Let $\mathbb { Z } ^ { + }$ denote the set of positive integers. We want to find all functions $g : \mathbb { Z } ^ { + } \rightarrow \mathbb { Z } ^ { + }$ such that the following equation holds for any $m, n$ in $\mathbb { Z } ^ { + }$. $$g ( n + m ) = g ( n ) + n m ( n + m ) + g ( m )$$ Prove that $g ( n )$ must be of the form $\sum _ { i = 0 } ^ { d } c _ { i } n ^ { i }$ and find the precise necessary and sufficient condition(s) on $d$ and on the coefficients $c _ { 0 } , \ldots , c _ { d }$ for $g$ to satisfy the required equation.

For a natural number $p \geqq 2$, a sequence $\left\{ a _ { n } \right\}$ satisfies the following three conditions. Which of the following in are correct? [4 points] Conditions (가) $a _ { 1 } = 0$ (나) $a _ { k + 1 } = a _ { k } + 1 \quad ( 1 \leqq k \leqq p - 1 )$ (다) $a _ { k + p } = a _ { k } \quad ( k = 1,2,3 , \cdots )$ 〈Remarks〉 ㄱ. $a _ { 2 k } = 2 a _ { k }$ ㄴ. $a _ { 1 } + a _ { 2 } + \cdots + a _ { p } = \frac { p ( p - 1 ) } { 2 }$ ㄷ. $a _ { p } + a _ { 2 p } + \cdots + a _ { k p } = k ( p - 1 )$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Let $a _ { n }$ be 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 with equal quotient and remainder 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]

The sequence $\left\{ a_n \right\}$ satisfies $a_1 = 4$ and $$a_{n+1} = n \cdot 2^n + \sum_{k=1}^{n} \frac{a_k}{k} \quad (n \geq 1)$$ The following is the process of finding the general term $a_n$.
From the given equation, $$a_n = (n-1) \cdot 2^{n-1} + \sum_{k=1}^{n-1} \frac{a_k}{k} \quad (n \geq 2)$$ Therefore, for natural numbers $n \geq 2$, $$a_{n+1} - a_n = \text{(가)} + \frac{a_n}{n}$$ so $$a_{n+1} = \frac{(n+1)a_n}{n} + \text{(가)}$$ If $b_n = \frac{a_n}{n}$, then $$b_{n+1} = b_n + \frac{(\text{가})}{n+1} \quad (n \geq 2)$$ and since $b_2 = 3$, $$b_n = \text{(나)} \quad (n \geq 2)$$ Therefore, $$a_n = \begin{cases} 4 & (n = 1) \\ n \times (\text{나}) & (n \geq 2) \end{cases}$$ If the expressions for (가) and (나) are $f(n)$ and $g(n)$, respectively, what is the value of $f(4) + g(7)$? [4 points]
(1) 90
(2) 95
(3) 100
(4) 105
(5) 110

The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 4$ and satisfies
$$a _ { n + 1 } = n \cdot 2 ^ { n } + \sum _ { k = 1 } ^ { n } \frac { a _ { k } } { k } \quad ( n \geq 1 )$$
The following is the process of finding the general term $a _ { n }$.
From the given equation,
$$a _ { n } = ( n - 1 ) \cdot 2 ^ { n - 1 } + \sum _ { k = 1 } ^ { n - 1 } \frac { a _ { k } } { k } \quad ( n \geq 2 )$$
Therefore, for natural numbers $n \geq 2$,
$$a _ { n + 1 } - a _ { n } = \text { (a) } + \frac { a _ { n } } { n }$$
so
$$a _ { n + 1 } = \frac { ( n + 1 ) a _ { n } } { n } + \text { (a) }$$
If $b _ { n } = \frac { a _ { n } } { n }$, then
$$b _ { n + 1 } = b _ { n } + \frac { ( \text { a } ) } { n + 1 } ( n \geq 2 )$$
and since $b _ { 2 } = 3$,
$$b _ { n } = \text { (b) } \quad ( n \geq 2 )$$
Therefore,
$$a _ { n } = \left\{ \begin{array} { c c } 4 & ( n = 1 ) \\ n \times ( \boxed { ( \text{b} ) } ) & ( n \geq 2 ) \end{array} \right.$$
Let $f ( n )$ and $g ( n )$ be the expressions that fit (a) and (b), respectively. What is the value of $f ( 4 ) + g ( 7 )$? [4 points]
(1) 90
(2) 95
(3) 100
(4) 105
(5) 110

A sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 2$ and satisfies for all natural numbers $n$: $$a _ { n + 1 } = \left\{ \begin{array} { c c } \frac { a _ { n } } { 2 - 3 a _ { n } } & ( \text{when } n \text{ is odd} ) \\ 1 + a _ { n } & ( \text{when } n \text{ is even} ) \end{array} \right.$$ What is the value of $\sum _ { n = 1 } ^ { 40 } a _ { n }$? [3 points]
(1) 30
(2) 35
(3) 40
(4) 45
(5) 50

All terms of a sequence $\left\{ a_{n} \right\}$ are integers and satisfy the following conditions. What is the sum of the values of $\left| a_{1} \right|$? [4 points] (가) For all natural numbers $n$, $$a_{n+1} = \begin{cases} a_{n} - 3 & \left(\left| a_{n} \right| \text{ is odd}\right) \\ \frac{1}{2}a_{n} & \left(a_{n} = 0 \text{ or } \left| a_{n} \right| \text{ is even}\right) \end{cases}$$ (나) The minimum value of the natural number $m$ such that $\left| a_{m} \right| = \left| a_{m+2} \right|$ is 3.

The sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions.

• $a _ { 1 } = 7$
• For natural numbers $n \geq 2$,

$$\sum _ { k = 1 } ^ { n } a _ { k } = \frac { 2 } { 3 } a _ { n } + \frac { 1 } { 6 } n ^ { 2 } - \frac { 1 } { 6 } n + 10$$
The following is the process of finding the value of $\sum _ { k = 1 } ^ { 12 } a _ { k } + \sum _ { k = 1 } ^ { 5 } a _ { 2 k + 1 }$.
For natural numbers $n \geq 2$, since $a _ { n + 1 } = \sum _ { k = 1 } ^ { n + 1 } a _ { k } - \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = \frac { 2 } { 3 } \left( a _ { n + 1 } - a _ { n } \right) + \text { (가) }$$ and simplifying this equation gives $$2 a _ { n } + a _ { n + 1 } = 3 \times \text { (가) } \quad \cdots \cdots \text { (ㄱ) }$$ From $$\sum _ { k = 1 } ^ { n } a _ { k } = \frac { 2 } { 3 } a _ { n } + \frac { 1 } { 6 } n ^ { 2 } - \frac { 1 } { 6 } n + 10 \quad ( n \geq 2 )$$ substituting $n = 2$ gives $$a _ { 2 } = \text { (나) }$$ By (ㄱ) and (ㄴ), $$\begin{aligned} \sum _ { k = 1 } ^ { 12 } a _ { k } + \sum _ { k = 1 } ^ { 5 } a _ { 2 k + 1 } & = a _ { 1 } + a _ { 2 } + \sum _ { k = 1 } ^ { 5 } \left( 2 a _ { 2 k + 1 } + a _ { 2 k + 2 } \right) \\ & = \text { (다) } \end{aligned}$$ Let $f ( n )$ be the expression that fits in (가), and let $p$ and $q$ be the numbers that fit in (나) and (다), respectively. Find the value of $\frac { p \times q } { f ( 12 ) }$. [4 points]

After completing its lunar exploration mission, the Chang'e-2 satellite continued deep space exploration and became China's first artificial planet orbiting the sun. To study the ratio of Chang'e-2's orbital period around the sun to Earth's orbital period around the sun, the sequence $\{b_n\}$ is used: $b_1 = 1 + \frac{1}{a_1}, b_2 = 1 + \frac{1}{a_1 + \frac{1}{a_2}}, b_3 = 1 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3}}}, \cdots$, and so on, where $a_k \in \mathbf{N}^* (k = 1,2,\cdots)$. Then
A. $b_1 < b_5$
B. $b_3 < b_8$
C. $b_6 < b_2$
D. $b_4 < b_7$

Given $M = \left\{ k \mid a _ { k } = b _ { k } \right\}$, where $a _ { n }, b _ { n }$ are not constant sequences and all terms are distinct. Which of the following is correct? \_\_\_
(1) If $a _ { n }, b _ { n }$ are both arithmetic sequences, then $M$ has at most one element;
(2) If $a _ { n }, b _ { n }$ are both geometric sequences, then $M$ has at most three elements;
(3) If $a _ { n }$ is an arithmetic sequence and $b _ { n }$ is a geometric sequence, then $M$ has at most three elements;
(4) If $a _ { n }$ is monotonically increasing and $b _ { n }$ is monotonically decreasing, then $M$ has at most one element.

We define a sequence of polynomials $\left( A _ { n } \right) _ { n \in \mathbb { N } } = \left( A _ { n } ( X ) \right) _ { n \in \mathbb { N } }$ satisfying the following conditions $$A _ { 0 } = 1 , A _ { n + 1 } ^ { \prime } = A _ { n } \text { and } \int _ { 0 } ^ { 1 } A _ { n + 1 } ( t ) d t = 0 \text { for every } n \in \mathbb { N }$$ The polynomials $B _ { n } = n ! A _ { n }$ are called Bernoulli polynomials.
a) Show that the sequence $\left( A _ { n } \right) _ { n \in \mathbb { N } }$ is uniquely determined by the conditions above; specify the degree of $A _ { n }$; calculate $A _ { 1 } , A _ { 2 }$ and $A _ { 3 }$.
b) Show that $A _ { n } ( t ) = ( - 1 ) ^ { n } A _ { n } ( 1 - t )$ for every $n \in \mathbb { N }$ and every $t \in \mathbb { R }$.
c) For every integer $n \geqslant 2$, show that $A _ { n } ( 0 ) = A _ { n } ( 1 )$ and that $A _ { 2 n - 1 } ( 0 ) = 0$.
d) We provisionally set $c _ { n } = A _ { n } ( 0 )$ for every natural integer $n$. Show that for every $n \in \mathbb { N }$, $$A _ { n } ( X ) = c _ { 0 } \frac { X ^ { n } } { n ! } + \cdots + c _ { n - 2 } \frac { X ^ { 2 } } { 2 ! } + c _ { n - 1 } X + c _ { n }$$ then that if $n \geqslant 1$, $$\frac { c _ { 0 } } { ( n + 1 ) ! } + \cdots + \frac { c _ { n - 2 } } { 3 ! } + \frac { c _ { n - 1 } } { 2 ! } + c _ { n } = 0$$
e) Deduce that for every $n \in \mathbb { N }$, we actually have $c _ { n } = a _ { n }$.

Let $x$ be a linear recurrent sequence. Show that the set $J_x$ of polynomials $A$ such that $A(\sigma)(x) = 0$ is an ideal of $\mathbb{K}[X]$, not reduced to $\{0\}$.
We recall that this implies two things:

• on the one hand, there exists in $J_x$ a unique monic polynomial $B$ of minimal degree;
• on the other hand, the elements of $J_x$ are the multiples of $B$.

By definition, we say that $B$ is the minimal polynomial of the sequence $x$, that the degree of $B$ is the minimal order of $x$, and that the relation $B(\sigma)(x) = 0$ is the minimal recurrence relation of $x$.

In $\mathbb{K}^{\mathbb{N}}$, what are the linear recurrent sequences of order 0? of order 1?
What are the sequences in $\mathbb{K}^{\mathbb{N}}$ whose minimal polynomial is $(X-1)^2$?

We consider the sequence $x$ defined by $x_0 = 0, x_1 = -1, x_2 = 2$ and by the linear recurrence relation of order 3: $\forall n \in \mathbb{N}, x_{n+3} = -3x_{n+2} - 3x_{n+1} - x_n$.
Determine the minimal polynomial (and thus the minimal order) of the sequence $x$.

Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic.
Prove that $\mathcal{R}_A(\mathbb{K})$ is a vector subspace of dimension $p$ of $\mathbb{K}^{\mathbb{N}}$ and that it is stable under $\sigma$ (we do not ask here to determine a basis of $\mathcal{R}_A(\mathbb{K})$, as this is the object of the following questions).

Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic.
Determine $\mathcal{R}_A(\mathbb{K})$ when $A = X^p$ (with $p \geqslant 1$) and give a basis for it.

Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic. In this question, we assume $p \geqslant 1$ and $A = (X - \lambda)^p$, with $\lambda$ in $\mathbb{K}^*$.
We denote by $E_A(\mathbb{K})$ the set of $x$ in $\mathbb{K}^{\mathbb{N}}$ with general term $x_n = Q(n)\lambda^n$, where $Q$ is in $\mathbb{K}_{p-1}[X]$.
a) Show that $E_A(\mathbb{K})$ is a vector subspace of $\mathbb{K}^{\mathbb{N}}$ and specify its dimension.
b) Show the equality $\mathcal{R}_A(\mathbb{K}) = E_A(\mathbb{K})$.

In this question, we assume that the polynomial $A$ is split over $\mathbb{K}$. More precisely, we write $A = X^{m_0} \prod_{k=1}^{d} (X - \lambda_k)^{m_k}$, where:

• the scalars $\lambda_1, \lambda_2, \ldots, \lambda_d$ are the distinct non-zero roots of $A$ in $\mathbb{K}$, and $m_1, m_2, \ldots, m_d$ are their respective multiplicities (greater than or equal to 1). If $A$ has no non-zero root, we adopt the convention that $d = 0$ and that $\prod_{k=1}^{d}(X-\lambda_k)^{m_k} = 1$;
• the integer $m_0$ is the multiplicity of 0 as a possible root of $A$. If 0 is not a root of $A$, we adopt the convention $m_0 = 0$.

With these notations, we have $\sum_{k=0}^{d} m_k = \deg A = p$.
Using the kernel decomposition theorem, show that $\mathcal{R}_A(\mathbb{K})$ is the set of sequences $x = (x_n)_{n \geqslant 0}$ in $\mathbb{K}^{\mathbb{N}}$ such that: $$\forall n \geqslant m_0, \quad x_n = \sum_{k=1}^{d} Q_k(n) \lambda_k^n$$ where, for all $k$ in $\{1, \ldots, d\}$, $Q_k$ is in $\mathbb{K}[X]$ with $\deg Q_k < m_k$.
Remark: if $d = 0$, the sum $\sum_{k=1}^{d} Q_k(n)\lambda_k^n$ is by convention equal to 0.

Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. The kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.
With these notations, show that the minimal polynomial of $x$ is $B = X^p + b_{p-1}X^{p-1} + \cdots + b_1 X + b_0$.

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
Specify the rank of $H_n(x)$ for any integer $n$ in $\mathbb{N}^*$ and indicate the minimal order of the sequence $x$.

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
Determine the minimal recurrence relation of the sequence $x$.

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
Give a formula allowing for any $n \geqslant 1$ to directly compute $x_n$.

Show that $V$ is a vector subspace of $\mathbf{C}^{\mathbf{Z}}$. Given $f \in \mathbf{C}^{\mathbf{Z}}$, we define $E(f) \in \mathbf{C}^{\mathbf{Z}}$ by $E(f)(k) = f(k+1), k \in \mathbf{Z}$.

Show that $E \in \mathcal{L}(\mathbf{C}^{\mathbf{Z}})$ and that $V$ is stable under $E$.

Show that $E \in \mathrm{GL}(V)$.