }$
(B) $\frac { x y } { \sqrt { 1 - x ^ { 2 } } }$
(C) $\frac { y } { 1 + x ^ { 2 } }$
(D) $e ^ { \arcsin x }$
(E) $\frac { e ^ { \arcsin x } } { 1 + x ^ { 2 } }$

Exercise 4 (Candidates who have not followed the specialization course)
The common spruce is a species of coniferous tree that can measure up to 40 meters in height and live more than 150 years. The objective of this exercise is to estimate the age and height of a spruce based on the diameter of its trunk measured at $1.30 \mathrm {~m}$ from the ground.

Part A - Modeling the age of a spruce

For a spruce whose age is between 20 and 120 years, the relationship between its age (in years) and the diameter of its trunk (in meters) measured at $1.30 \mathrm {~m}$ from the ground is modeled by the function $f$ defined on the interval $] 0 ; 1 [$ by: $$f ( x ) = 30 \ln \left( \frac { 20 x } { 1 - x } \right)$$ where $x$ denotes the diameter expressed in meters and $f ( x )$ the age in years.

1. Prove that the function $f$ is strictly increasing on the interval $] 0 ; 1 [$.
2. Determine the values of the trunk diameter $x$ such that the age calculated in this model remains consistent with its validity conditions, that is, between 20 and 120 years.

Part B

The average height of spruces in representative samples of trees aged 50 to 150 years was measured. The following table, created using a spreadsheet, groups these results and allows calculation of the average growth rate of a spruce.

	A	B	C	D	E	F	G	H	I	J	K	L	M
1	Ages (in years)	50	70	80	85	90	95	100	105	110	120	130	150
2	Heights (in meters)	11.2	15.6	18.05	19.3	20.55	21.8	23	24.2	25.4	27.6	29.65	33
3	Growth rate (in meters per year)		0.22	0.245	0.25

1. a. Interpret the number 0.245 in cell D3. b. What formula should be entered in cell C3 to complete line 3 by copying cell C3 to the right?
2. Determine the expected height of a spruce whose trunk diameter measured at $1.30 \mathrm {~m}$ from the ground is 27 cm.
3. The quality of the wood is better when the growth rate is maximal. a. Determine an age interval during which the wood quality is best by explaining the approach. b. Is it consistent to ask loggers to cut trees when their diameter measures approximately 70 cm?

Consider in $\mathbb{R}$ the equation: $$\ln(6x - 2) + \ln(2x - 1) = \ln(x)$$ Statement 3: the equation has two solutions in the interval $]\frac{1}{2}; +\infty[$. Indicate whether Statement 3 is true or false, justifying your answer.

In a factory, a kiln bakes ceramics at a temperature of $1000^{\circ}\mathrm{C}$. At the end of baking, it is turned off and cools down. The temperature of the kiln is expressed in degrees Celsius (${}^{\circ}\mathrm{C}$). The kiln door can be opened safely for the ceramics as soon as its temperature is below $70^{\circ}\mathrm{C}$.
In this part, we denote $t$ the time (in hours) elapsed since the moment the kiln was turned off. The temperature of the kiln (in degrees Celsius) at time $t$ is given by the function $f$ defined, for every positive real number $t$, by: $$f(t) = a\mathrm{e}^{-\frac{t}{5}} + b,$$ where $a$ and $b$ are two real numbers. We admit that $f$ satisfies the following relation: $f'(t) + \frac{1}{5}f(t) = 4$.

1. Determine the values of $a$ and $b$ knowing that initially, the temperature of the kiln is $1000^{\circ}\mathrm{C}$, that is $f(0) = 1000$.
2. For the following, we admit that, for every positive real number $t$: $$f(t) = 980\mathrm{e}^{-\frac{t}{5}} + 20.$$ a. Determine the limit of $f$ as $t$ tends to $+\infty$. b. Study the variations of $f$ on $[0; +\infty[$. Deduce its complete table of variations. c. With this model, after how many minutes can the kiln be opened safely for the ceramics?
3. The average temperature (in degrees Celsius) of the kiln between two times $t_1$ and $t_2$ is given by: $\frac{1}{t_2 - t_1}\int_{t_1}^{t_2} f(t)\,\mathrm{d}t$. a. Using the graphical representation of $f$, give an estimate of the average temperature $\theta$ of the kiln over the first 15 hours of cooling. Explain your approach. b. Calculate the exact value of this average temperature $\theta$ and give its value rounded to the nearest degree Celsius.
4. In this question, we are interested in the temperature drop (in degrees Celsius) of the kiln over the course of one hour, that is between two times $t$ and $(t+1)$. This drop is given by the function $d$ defined, for every positive real number $t$, by: $d(t) = f(t) - f(t+1)$. a. Verify that, for every positive real number $t$: $d(t) = 980\left(1 - \mathrm{e}^{-\frac{1}{5}}\right)\mathrm{e}^{-\frac{t}{5}}$. b. Determine the limit of $d(t)$ as $t$ tends to $+\infty$. What interpretation can be given to this?

EXERCISE A - Natural logarithm function
Part A:
In a country, a disease affects the population with a probability of 0.05. There is a screening test for this disease. We consider a sample of $n$ people ($n \geqslant 20$) taken at random from the population, assimilated to a draw with replacement. The sample is tested using this method: the blood of these $n$ individuals is mixed, the mixture is tested. If the test is positive, an individual analysis of each person is performed. Let $X_n$ be the random variable that gives the number of analyses performed.

1. Show that $X_n$ takes the values 1 and $(n+1)$.
2. Prove that $P(X_n = 1) = 0.95^n$.

Establish the distribution of $X_n$ by copying on the answer sheet and completing the following table:

$x_i$	1	$n+1$
$P(X_n = x_i)$

\setcounter{enumi}{2}
1. What does the expectation of $X_n$ represent in the context of the experiment?

Show that $E(X_n) = n + 1 - n \times 0.95^n$.
Part B:

1. Consider the function $f$ defined on $[20;+\infty[$ by $f(x) = \ln(x) + x\ln(0.95)$.

Show that $f$ is decreasing on $[20;+\infty[$.

\setcounter{enumi}{1}
1. We recall that $\lim_{x\rightarrow+\infty} \frac{\ln x}{x} = 0$. Show that $\lim_{x\rightarrow+\infty} f(x) = -\infty$.
2. Show that $f(x) = 0$ has a unique solution $a$ on $[20;+\infty[$. Give an approximation to 0.1 of this solution.
3. Deduce the sign of $f$ on $[20;+\infty[$.

Part C:
We seek to compare two types of screening. The first method is described in Part A, the second, more classical, consists of testing all individuals. The first method makes it possible to reduce the number of analyses as soon as $E(X_n) < n$. Using Part B, show that the first method reduces the number of analyses for samples containing at most 87 people.

Exercise 1 — Multiple Choice Questionnaire (Logarithmic function)
For each of the following questions, only one of the four proposed answers is correct. The six questions are independent.

1. Consider the function $f$ defined for all real $x$ by $f(x) = \ln\left(1 + x^2\right)$.
   On $\mathbb{R}$, the equation $f(x) = 2022$ a. has no solution. b. has exactly one solution. c. has exactly two solutions. d. has infinitely many solutions.
   
2. Let the function $g$ defined for all strictly positive real $x$ by: $$g(x) = x\ln(x) - x^2$$ We denote $\mathscr{C}_g$ its representative curve in a coordinate system of the plane. a. The function $g$ is convex on $]0; +\infty[$. b. The function $g$ is concave on $]0; +\infty[$. c. The curve $\mathscr{C}_g$ has exactly one inflection point on $]0; +\infty[$. d. The curve $\mathscr{C}_g$ has exactly two inflection points on $]0; +\infty[$.
   
3. Consider the function $f$ defined on $]-1; 1[$ by $$f(x) = \frac{x}{1 - x^2}$$ An antiderivative of the function $f$ is the function $g$ defined on the interval $]-1; 1[$ by: a. $g(x) = -\frac{1}{2}\ln\left(1 - x^2\right)$ b. $g(x) = \frac{1 + x^2}{\left(1 - x^2\right)^2}$ c. $g(x) = \frac{x^2}{2\left(x - \frac{x^3}{3}\right)}$ d. $g(x) = \frac{x^2}{2}\ln\left(1 - x^2\right)$
   
4. The function $x \longmapsto \ln\left(-x^2 - x + 6\right)$ is defined on a. $]-3; 2[$ b. $]-\infty; 6]$ c. $]0; +\infty[$ d. $]2; +\infty[$
   
5. Consider the function $f$ defined on $]0.5; +\infty[$ by $$f(x) = x^2 - 4x + 3\ln(2x - 1)$$ An equation of the tangent line to the representative curve of $f$ at the point with abscissa 1 is: a. $y = 4x - 7$ b. $y = 2x - 4$ c. $y = -3(x - 1) + 4$ d. $y = 2x - 1$
   
6. The set $S$ of solutions in $\mathbb{R}$ of the inequality $\ln(x + 3) < 2\ln(x + 1)$ is: a. $S = ]-\infty; -2[ \cup ]1; +\infty[$ b. $S = ]1; +\infty[$ c. $S = \varnothing$ d. $S = ]-1; 1[$

This exercise is a multiple choice questionnaire. For each of the six following questions, only one of the four proposed answers is correct.
A wrong answer, multiple answers, or absence of an answer to a question earns neither points nor deducts points.

1. Consider the function $g$ defined and differentiable on $]0;+\infty[$ by: $$g(x) = \ln\left(x^2 + x + 1\right).$$ For every strictly positive real number $x$: a. $g^{\prime}(x) = \frac{1}{2x+1}$ b. $g^{\prime}(x) = \frac{1}{x^2+x+1}$ c. $g^{\prime}(x) = \ln(2x+1)$ d. $g^{\prime}(x) = \frac{2x+1}{x^2+x+1}$
   
2. The function $x \longmapsto \ln(x)$ admits as an antiderivative on $]0;+\infty[$ the function: a. $x \longmapsto \ln(x)$ b. $x \longmapsto \frac{1}{x}$ c. $x \longmapsto x\ln(x) - x$ d. $x \longmapsto \frac{\ln(x)}{x}$
   
3. Consider the sequence $(a_n)$ defined for all $n$ in $\mathbb{N}$ by: $$a_n = \frac{1 - 3^n}{1 + 2^n}.$$ The limit of the sequence $(a_n)$ is equal to: a. $-\infty$ b. $-1$ c. $1$ d. $+\infty$
   
4. Consider a function $f$ defined and differentiable on $[-2;2]$. The variation table of the function $f^{\prime}$ derivative of the function $f$ on the interval $[-2;2]$ is given by:
   

   $x$	$-2$	$-1$	$0$	$2$
   variations of $f^{\prime}$	$1$			$>_{-2}^{-1}$

   
   The function $f$ is: a. convex on $[-2;-1]$ b. concave on $[0;1]$ c. convex on $[-1;2]$ d. concave on $[-2;0]$
   
5. The representative curve of the derivative $f^{\prime}$ of a function $f$ defined on the interval $[-2;4]$ is given above. By graphical reading of the curve of $f^{\prime}$, determine the correct statement for $f$: a. $f$ is decreasing on $[0;2]$ b. $f$ is decreasing on $[-1;0]$ c. $f$ admits a maximum at $1$ on $[0;2]$ d. $f$ admits a maximum at $3$ on $[2;4]$
   
6. A stock is quoted at $57\,€$. Its value increases by $3\%$ every month. The Python function \texttt{seuil()} which returns the number of months to wait for its value to exceed $200\,€$ is: a. \begin{verbatim} def seuil() : m=0 v=57 while v < 200 : m=m+1 v = v*1.03 return m \end{verbatim} b. \begin{verbatim} def seuil() : m=0 v=57 while v > 200 : m=m+1 v = v*1.03 return m \end{verbatim} c. \begin{verbatim} def seuil() : v=57 for i in range (200) : v = v*1.03 return v \end{verbatim} d. \begin{verbatim} def seuil() : m=0 v=57 if v<200: m=m+1 else : v = v*1.03 return m \end{verbatim}

This exercise is a multiple choice questionnaire. For each of the six following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns no points and loses no points.

1. Consider the function $f$ defined and differentiable on $] 0 ; + \infty [$ by: $$f ( x ) = x \ln ( x ) - x + 1 .$$ Among the four expressions below, which one is the derivative of $f$?
   

   a. $\ln ( x )$

   b. $\frac { 1 } { x } - 1$

   c. $\ln ( x ) - 2$

   d. $\ln ( x ) - 1$

   
   
2. Consider the function $g$ defined on $] 0$; $+ \infty \left[ \text{ by } g ( x ) = x ^ { 2 } [ 1 - \ln ( x ) ] \right.$. Among the four statements below, which one is correct?
   

   a. $\lim _ { x \rightarrow 0 } g ( x ) = + \infty$	b. $\lim _ { x \rightarrow 0 } g ( x ) = - \infty$	c. $\lim _ { x \rightarrow 0 } g ( x ) = 0$	\begin{tabular}{ l } d. The function $g$
   does not have a li-
   mit at 0.

   \hline \end{tabular}
   
3. Consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 3 } - 0,9 x ^ { 2 } - 0,1 x$. The number of solutions to the equation $f ( x ) = 0$ on $\mathbb { R }$ is:
   

   a. 0

   b. 1

   c. 2

   d. 3

   
   
4. If $H$ is an antiderivative of a function $h$ defined and continuous on $\mathbb { R }$, and if $k$ is the function defined on $\mathbb { R }$ by $k ( x ) = h ( 2 x )$, then an antiderivative $K$ of $k$ is defined on $\mathbb { R }$ by:
   

   a. $K ( x ) = H ( 2 x )$

   b. $K ( x ) = 2 H ( 2 x )$

   c. $K ( x ) = \frac { 1 } { 2 } H ( 2 x )$

   d. $K ( x ) = 2 H ( x )$

   
   
5. The equation of the tangent line at the point with abscissa 1 to the curve of the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$ is:
   

   a. $y = \mathrm { e } x + \mathrm { e }$

   b. $y = 2 \mathrm { e } x - \mathrm { e }$

   c. $y = 2 \mathrm { e } x + \mathrm { e }$

   d. $y = \mathrm { e } x$

   
   
6. The integers $n$ that are solutions to the inequality $( 0,2 ) ^ { n } < 0,001$ are all integers $n$ such that:
   

   a. $n \leqslant 4$

   b. $n \leqslant 5$

   c. $n \geqslant 4$

   d. $n \geqslant 5$

   
   

Exercise 2 — 6 points

Theme: Exponential function Main topics covered: Sequences; Functions, Logarithm function. This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer, or no answer to a question earns or loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.

1. A container initially containing 1 litre of water is left in the sun. Every hour, the volume of water decreases by $15 \%$. After how many whole hours does the volume of water become less than a quarter of a litre? a. 2 hours b. 8 hours. c. 9 hours d. 13 hours
   
2. We consider the function $f$ defined on the interval $] 0$; $+ \infty [ \operatorname { by } f ( x ) = 4 \ln ( 3 x )$. For every real $x$ in the interval $] 0$; $+ \infty [$, we have: a. $f ( 2 x ) = f ( x ) + \ln ( 24 )$ b. $f ( 2 x ) = f ( x ) + \ln ( 16 )$ c. $f ( 2 x ) = \ln ( 2 ) + f ( x )$ d. $f ( 2 x ) = 2 f ( x )$
   
3. We consider the function $g$ defined on the interval $] 1 ; + \infty [$ by: $$g ( x ) = \frac { \ln ( x ) } { x - 1 } .$$ We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathscr { C } _ { g }$ has: a. a vertical asymptote and a horizontal asymptote. b. a vertical asymptote and no horizontal asymptote. c. no vertical asymptote and a horizontal asymptote. d. no vertical asymptote and no horizontal asymptote.
   In the rest of the exercise, we consider the function $h$ defined on the interval ]0;2] by: $$h ( x ) = x ^ { 2 } [ 1 + 2 \ln ( x ) ] .$$ We denote $\mathscr { C } _ { h }$ the representative curve of $h$ in a coordinate system of the plane. We assume that $h$ is twice differentiable on the interval ]0; 2]. We denote $h ^ { \prime }$ its derivative and $h ^ { \prime \prime }$ its second derivative. We assume that, for every real $x$ in the interval ] 0 ; 2], we have: $$h ^ { \prime } ( x ) = 4 x ( 1 + \ln ( x ) ) .$$
   
4. On the interval $\left. ] \frac { 1 } { \mathrm { e } } ; 2 \right]$, the function $h$ equals zero: a. exactly 0 times. b. exactly 1 time. c. exactly 2 times. d. exactly 3 times.
   
5. An equation of the tangent line to $\mathscr { C } _ { h }$ at the point with abscissa $\sqrt { \mathrm { e } }$ is: a. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x$ b. $y = ( 6 \sqrt { \mathrm { e } } ) \cdot x + 2 \mathrm { e }$ c. $y = 6 \mathrm { e } ^ { \frac { x } { 2 } }$ d. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x - 4 \mathrm { e }$.
   
6. On the interval $] 0 ; 2 ]$, the number of inflection points of the curve $\mathscr { C } _ { h }$ is equal to: a. 0 b. 1 c. 2 d. 3
   
7. We consider the sequence $\left( u _ { n } \right)$ defined for every natural number $n$ by $$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 3 \quad \text { and } \quad u _ { 0 } = 6 .$$ We can affirm that: a. the sequence $\left( u _ { n } \right)$ is strictly increasing. b. the sequence $( u _ { n } )$ is strictly decreasing. c. the sequence $( u _ { n } )$ is not monotonic. d. the sequence $( u _ { n } )$ is constant.

Exercise 4 (7 points) Theme: natural logarithm function, probabilities This exercise is a multiple choice questionnaire (MCQ) comprising six questions. The six questions are independent. For each question, only one of the four answers is correct. The candidate will indicate on his answer sheet the number of the question followed by the letter corresponding to the correct answer. No justification is required.
A wrong answer, a multiple answer or no answer gives neither points nor deducts any points.
Question 1 The real number $a$ defined by $a = \ln ( 9 ) + \ln \left( \frac { \sqrt { 3 } } { 3 } \right) + \ln \left( \frac { 1 } { 9 } \right)$ is equal to: a. $1 - \frac { 1 } { 2 } \ln ( 3 )$ b. $\frac { 1 } { 2 } \ln ( 3 )$ c. $3 \ln ( 3 ) + \frac { 1 } { 2 }$ d. $- \frac { 1 } { 2 } \ln ( 3 )$
Question 2 We denote by $(E)$ the following equation $\ln x + \ln ( x - 10 ) = \ln 3 + \ln 7$ with unknown real $x$. a. 3 is a solution of $(E)$. b. $5 - \sqrt { 46 }$ is a solution of $(E)$. c. The equation $(E)$ admits a unique real solution. d. The equation $(E)$ admits two real solutions.
Question 3 The function $f$ is defined on the interval $] 0 ; + \infty [$ by the expression $f ( x ) = x ^ { 2 } ( - 1 + \ln x )$. We denote by $\mathscr { C } _ { f }$ its representative curve in the plane with a coordinate system. a. For every real $x$ in the interval $] 0 ; + \infty [$ , $f ^ { \prime } ( x ) = 2 x + \frac { 1 } { x }$. b. The function $f$ is increasing on the interval $] 0 ; + \infty [$. c. $f ^ { \prime } ( \sqrt { \mathrm { e } } )$ is different from 0. d. The line with equation $y = - \frac { 1 } { 2 } e$ is tangent to the curve $\mathscr { C } _ { f }$ at the point with abscissa $\sqrt { e }$.
Question 4
A bag contains 20 yellow tokens and 30 blue tokens. We draw successively and with replacement 5 tokens from the bag. The probability of drawing exactly 2 yellow tokens, rounded to the nearest thousandth, is: a. 0.683 b. 0.346 c. 0.230 d. 0.165
Question 5
A bag contains 20 yellow tokens and 30 blue tokens. We draw successively and with replacement 5 tokens from the bag. The probability of drawing at least one yellow token, rounded to the nearest thousandth, is: a. 0.078 b. 0.259 c. 0.337 d. 0.922
Question 6
A bag contains 20 yellow tokens and 30 blue tokens. We perform the following random experiment: we draw successively and with replacement five tokens from the bag. We note the number of yellow tokens obtained after these five draws. If we repeat this random experiment a very large number of times then, on average, the number of yellow tokens is equal to: a. 0.4 b. 1.2 c. 2 d. 2.5

Question 170
O logaritmo de 1 000 na base 10 é
(A) 1 (B) 2 (C) 3 (D) 4 (E) 10

A equação $\log_2(x+1) = 3$ tem como solução
(A) $x = 5$ (B) $x = 6$ (C) $x = 7$ (D) $x = 8$ (E) $x = 9$

O valor de $\log_{10} 1000 + \log_{10} 0{,}01$ é
(A) $-1$ (B) $0$ (C) $1$ (D) $2$ (E) $3$

In September 1987, Goiânia was the site of the largest radioactive accident that occurred in Brazil, when a sample of caesium-137, removed from an abandoned radiotherapy device, was inadvertently handled by part of the population. The half-life of a radioactive material is the time required for the mass of that material to be reduced to half. The half-life of caesium-137 is 30 years and the amount of remaining mass of a radioactive material, after $t$ years, is calculated by the expression $M(t) = A \cdot (2.7)^{kt}$, where $A$ is the initial mass and $k$ is a negative constant.
Consider 0.3 as an approximation for $\log_{10} 2$.
What is the time required, in years, for an amount of caesium-137 mass to be reduced to 10\% of the initial amount?
(A) 27 (B) 36 (C) 50 (D) 54 (E) 100

QUESTION 149
The value of $\log_2 32$ is
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

To take the trip of her dreams, a person needed to take out a loan in the amount of $\mathrm{R}\$ 5000.00$. To pay the installments, she has at most $\mathrm{R}\$ 400.00$ monthly. For this loan amount, the installment value ($P$) is calculated as a function of the number of installments ($n$) according to the formula
$$P = \frac { 5000 \times 1.013 ^ { n } \times 0.013 } { \left( 1.013 ^ { n } - 1 \right) }$$
If necessary, use 0.005 as an approximation for $\log 1.013$; 2.602 as an approximation for $\log 400$; 2.525 as an approximation for $\log 335$.
According to the given formula, the smallest number of installments whose values do not compromise the limit defined by the person is
(A) 12.
(B) 14.
(C) 15.
(D) 16.
(E) 17.

The equation $\log_2(x+1) = 3$ has solution:
(A) $x = 6$
(B) $x = 7$
(C) $x = 8$
(D) $x = 9$
(E) $x = 10$

The value of $\log_3 81$ is:
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

An entrepreneur uses machines whose internal pressure $P$, in atmosphere, depends on the continuous time of use $t$, in hour, and on a positive parameter $K$, which defines the model of the machine, according to the expression: $$P = 4 \cdot \log[-K \cdot (t + 1) \cdot (t - 19)]$$
The manufacturer of these machines recommends to the user that the internal pressure of this type of machine does not exceed 10 atmospheres during its operation.
The entrepreneur intends to buy new machines of this type that should operate, daily, for a continuous period of 10 hours. For this, he needs to define the model of machine to be acquired by choosing the largest possible value of the parameter $K$, in accordance with the manufacturer's recommendation. The largest value to be chosen for $K$ is
(A) $10^{0.5}$
(B) $10^{8}$
(C) $\dfrac{10^{2.5}}{84}$
(D) $\dfrac{10^{2.5}}{99}$
(E) $25 \times 10^{-2}$

If $a , b , c$ are real numbers $> 1$, then show that $$\frac { 1 } { 1 + \log _ { a ^ { 2 } b } \frac { c } { a } } + \frac { 1 } { 1 + \log _ { b ^ { 2 } c } \frac { a } { b } } + \frac { 1 } { 1 + \log _ { c ^ { 2 } a } \frac { b } { c } } = 3$$

Consider the two equations numbered [1] and [2]:
$$\begin{aligned} \log _ { 2021 } a & = 2022 - a \\ 2021 ^ { b } & = 2022 - b \end{aligned}$$
(a) Equation [1] has a unique solution.
(b) Equation [2] has a unique solution.
(c) There exists a solution $a$ for [1] and a solution $b$ for [2] such that $a = b$.
(d) There exists a solution $a$ for [1] and a solution $b$ for [2] such that $a + b$ is an integer.

Statements
(17) Let $a = \frac{1}{\ln 3}$. Then $3^a = e$. (18) $\sin(0.02) < 2\sin(0.01)$. (19) $\arctan(0.01) > 0.01$. (20) $4\int_0^1 \arctan(x)\, dx = \pi - \ln 4$.

From the following , select all correct statements. [3 points]
ㄱ. $2 ^ { \log _ { 2 } 1 + \log _ { 2 } 2 + \log _ { 2 } 3 + \cdots + \log _ { 2 } 10 } = 10$ ㄴ. $\log _ { 2 } \left( 2 ^ { 1 } \times 2 ^ { 2 } \times 2 ^ { 3 } \times \cdots \times 2 ^ { 10 } \right) ^ { 2 } = 55 ^ { 2 }$ ㄷ. $\left( \log _ { 2 } 2 ^ { 1 } \right) \left( \log _ { 2 } 2 ^ { 2 } \right) \left( \log _ { 2 } 2 ^ { 3 } \right) \cdots \left( \log _ { 2 } 2 ^ { 10 } \right) = 55$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Which of the following in $\langle$Remarks$\rangle$ are correct? [3 points]
$\langle$Remarks$\rangle$ ㄱ. $2 ^ { \log _ { 2 } 1 + \log _ { 2 } 2 + \log _ { 2 } 3 + \cdots + \log _ { 2 } 10 } = 10 !$ ㄴ. $\log _ { 2 } \left( 2 ^ { 1 } \times 2 ^ { 2 } \times 2 ^ { 3 } \times \cdots \times 2 ^ { 10 } \right) ^ { 2 } = 55 ^ { 2 }$ ㄷ. $\left( \log _ { 2 } 2 ^ { 1 } \right) \left( \log _ { 2 } 2 ^ { 2 } \right) \left( \log _ { 2 } 2 ^ { 3 } \right) \cdots \left( \log _ { 2 } 2 ^ { 10 } \right) = 55$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ