23. Solution: (1) Since $a^2 + b^2 \geq 2ab$, $b^2 + c^2 \geq 2bc$, $c^2 + a^2 \geq 2ac$, and $abc = 1$, we have $a^2 + b^2 + c^2 \geq ab + bc + ca = \frac{ab + bc + ca}{abc} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. Therefore $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq a^2 + b^2 + c^2$.
(2) Since $a, b, c$ are positive numbers and $abc = 1$, we have $(a+b)^3 + (b+c)^3 + (c+a)^3 \geq 3\sqrt[3]{(a+b)^3(b+c)^3(a+c)^3}$ $= 3(a+b)(b+c)(a+c)$ $\geq 3 \times (2\sqrt{ab}) \times (2\sqrt{bc}) \times (2\sqrt{ac})$ $= 24$. Therefore $(a+b)^3 + (b+c)^3 + (c+a)^3 \geq 24$.

Consider the following four propositions:
$p _ { 1 }$ : Three lines that are pairwise intersecting and do not pass through the same point must lie in the same plane.
$p _ { 2 }$ : Through any three points in space, there is exactly one plane.
$p _ { 3 }$ : If two lines in space do not intersect, then these two lines are parallel.
$p _ { 4 }$ : If line $l \subset$ plane $\alpha$ and line $m \perp$ plane $\alpha$ , then $m \perp l$ .
The sequence numbers of all true propositions among the following statements are $\_\_\_\_$.
(1) $p _ { 1 } \wedge p _ { 4 }$
(2) $p _ { 1 } \wedge p _ { 2 }$
(3) $\neg p _ { 2 } \vee p _ { 3 }$
(4) $\neg p _ { 3 } \vee \neg p _ { 4 }$

[Elective 4-5: Inequalities] Let $a , b , c \in \mathbf { R } , a + b + c = 0 , a b c = 1$ .
(1) Prove: $a b + b c + c a < 0$;
(2) Let $\max \{ a , b , c \}$ denote the maximum value among $a , b , c$. Prove: $\max \{ a , b , c \} \geqslant \sqrt[3]{\frac{3}{2}}$.

[Elective 4-5: Inequalities] Given that $a , b , c$ are all positive numbers and $a ^ { 2 } + b ^ { 2 } + 4 c ^ { 2 } = 3$ , prove:
(1) $a + b + 2 c \leq 3$ ;
(2) If $b = 2 c$ , then $\frac { 1 } { a } + \frac { 1 } { b } + \frac { 4 } { c } \geq 3$ .

[Elective 4-5: Inequalities] (10 points) Given that $a$, $b$, $c$ are all positive numbers and $a ^ { 2 } + b ^ { 2 } + 4 c ^ { 2 } = 3$, prove that:
(1) $a + b + 2 c \leq 3$;
(2) If $b = 2 c$, then $\frac { 1 } { a } + \frac { 1 } { c } \geq 3$.

Given proposition $p : \forall x \in \mathbf { R } , | x + 1 | > 1$; proposition $q : \exists x > 0 , x ^ { 3 } = x$, then
A. Both $p$ and $q$ are true propositions
B. Both $\neg p$ and $q$ are true propositions
C. Both $p$ and $\neg q$ are true propositions
D. Both $\neg p$ and $\neg q$ are true propositions

Let the set $M = \{ ( i , j , s , t ) \mid i \in \{ 1,2 \} , j \in \{ 3,4 \} , s \in \{ 5,6 \} , t \in \{ 7,8 \} \}$. For a given finite sequence $A$ and sequence $\Omega : \omega _ { 1 } , \omega _ { 2 } , \cdots , \omega _ { k } , \omega _ { k } = \left( i _ { k } , j _ { k } , s _ { k } , t _ { k } \right) \in M$, define transformation $T$: add 1 to columns $i _ { 1 } , j _ { 1 } , s _ { 1 } , t _ { 1 }$ of sequence $A$ to obtain sequence $T _ { 1 } ( A )$; add 1 to columns $i _ { 2 } , j _ { 2 } , s _ { 2 } , t _ { 2 }$ of sequence $T _ { 1 } ( A )$ to obtain sequence $T _ { 2 } T _ { 1 } ( A )$; repeat the above operations to obtain sequence $T _ { k } \cdots T _ { 2 } T _ { 1 } ( A )$, denoted as $\Omega ( A )$.
(3) If $a _ { 1 } + a _ { 3 } + a _ { 5 } + a _ { 7 }$ is even, prove that ``$\Omega ( A )$ is a constant sequence'' is a necessary and sufficient condition for ``$a _ { 1 } + a _ { 2 } = a _ { 3 } + a _ { 4 } = a _ { 5 } + a _ { 6 } = a _ { 7 } + a _ { 8 }$''.

Establish that $\tau = \tau _ { 0 } \cup \tau _ { 1 }$.

We seek to show that the inequality $|\sin(n\theta)| \leqslant n \sin(\theta)$ is satisfied for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2}\right]$.
a) Show that $\sin(n\theta) \leqslant n \sin(\theta)$ for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2n}\right]$.
b) Show that, for all $\theta \in \left[0, \frac{\pi}{2}\right]$, we have $\sin(\theta) \geqslant \frac{2}{\pi} \theta$.
c) Deduce that: $$\forall \theta \in \left[\frac{\pi}{2n}, \frac{\pi}{2}\right], \quad 1 \leqslant n \sin(\theta)$$
d) Conclude.
e) For which values of $\theta \in \left[0, \frac{\pi}{2}\right]$ do we have $|\sin(n\theta)| = n \sin(\theta)$?

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order.
Show that, for all $x \in [x_{n,1}, x_{n,n}]$, we have $$\sqrt{1 - x^2} \geq \frac{1}{n}$$

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$.
a) Let $n \in \mathbb{N}^*$ and $P \in E_{n-1}$ such that $\sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)| \leq 1$.
Show that $$\sup_{x \in [-1,1]} |P(x)| \leq n$$ (Distinguish three cases according to whether $x$ belongs to one of the intervals $[-1, x_{n,1}[$, $[x_{n,1}, x_{n,n}]$ or $]x_{n,n}, 1]$.)
b) Deduce that for all $n \in \mathbb{N}^*$ and for all $P \in E_{n-1}$, we have: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)|.$$

Let $T$ be a trigonometric polynomial of the form $$T(\theta) = a_0 + \sum_{k=1}^{n} \left[ a_k \cos(k\theta) + b_k \sin(k\theta) \right]$$ where $a_0, a_1, b_1, \ldots, a_n, b_n \in \mathbb{R}$.
a) Let $k \in \mathbb{N}^*$. Show that there exists a polynomial function $B_k$ of degree $(k-1)$ such that: $$\forall \theta \in \mathbb{R}, \quad \sin(k\theta) = B_k(\cos(\theta)) \sin(\theta).$$
b) Let $\theta_0 \in \mathbb{R}$. Show that there exists a polynomial function $P \in E_{n-1}$ such that, for all $\theta \in \mathbb{R}$, we have: $$T(\theta_0 + \theta) - T(\theta_0 - \theta) = 2 P(\cos\theta) \sin\theta$$
c) Deduce that: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$
d) Show that: $$\sup_{\theta \in \mathbb{R}} \left| T'(\theta) \right| \leq n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$

Let $P \in E_n$. Show that: $$\sup_{x \in [-1,1]} \left| P'(x) \right| \leq n^2 \sup_{x \in [-1,1]} |P(x)|$$ (One may use the trigonometric polynomial $T(\theta) = P(\cos(\theta))$.)

If $A$ is a subset of $E$, we denote $A^{\perp\varphi} = \{ x \in E \mid \forall a \in A,\ \varphi(x,a) = 0 \}$. Show that $A^{\perp\varphi}$ is a vector subspace of $E$.

We say that $\varphi$ is non-degenerate if and only if $E^{\perp\varphi} = \{0\}$.
Show that $\varphi$ is non-degenerate if and only if $h$ is an isomorphism.

Let $q \in Q(E)$.
Show that there exists a unique symmetric bilinear form on $E$, denoted $\varphi$, such that $q = q_\varphi$.

Let $q$ be a quadratic form on $E$. Let $E'$ be a second $\mathbb{K}$-vector space of dimension $n$, and let $q'$ be a quadratic form on $E'$.
We call an isometry from $(E,q)$ to $(E',q')$ any isomorphism $f$ from $E$ to $E'$ satisfying: for all $x \in E$, $q'(f(x)) = q(x)$. We will say that $(E,q)$ and $(E',q')$ are isometric if and only if there exists an isometry from $(E,q)$ to $(E',q')$.
Show that $(E,q)$ and $(E',q')$ are isometric if and only if there exists a basis $e$ of $E$ and a basis $e'$ of $E'$ such that $\operatorname{mat}(q,e) = \operatorname{mat}(q',e')$.

Let $p \in \mathbb{N}^*$. We denote by $c = (c_1, \ldots, c_{2p})$ the canonical basis of $\mathbb{K}^{2p}$. $$\text{For all } x = \sum_{i=1}^{2p} x_i c_i \in \mathbb{K}^{2p}, \text{ we set } q_p(x) = 2\sum_{i=1}^{p} x_i x_{i+p}.$$
a) Show that $q_p$ is a quadratic form on $\mathbb{K}^{2p}$ and compute $\operatorname{mat}(q_p, c)$.
b) We call an Artin space (or artinian space) of dimension $2p$ any pair $(F,q)$, where $F$ is a $\mathbb{K}$-vector space of dimension $2p$, and where $q$ is a quadratic form on $F$ such that $(F,q)$ and $(\mathbb{K}^{2p}, q_p)$ are isometric. Show that in this case, $q$ is non-degenerate. When $p=1$, we say that $(F,q)$ is an artinian plane.
c) We assume that $\mathbb{K} = \mathbb{C}$ and for all $$x = \sum_{k=1}^{2p} x_k c_k \in \mathbb{C}^{2p}, \text{ we set } q(x) = \sum_{k=1}^{2p} x_k^2.$$ Show that $(\mathbb{C}^{2p}, q)$ is an Artin space.
d) We assume that $\mathbb{K} = \mathbb{R}$ and for all $$x = \sum_{i=1}^{2p} x_i c_i \in \mathbb{R}^{2p}, \text{ we set } q'(x) = \sum_{i=1}^{p} x_i^2 - \sum_{i=p+1}^{2p} x_i^2.$$ Show that $(\mathbb{R}^{2p}, q')$ is an Artin space.
e) If $(F,q)$ is an Artin space of dimension $2p$, show that there exists a vector subspace $G$ of $F$ of dimension $p$ such that the restriction of $q$ to $G$ is identically zero.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We still denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$. Let $p \in \{1, \ldots, n\}$. We denote by $F$ the space spanned by $e_1, \ldots, e_p$.
a) Show that $F^\perp$ is the preimage under $h$ of $\operatorname{Vect}(e_{p+1}^*, \ldots, e_n^*)$, where $h$ is defined in I.A.1.
b) Show that $\operatorname{dim}(F) + \operatorname{dim}(F^\perp) = n$.
c) Show that $(F^\perp)^\perp = F$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ and $G$ be two vector subspaces of $E$.
a) Show that $(F+G)^\perp = F^\perp \cap G^\perp$.
b) Show that $(F \cap G)^\perp = F^\perp + G^\perp$.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ be a vector subspace of $E$. We denote by $\varphi_F$ the restriction of $\varphi$ to $F^2$. We will say that $F$ is singular if and only if $\varphi_F$ is degenerate.
Show that $F$ is non-singular if and only if one of the following properties is verified:

• $F \cap F^\perp = \{0\}$;
• $E = F \oplus F^\perp$;
• $F^\perp$ is non-singular.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We say that two vector subspaces $F$ and $G$ of $E$ are orthogonal if and only if for all $(x,y) \in F \times G$, $\varphi(x,y) = 0$.
If $F$ and $G$ are two vector subspaces of $E$ that are orthogonal and non-singular, show that $F \oplus G$ is non-singular.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $E = \mathbb{R}^2$ and for all $(x,y) \in \mathbb{R}^2$, $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$.
Determine a $q$-orthogonal basis and a $q'$-orthogonal basis.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
With $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$ on $\mathbb{R}^2$ as defined in question II.B.1, does there exist a basis of $\mathbb{R}^2$ orthogonal for both $q$ and $q'$?

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $x \in E$ such that $q(x) = 0$ and such that $x \neq 0$.
We propose to demonstrate that there exists a plane $\Pi \subset E$ containing $x$ such that $(\Pi, q_{/\Pi})$ is an artinian plane (where $q_{/\Pi}$ denotes the restriction of the application $q$ to the plane $\Pi$).
a) Demonstrate that there exists $z \in E$ such that $\varphi(x,z) = 1$.
b) We set $y = z - \frac{q(z)}{2}x$. Compute $q(y)$.
c) Conclude.