(For candidates who have NOT followed the specialization course)
We consider the sequence $(u_n)$ defined by $$u_0 = 0 \quad \text{and, for every natural integer } n, u_{n+1} = u_n + 2n + 2.$$

1. Calculate $u_1$ and $u_2$.
2. We consider the following two algorithms:
   

   \multicolumn{2}{|l|}{Algorithm 1}	\multicolumn{2}{|l|}{Algorithm 2}
   Variables :	$n$ is a natural integer $u$ is a real number	Variables :	$n$ is a natural integer $u$ is a real number
   Input : Processing:		Input : Processing:
   	\begin{tabular}{l} Enter the value of $n$ $u$ takes the value 0
   For $i$ ranging from 1 to $n$ : $u$ takes the value $u + 2i + 2$

   & &

   Enter the value of $n$ $u$ takes the value 0

   For $i$ ranging from 0 to $n - 1$ : $u$ takes the value $u + 2i + 2$

   \hline & End For & & End For \hline Output : & Display $u$ & Output : & Display $u$ \hline \end{tabular}
   Of these two algorithms, which one allows the output to display the value of $u_n$, with the value of the natural integer $n$ being entered by the user?
3. Using the algorithm, we obtained the table and the scatter plot below where $n$ is on the horizontal axis and $u_n$ is on the vertical axis.
   

   $n$	$u_n$
   0	0
   1	2
   2	6
   3	12
   4	20
   5	30
   6	42
   7	56
   8	72
   9	90
   10	110
   11	132
   12	156

   
   a. What conjecture can be made about the direction of variation of the sequence $\left( u_n \right)$? Prove this conjecture. b. The parabolic shape of the scatter plot leads to conjecturing the existence of three real numbers $a, b$ and $c$ such that, for every natural integer $n, u_n = an^2 + bn + c$. Within the framework of this conjecture, find the values of $a, b$ and $c$ using the information provided.
4. We define, for every natural integer $n$, the sequence $\left( v_n \right)$ by: $v_n = u_{n+1} - u_n$. a. Express $v_n$ as a function of the natural integer $n$. What is the nature of the sequence $\left( v_n \right)$? b. We define, for every natural integer $n, S_n = \sum_{k=0}^{n} v_k = v_0 + v_1 + \cdots + v_n$. Prove that, for every natural integer $n, S_n = (n+1)(n+2)$. c. Prove that, for every natural integer $n, S_n = u_{n+1} - u_0$, then express $u_n$ as a function of $n$.

Question 149
A sequência $(a_n)$ é definida por $a_1 = 2$ e $a_{n+1} = 3a_n - 1$, para todo $n \geq 1$. O valor de $a_4$ é
(A) 14 (B) 20 (C) 38 (D) 41 (E) 56

[12 points] Let $f$ be a function from natural numbers to natural numbers that satisfies
$$\begin{aligned} & f ( n ) = n - 2 \quad \text { for } n > 3000 \\ & f ( n ) = f ( f ( n + 5 ) ) \quad \text { for } n \leq 3000 \end{aligned}$$
Show that $f ( 2022 )$ is uniquely decided and find its value.

For a sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 2$ and $a _ { n + 1 } = 2 a _ { n } + 2$, what is the value of $a _ { 10 }$? [3 points]
(1) 1022
(2) 1024
(3) 2021
(4) 2046
(5) 2082

Let $a_n$ denote the sum of all natural numbers such that when divided by a natural number $n$ ($n \geqq 2$), the quotient and remainder are equal. For example, when divided by 4, the natural numbers for which the quotient and remainder are equal are $5, 10, 15$, so $a_4 = 5 + 10 + 15 = 30$. Find the minimum value of the natural number $n$ satisfying $a_n > 500$. [4 points]

[Discrete Mathematics] A sequence $\left\{ a _ { n } \right\}$ satisfies $$\left\{ \begin{array} { l } a _ { 1 } = 2 , a _ { 2 } = 5 \\ a _ { n } = 2 a _ { n - 1 } + a _ { n - 2 } \end{array} \quad ( n \geqq 3 ) \right.$$ What is the value of $a _ { 5 }$? [3 points]
(1) 70
(2) 72
(3) 74
(4) 76
(5) 78

The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 2$, and for all natural numbers $n$, $$a _ { n + 1 } = \begin{cases} a _ { n } - 1 & \text{(when } a _ { n } \text{ is even)} \\ a _ { n } + n & \text{(when } a _ { n } \text{ is odd)} \end{cases}$$ Find the value of $a _ { 7 }$. [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

A sequence $\left\{ a _ { n } \right\}$ satisfies $0 < a _ { 1 } < 1$ and the following conditions for all natural numbers $n$: (가) $a _ { 2 n } = a _ { 2 } \times a _ { n } + 1$ (나) $a _ { 2 n + 1 } = a _ { 2 } \times a _ { n } - 2$ If $a _ { 7 } = 2$, what is the value of $a _ { 25 }$? [4 points]
(1) 78
(2) 80
(3) 82
(4) 84
(5) 86

The sequence $\left\{ a _ { n } \right\}$ satisfies $0 < a _ { 1 } < 1$ and the following conditions for all natural numbers $n$. (가) $a _ { 2 n } = a _ { 2 } \times a _ { n } + 1$ (나) $a _ { 2 n + 1 } = a _ { 2 } \times a _ { n } - 2$ When $a _ { 8 } - a _ { 15 } = 63$, what is the value of $\frac { a _ { 8 } } { a _ { 1 } }$? [4 points]
(1) 91
(2) 92
(3) 93
(4) 94
(5) 95

For all sequences $\left\{ a _ { n } \right\}$ with all natural number terms satisfying the following conditions, let $M$ and $m$ be the maximum and minimum values of $a _ { 9 }$, respectively. What is the value of $M + m$? [4 points] (가) $a _ { 7 } = 40$ (나) For all natural numbers $n$, $$a _ { n + 2 } = \begin{cases} a _ { n + 1 } + a _ { n } & ( \text{when } a _ { n + 1 } \text{ is not a multiple of } 3 ) \\ \frac { 1 } { 3 } a _ { n + 1 } & ( \text{when } a _ { n + 1 } \text{ is a multiple of } 3 ) \end{cases}$$ (1) 216
(2) 218
(3) 220
(4) 222
(5) 224

A sequence $\{a_n\}$ with a natural number as its first term satisfies $$a_{n+1} = \begin{cases} 2^{a_n} & (\text{if } a_n \text{ is odd}) \\ \frac{1}{2}a_n & (\text{if } a_n \text{ is even}) \end{cases}$$ for all natural numbers $n$. Find the sum of all values of $a_1$ such that $a_6 + a_7 = 3$. [4 points]
(1) 139
(2) 146
(3) 153
(4) 160
(5) 167

The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 1$ and satisfies $$a _ { n + 1 } = n ^ { 2 } a _ { n } + 1$$ for all natural numbers $n$. Find the value of $a _ { 3 }$. [3 points]

We denote $V_n(z) = U_{n+1}(z, -1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, where $U(a,b) = (U_n(a,b))_{n \in \mathbb{N}}$ is the unique sequence satisfying $E_{a,b}$ with initial conditions $U_0(a,b) = 0$ and $U_1(a,b) = 1$. Explicitly write $V_1(z)$, $V_2(z)$ and $V_3(z)$ and determine their roots in $\mathbb{C}$.

Mathematics Specialty - EXERCISE I (20 points)

First Part

Consider the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ defined by $u _ { 0 } = 2$ and for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$ where $f$ is the function defined for every positive real $x$ by $f ( x ) = \frac { 3 x + 2 } { x + 4 }$. We admit that, for every natural number $\boldsymbol { n }$, $\boldsymbol { u } _ { \boldsymbol { n } }$ is greater than or equal to $1$.
I-1-a- Calculate the exact values of $u _ { 1 }$ and $u _ { 2 }$. Give the result as an irreducible fraction.
I-1-b- The graph below gives the representative curve in an orthonormal coordinate system of the function $f$. From this graph, what can be conjectured about the variations and convergence of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$? Specify the possible limit.

Second Part - Method 1

I-2-a- Show that, for every natural number $n$, $u _ { n + 1 } - u _ { n } = \frac { \left( 1 - u _ { n } \right) \left( u _ { n } + 2 \right) } { u _ { n } + 4 }$. I-2-b- Deduce the direction of variation of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$. Justify your answer. I-3- Prove that the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ is convergent. Let $l$ denote its limit. I-4- Determine the value of $l$. Justify your answer.

Third Part - Method 2

Consider the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ defined for every natural number $n$ by: $v _ { n } = \frac { u _ { n } - 1 } { u _ { n } + 2 }$. I-5- Calculate $v _ { 0 }$. I-6-a- Determine the constant $k$ in $] 0 ; 1 [$ such that $v _ { n + 1 } = k \times v _ { n }$ for every natural number $n$. Justify your answer. What can be deduced about the nature of the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$? For questions $\mathbf { I - 6 - b }$ and $\mathbf { I - 6 - c }$, answers may be expressed as a function of $k$ or its value. I-6-b- Deduce the expression of $v _ { n }$ as a function of $n$. I-6-c- Deduce the convergence of the sequence $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ and its limit. Justify your answer. I-7-a- Express $u _ { n }$ as a function of $v _ { n }$ for every natural number $n$. I-7-b- Deduce the convergence of the sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ and its limit. Justify your answer.

Consider the real-valued function $h : \{ 0,1,2 , \ldots , 100 \} \rightarrow \mathbb { R }$ such that $h ( 0 ) = 5 , h ( 100 ) = 20$ and satisfying $h ( i ) = \frac { 1 } { 2 } ( h ( i + 1 ) + h ( i - 1 ) )$, for every $i = 1,2 , \ldots , 99$. Then, the value of $h ( 1 )$ is:
(A) 5.15
(B) 5.5
(C) 6
(D) 6.15.

Let $\mathbb { Z }$ denote the set of integers. Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be such that $f ( x ) f ( y ) = f ( x + y ) + f ( x - y )$ for all $x , y \in \mathbb { Z }$. If $f ( 1 ) = 3$, then $f ( 7 )$ equals
(A) 840
(B) 844
(C) 843
(D) 842

49. The value of $b _ { 6 }$ is
(A) 7
(B) 8
(C) 9
(D) 11

ANSWER : B

1. Which of the following is correct?
   (A) $a _ { 17 } = a _ { 16 } + a _ { 15 }$
   (B) $c _ { 17 } \neq c _ { 16 } + c _ { 15 }$
   (C) $b _ { 17 } \neq b _ { 16 } + c _ { 16 }$
   (D) $a _ { 17 } = c _ { 17 } + b _ { 16 }$

ANSWER:A

Paragraph for Questions 51 and 52

Let $f ( x ) = ( 1 - x ) ^ { 2 } \sin ^ { 2 } x + x ^ { 2 }$ for all $x \in \mathbb { R }$, and let $g ( x ) = \int _ { 1 } ^ { x } \left( \frac { 2 ( t - 1 ) } { t + 1 } - \ln t \right) f ( t ) d t$ for all $x \in ( 1 , \infty )$. 51. Which of the following is true?
(A) $g$ is increasing on $( 1 , \infty )$
(B) $g$ is decreasing on $( 1 , \infty )$
(C) $g$ is increasing on $( 1,2 )$ and decreasing on $( 2 , \infty )$
(D) $g$ is decreasing on $( 1,2 )$ and increasing on $( 2 , \infty )$

ANSWER : B

1. Consider the statements : $\mathbf { P }$ : There exists some $x \in \mathbb { R }$ such that $f ( x ) + 2 x = 2 \left( 1 + x ^ { 2 } \right)$ Q: There exists some $x \in \mathbb { R }$ such that $2 f ( x ) + 1 = 2 x ( 1 + x )$ Then
   (A) both $\mathbf { P }$ and $\mathbf { Q }$ are true
   (B) $\mathbf { P }$ is true and $\mathbf { Q }$ is false
   (C) $\mathbf { P }$ is false and $\mathbf { Q }$ is true
   (D) both $\mathbf { P }$ and $\mathbf { Q }$ are false

ANSWER : C

MATHEMATICS

Paragraph for Questions 53 and 54

A tangent $P T$ is drawn to the circle $x ^ { 2 } + y ^ { 2 } = 4$ at the point $P ( \sqrt { 3 } , 1 )$. A straight line $L$, perpendicular to $P T$ is a tangent to the circle $( x - 3 ) ^ { 2 } + y ^ { 2 } = 1$. 53. A possible equation of $L$ is
(A) $x - \sqrt { 3 } y = 1$
(B) $x + \sqrt { 3 } y = 1$
(C) $x - \sqrt { 3 } y = - 1$
(D) $x + \sqrt { 3 } y = 5$

ANSWER : A

1. A common tangent of the two circles is
   (A) $x = 4$
   (B) $y = 2$
   (C) $x + \sqrt { 3 } y = 4$
   (D) $x + 2 \sqrt { 2 } y = 6$

ANSWER : D

SECTION III : Multiple Correct Answer(s) Type

This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct. 55. For every integer $n$, let $a _ { n }$ and $b _ { n }$ be real numbers. Let function $f : \mathbb { R } \rightarrow \mathbb { R }$ be given by $f ( x ) = \left\{ \begin{array} { l l } a _ { n } + \sin \pi x , & \text { for } x \in [ 2 n , 2 n + 1 ] \\ b _ { n } + \cos \pi x , & \text { for } x \in ( 2 n - 1,2 n ) \end{array} \right.$, for all integers $n$. If $f$ is continuous, then which of the following hold(s) for all $n$ ?
(A) $a _ { n - 1 } - b _ { n - 1 } = 0$
(B) $a _ { n } - b _ { n } = 1$
(C) $a _ { n } - b _ { n + 1 } = 1$
(D) $a _ { n - 1 } - b _ { n } = - 1$

ANSWER : BD

1. If $f ( x ) = \int _ { 0 } ^ { x } e ^ { t ^ { 2 } } ( t - 2 ) ( t - 3 ) d t$ for all $x \in ( 0 , \infty )$, then
   (A) $f$ has a local maximum at $x = 2$
   (B) $f$ is decreasing on $( 2,3 )$
   (C) there exists some $c \in ( 0 , \infty )$ such that $f ^ { \prime \prime } ( c ) = 0$
   (D) $f$ has a local minimum at $x = 3$

ANSWER : ABCD 57. If the straight lines $\frac { x - 1 } { 2 } = \frac { y + 1 } { k } = \frac { z } { 2 }$ and $\frac { x + 1 } { 5 } = \frac { y + 1 } { 2 } = \frac { z } { k }$ are coplanar, then the plane(s) containing these two lines is(are)
(A) $y + 2 z = - 1$
(B) $y + z = - 1$
(C) $y - z = - 1$
(D) $y - 2 z = - 1$

ANSWER : BC

1. Let $X$ and $Y$ be two events such that $P ( X \mid Y ) = \frac { 1 } { 2 } , P ( Y \mid X ) = \frac { 1 } { 3 }$ and $P ( X \cap Y ) = \frac { 1 } { 6 }$. Which of the following is (are) correct?
   (A) $P ( X \cup Y ) = \frac { 2 } { 3 }$
   (B) $X$ and $Y$ are independent
   (C) $X$ and $Y$ are not independent
   (D) $P \left( X ^ { \mathrm { c } } \cap Y \right) = \frac { 1 } { 3 }$

ANSWER : AB

MATHEMATICS

1. If the adjoint of a $3 \times 3$ matrix $P$ is $\left[ \begin{array} { l l l } 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{array} \right]$, then the possible value(s) of the determinant of $P$ is (are)
   (A) - 2
   (B) - 1
   (C) 1
   (D) 2

ANSWER : AD

1. Let $f : ( - 1,1 ) \rightarrow \mathbb { R }$ be such that $f ( \cos 4 \theta ) = \frac { 2 } { 2 - \sec ^ { 2 } \theta }$ for $\theta \in \left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$. Then the value(s) of $f \left( \frac { 1 } { 3 } \right)$ is (are)
   (A) $1 - \sqrt { \frac { 3 } { 2 } }$
   (B) $1 + \sqrt { \frac { 3 } { 2 } }$
   (C) $1 - \sqrt { \frac { 2 } { 3 } }$
   (D) $1 + \sqrt { \frac { 2 } { 3 } }$

Zero Marks to all

Let $p , q$ be integers and let $\alpha , \beta$ be the roots of the equation, $x ^ { 2 } - x - 1 = 0$, where $\alpha \neq \beta$. For $n = 0,1,2 , \ldots$, let $a _ { n } = p \alpha ^ { n } + q \beta ^ { n }$.
FACT: If $a$ and $b$ are rational numbers and $a + b \sqrt { 5 } = 0$, then $a = 0 = b$.
If $a _ { 4 } = 28$, then $p + 2 q =$
[A] 21
[B] 14
[C] 7
[D] 12

Let $\alpha$ and $\beta$ be the roots of equation $x ^ { 2 } - 6 x - 2 = 0$. If $a _ { n } = \alpha ^ { n } - \beta ^ { n } , \forall n \geq 1$, then the value of $\frac { a _ { 10 } - 2 a _ { 8 } } { 2 a _ { 9 } }$ is equal to
(1) - 3
(2) 6
(3) - 6
(4) 3

Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - 6 x - 2 = 0$. If $a _ { n } = \alpha ^ { n } - \beta ^ { n }$ for $n \geqslant 1$, then the value of $\frac { a _ { 10 } - 2 a _ { 8 } } { 3 a _ { 9 } }$ is:
(1) 1
(2) 3
(3) 2
(4) 4

Let $\alpha , \beta$ be roots of $x ^ { 2 } + \sqrt { 2 } x - 8 = 0$. If $\mathrm { U } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { n }$, then $\frac { \mathrm { U } _ { 10 } + \sqrt { 2 } \mathrm { U } _ { 9 } } { 2 \mathrm { U } _ { 8 } }$ is equal to $\_\_\_\_$

The function f satisfies the equation
$$f ( n ) = 2 \cdot f ( n - 1 ) + 1$$
for integers $n \geq 1$. Given that $f ( 0 ) = 1$, what is $f ( 2 )$?
A) 8
B) 7
C) 6
D) 5
E) 4

The sequence $(a _ { k })$ is defined as
$$\begin{aligned} & a _ { 1 } = 40 \\ & a _ { k + 1 } = a _ { k } - k \quad ( k = 1,2,3 , \ldots ) \end{aligned}$$
Accordingly, what is the term $\mathrm { a } _ { 8 }$?
A) 4
B) 7
C) 12
D) 15
E) 19

Let $a _ { 1 } , a _ { 2 }$ be real numbers. The sequence $\left( a _ { n } \right)$ satisfies the relation
$$a _ { n + 2 } = a _ { n + 1 } + a _ { n } \quad ( n = 1,2 , \cdots )$$
Given that $a _ { 8 } = 6$, what is the sum $a _ { 6 } + a _ { 9 }$?
A) 9
B) 10
C) 12
D) 15
E) 16

The sequence $(a_n)$ of real numbers satisfies for every positive integer $n$
$$a_{n+1} = a_n + \frac{(-1)^n \cdot a_n}{2}$$
the equality. If $a_5 = 18$, what is $a_1$?
A) 4
B) 8
C) 16
D) 32
E) 64