The objective of this question is part of proving that $\lambda \leqslant \mathrm { e }$. We assume by contradiction that $\lambda > \mathrm { e }$.
Verify that, for all $k$ in $\mathbb { N } , \frac { 1 } { \mathrm { e } } \leqslant \left( \frac { k + 1 } { k + 2 } \right) ^ { k + 1 }$.

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.

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.