Show that $$\forall x , y \in \mathbf { R } _ { + } , \quad x y \leq \frac { x ^ { p } } { p } + \frac { y ^ { q } } { q }$$ where $p , q \in ] 1 , + \infty [$ such that $\frac { 1 } { p } + \frac { 1 } { q } = 1$.

What inequality do we recover when $p = q = 2$ ? Give a direct proof of it.

13. Let $\mathscr { B } = \left( a _ { 0 } , \ldots , a _ { n } \right)$ be the unique orthogonal basis of $\left( \mathbb { R } _ { n } [ X ] , \langle \cdot , \cdot \rangle \right)$ such that $a _ { i }$ is a monic polynomial of degree $i$ for all $0 \leqslant i \leqslant n$. Show that, for all $0 \leqslant j \leqslant n - 1$, the coefficients of the polynomial $\prod _ { \ell = j + 1 } ^ { n } \left( X - r _ { \ell } \right)$ in the basis $\mathscr { B }$ are strictly positive real numbers. Hint: one may denote $\left( q _ { j , 0 } , \ldots , q _ { j , n - j } \right)$ the basis of $\left( \mathbb { R } _ { n - j } [ X ] , \langle \cdot , \cdot \rangle _ { j } \right)$ obtained in questions 8a and 8b and reason by descending induction on $j$.

Third Part

Let $\lambda$ be a strictly positive real number. For all real $x$ and $r$ such that $| x | < 1$ and $| r | < 1$, we set
$$F _ { \lambda } ( x , r ) = \left( 1 - 2 r x + r ^ { 2 } \right) ^ { - \lambda }$$
Show that the function $F _ { \lambda }$ is of class $\mathscr { C } ^ { \infty }$ on $] - 1,1 \left[ ^ { 2 } \right.$.

Prove that $2^n < \dbinom{2n}{n} < \dfrac{2^n}{\prod_{j=0}^{n-1}\left(1 - \frac{j}{n}\right)}$ for all positive integers $n$.

Let $a , b , c$ be three real numbers which are roots of a cubic polynomial, and satisfy $a + b + c = 6$ and $a b + b c + a c = 9$. Suppose $a < b < c$. Show that
$$0 < a < 1 < b < 3 < c < 4$$

In a sports tournament involving $N$ teams, each team plays every other team exactly once. At the end of every match, the winning team gets 1 point and the losing team gets 0 points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows:
$$x _ { 1 } \geq x _ { 2 } \geq \cdots \geq x _ { N }$$
Prove that for any $1 \leq k \leq N$,
$$\frac { N - k } { 2 } \leq x _ { k } \leq N - \frac { k + 1 } { 2 } .$$

Let $n \geq 2$ and let $a _ { 1 } \leq a _ { 2 } \leq \cdots \leq a _ { n }$ be positive integers such that $\sum _ { i = 1 } ^ { n } a _ { i } = \Pi _ { i = 1 } ^ { n } a _ { i }$. Prove that $\sum _ { i = 1 } ^ { n } a _ { i } \leq 2n$ and determine when equality holds.

15. If $\mathrm { a } , \mathrm { b } , \mathrm { c }$ are positive real numbers, then prove that $[ ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) ] ^ { 7 } > 7 ^ { 7 } \mathrm { a } ^ { 4 } \mathrm {~b} ^ { 4 } \mathrm { c } ^ { 4 }$.
Sol. $( 1 + a ) ( 1 + b ) ( 1 + c ) = 1 + a b + a + b + c + a b c + a c + b c$ $\Rightarrow \frac { ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) - 1 } { 7 } \geq ( \mathrm { ab } . \mathrm { a } . \mathrm { b } . \mathrm { c } . \mathrm { abc } . \mathrm { ac } . \mathrm { bc } ) ^ { 1 / 7 } \quad ($ using $A M \geq G M )$ $\Rightarrow ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) - 1 > 7 \left( \mathrm { a } ^ { 4 } \cdot \mathrm {~b} ^ { 4 } \cdot \mathrm { c } ^ { 4 } \right) ^ { 1 / 7 }$ $\Rightarrow ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) > 7 \left( \mathrm { a } ^ { 4 } \cdot \mathrm {~b} ^ { 4 } \cdot \mathrm { c } ^ { 4 } \right) ^ { 1 / 7 }$ $\Rightarrow ( 1 + \mathrm { a } ) ^ { 7 } ( 1 + \mathrm { b } ) ^ { 7 } ( 1 + \mathrm { c } ) ^ { 7 } > 7 ^ { 7 } \left( \mathrm { a } ^ { 4 } \cdot \mathrm {~b} ^ { 4 } \cdot \mathrm { c } ^ { 4 } \right)$.