$\int _ { 0 } ^ { 1 } \frac { 5 x + 8 } { x ^ { 2 } + 3 x + 2 } d x$ is
(A) $\ln ( 8 )$
(B) $\ln \left( \frac { 27 } { 2 } \right)$
(C) $\ln ( 18 )$
(D) $\ln ( 288 )$
(E) divergent

Let $p(x)$ be a polynomial of degree strictly less than 100 and such that it does not have $x^{3} - x$ as a factor. If $$\frac{d^{100}}{dx^{100}} \left( \frac{p(x)}{x^{3} - x} \right) = \frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$. Here $\frac{d^{100}}{dx^{100}}$ means taking the 100th derivative.

Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$.
Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation.
$\{Q \mid \mathbf{u} \cdot \mathbf{v} = 2024\}$. [1 point]
Options:

1. The empty set
2. A singleton set
3. A line
4. A pair of lines
5. A circle
6. A plane perpendicular to $\mathbf{u}$
7. A plane parallel to $\mathbf{u}$
8. An infinite cone
9. A finite cone
10. A sphere
11. None of the above

We consider the functions $\varphi : x \mapsto \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { 2 } }$ and $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$. Determine sequences $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ and $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ such that, for all $x \in ] - 1,1 [$, $$\varphi ( x ) = \sum _ { n = 0 } ^ { + \infty } u _ { n } x ^ { n } \quad \text { and } \quad \psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n } .$$ We will express explicitly as a function of $n$, according to the parity of $n$, the reals $u _ { n }$ and $v _ { n }$.

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\}.$$ We assume $k = 2$. We set $a = a_1, b = a_2, \omega_a = \exp(2\mathrm{i}\pi/a), \omega_b = \exp(2\mathrm{i}\pi/b)$. From a partial fraction decomposition of the fraction $\frac{1}{(1 - x^a)(1 - x^b)}$, show the formula $$P(n) = \frac{1}{2a} + \frac{1}{2b} + \frac{n}{ab} + \frac{1}{a} \sum_{j=1}^{a-1} \frac{\omega_a^{-jn}}{1 - \omega_a^{jb}} + \frac{1}{b} \sum_{k=1}^{b-1} \frac{\omega_b^{-kn}}{1 - \omega_b^{ka}}$$ for all integer $n \geq 0$.

We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$. We define the rational fraction $F ( X ) := \dfrac { X ^ { q - 1 } } { 1 + X ^ { p } }$.
Show that there exist constants $\left( a _ { 0 } , b _ { 0 } , \ldots , b _ { \lfloor p / 2 \rfloor - 1 } \right) \in \mathbf { C } ^ { \lfloor p / 2 \rfloor + 1 }$ such that $$F ( X ) = \frac { 1 - ( - 1 ) ^ { p } } { 2 } \cdot \frac { a _ { 0 } } { X + 1 } + \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \left( \frac { b _ { k } } { X - \omega _ { p , k } } + \frac { \overline { b _ { k } } } { X - \overline { \omega _ { p , k } } } \right) ,$$ where the $\omega _ { p , k }$ are constants which one will specify and $F ( X )$ is the rational fraction defined at the beginning of this part.
In the case where $p$ is even, we set $a _ { 0 } = 0$.

We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$. We define the rational fraction $F ( X ) := \dfrac { X ^ { q - 1 } } { 1 + X ^ { p } }$, and set $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$.
Calculate $a _ { 0 }$ in the case where $p$ is odd, then show that, for all integers $k \in \llbracket 0 , \lfloor p / 2 \rfloor - 1 \rrbracket$, $b _ { k }$ can be written in the form $$b _ { k } = - \frac { 1 } { p } e ^ { i q \theta _ { k } }$$

We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$. We define the rational fraction $F ( X ) := \dfrac { X ^ { q - 1 } } { 1 + X ^ { p } }$, and set $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$.
Deduce the partial fraction decomposition of $F ( X )$ in $\mathbf { R } ( X )$: $$F ( X ) = \frac { 1 - ( - 1 ) ^ { p } } { 2 } \cdot \frac { ( - 1 ) ^ { q - 1 } } { p } \cdot \frac { 1 } { X + 1 } - \frac { 2 } { p } \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } F _ { k } ( X )$$ where, for all $0 \leq k \leq \lfloor p / 2 \rfloor - 1$, $$F _ { k } ( X ) := \frac { \cos \left( q \theta _ { k } \right) X - \cos \left( ( q - 1 ) \theta _ { k } \right) } { X ^ { 2 } - 2 \cos \left( \theta _ { k } \right) X + 1 }$$

For $k \in N$, let $\frac { 1 } { \alpha ( \alpha + 1 ) ( \alpha + 2 ) \ldots ( \alpha + 20 ) } = \sum _ { K = 0 } ^ { 20 } \frac { A _ { k } } { \alpha + k }$, where $\alpha > 0$. Then the value of $100 \left( \frac { A _ { 14 } + A _ { 15 } } { A _ { 13 } } \right) ^ { 2 }$ is equal to $\underline{\hspace{1cm}}$.

$$\begin{aligned} & x = \frac { a - b } { a + b } \\ & y = \frac { b - c } { b + c } \end{aligned}$$
Given that, which of the following is the equivalent of the expression $\frac { 1 + y } { 1 - x }$ in terms of $a , b$ and $c$?
A) $\frac { b - c } { a - b }$
B) $\frac { b + c } { a - b }$
C) $\frac { a - b } { a + c }$
D) $\frac { a - c } { b - c }$
E) $\frac { a + b } { b + c }$

$$\begin{aligned} & a = \frac { x } { x - y } \\ & b = \frac { y } { x + y } \end{aligned}$$
Given this, what is the value of the expression $\frac { a + b - 1 } { a \cdot b }$?
A) $- 2$
B) $- 1$
C) $0$
D) $1$
E) $2$

Let $a, b, c$ and $d$ be positive real numbers; the value of the first notation equals the number $\frac{a+d}{b+c}$, and the value of the second notation equals the number $\frac{a \cdot d}{b \cdot c}$.
Given that the above holds, what is x?
A) 12 B) 16 C) 24 D) 36 E) 48