[12 points] Consider polynomials $p(x)$ with the following property, called $(\dagger)$. $(\dagger)$ If $r$ is a root of $p(x)$, then $r^{2} - 4$ is also a root of $p(x)$.
(i) We want to find every quadratic polynomial of the form $p(x) = x^{2} + bx + c$ such that $p(x)$ has two distinct roots, has integer coefficients and has property $(\dagger)$. Prove that there are exactly two such polynomials and list them.
(ii) It is also true that there are exactly two cubic polynomials of the form $p(x) = x^{3} + ax^{2} + bx + c$ with the property $(\dagger)$ such that $p(x)$ shares no root with the polynomials you found in part (i). Explain fully how you will prove this along with the method to find the polynomials, but do not try to explicitly find the polynomials.

Consider polynomials of the form $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$ where $a , b , c$ are integers. Name the three (possibly non-real) roots of $f ( x )$ to be $p , q , r$.
(a) If $f ( 1 ) = 2021$, then $f ( x ) = ( x - 1 ) \left( x ^ { 2 } + s x + t \right) + 2021$ where $s , t$ must be integers.
(b) There is such a polynomial $f ( x )$ with $c = 2021$ and $p = 2$.
(c) There is such a polynomial $f ( x )$ with $r = \frac { 1 } { 2 }$.
(d) The value of $p ^ { 2 } + q ^ { 2 } + r ^ { 2 }$ does not depend on the value of $c$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P_\alpha)$ the set of complex polynomials that commute with $P_\alpha$ under composition.
Show that the only complex numbers $\alpha$ such that $\mathcal{C}(P_\alpha)$ contains a polynomial of degree three are 0 and $-2$.

Consider $(c_0, \ldots, c_{d-1}) \in \left(\mathbb{R}_{+}^{*}\right)^d$ and $P$ the polynomial $$X^d - c_{d-1} X^{d-1} - \cdots - c_1 X - c_0.$$ Show that the polynomial $P$ has a unique root in $\mathbb{R}_{+}^{*}$.

We consider the general case, without having information on the stability and multiplicity of the roots of $p$, and we seek to calculate $\sigma(p)$.
We construct the two polynomials $f$ and $g$ satisfying $f = p \wedge p_0$ and $p = fg$.
Show that $\sigma(g) = \pi(J(g))$.

We consider the general case, without having information on the stability and multiplicity of the roots of $p$, and we seek to calculate $\sigma(p)$.
We construct the two polynomials $f$ and $g$ satisfying $f = p \wedge p_0$ and $p = fg$.
Propose a method allowing us to construct a finite number (possibly zero) of polynomials $g_1, \ldots, g_\ell$, whose roots are stable and of multiplicity 1, such that $f = g_1 g_2 \cdots g_\ell$. Express $\sigma(p)$ using $n$, $\deg g$, $\pi(J(g))$, $\ell$, $\pi(J(g))$ as well as $\pi(J(g_1')), \ldots, \pi(J(g_\ell'))$.

The number of integers $n$ for which the cubic equation $X ^ { 3 } - X + n = 0$ has 3 distinct integer solutions is:
(A) 0
(B) 1
(C) 2
(D) infinite.

Let $p$ and $q$ be two non-zero polynomials such that the degree of $p$ is less than or equal to the degree of $q$, and $p ( a ) q ( a ) = 0$ for $a = 0,1,2 , \ldots , 10$. Which of the following must be true?
(A) degree of $q \neq 10$
(B) degree of $p \neq 10$
(C) degree of $q \neq 5$
(D) degree of $p \neq 5$

Let $P(x)$ be an odd degree polynomial in $x$ with real coefficients. Show that the equation $P(P(x)) = 0$ has at least as many distinct real roots as the equation $P(x) = 0$.

If $z$ is a complex number, then the number of common roots of the equation $z^{1985} + z^{100} + 1 = 0$ and $z^3 + 2z^2 + 2z + 1 = 0$, is equal to:
(1) 1
(2) 2
(3) 0
(4) 3

Let the polynomial function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b , c$ are all rational numbers. Select the correct options.
(1) The graph of $y = f ( x )$ and the parabola $y = x ^ { 2 } + 100$ may have no intersection points
(2) If $f ( 0 ) f ( 1 ) < 0 < f ( 0 ) f ( 2 )$, then the equation $f ( x ) = 0$ must have three distinct real roots
(3) If $1 + 3 i$ is a complex root of the equation $f ( x ) = 0$, then the equation $f ( x ) = 0$ has a rational root
(4) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form an arithmetic sequence in order
(5) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form a geometric sequence in order

A game company will hold a lottery activity. The company announces that each lottery draw requires using one token, and the winning probability for each draw is $\frac{1}{10}$. A certain person decides to save a certain number of tokens and start drawing after the activity begins, stopping only when all tokens are used. Select the correct options.
(1) The expected value of the number of draws needed for the person to win once is 10
(2) The probability that the person wins at least once in two draws is 0.2
(3) The probability that the person does not win in 10 draws is less than the probability of winning in 1 draw
(4) The person must save at least 22 tokens to guarantee a winning probability greater than 0.9
(5) If the person saves sufficiently many tokens, the winning probability can be guaranteed to be 1