5. Determine the smallest real constant $C$ such that for all $a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \not necessarily distinct, there always exist four distinct indices $i, j, k, l$ such that:

$$
\left|\frac{a_{i}}{a_{j}}-\frac{a_{k}}{a_{l}}\right| \leq C
$$
Solution: After a few trials, we notice that $C=1 / 2$ is optimal.

- $C \geq 1 / 2$ : substituting $(1 / 2,1,1,1, n)$ and leaving $n \rightarrow \infty$, we obtain $C \geq 1 / 2$.
- C \leq 1 / 2$: suppose $a_{5} \geq a_{4} \geq \ldots \geq a_{1}$. Let's split $[0,1]$ into two disjoint sub-intervals: $I=[0,1 / 2)$ and $J=[1 / 2,1]$. Let's look at the fractions $0<\frac{a_{1}}{a_{3}}, \frac{a_{1}}{a_{5}}, \frac{a_{3}}{a_{5}}. \leq 1$.

If the three fractions are in the same interval $I$ or $J$, then we notice that this interval also contains $\frac{a_{2}}{a_{5}}$ because $\frac{a_{1}}{a_{5}} \leq \frac{a_{2}}{a_{5}} \leq \frac{a_{3}}{a_{5}}$. So $\frac{a_{2}}{a_{5}}$ and $\frac{a_{1}}{a_{3}}$ lie in the same interval and therefore at a distance of less than $1 / 2$.

Otherwise, $I$ and $J$ contain at least one of the fractions $\frac{a_{1}}{a_{3}}, \frac{a_{1}}{a_{5}}, \frac{a_{3}}{a_{5}}$. Now $\frac{a_{2}}{a_{4}}$ lies in either $I$ or $J$. Once again, we obtain two fractions with different indices in the same interval.

## Marking scheme:

- C \geq 1 / 2: 2 \mathrm{P}$.
- $C \leq 1 / 2: 5 \mathrm{P}$, including
- work with $I$ and $J: 1 \mathrm{P}$.
- order values and focus on fractions $\leq 1: 1 \mathrm{P}$.