# Mean and Median

*English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email sota.ai.community@gmail.com.*

All-Russian School Olympiad in Informatics 2025–2026, Regional Stage, Grades 9–11<br>
“Artificial Intelligence” profile, Tour 1, 17 January 2026<br>
Task C

**Points for the task:** 50<br>
**Answer submission format:** entering the answer<br>
**Number of attempts:** 10<br>
**Submission that counts:** the last one

## Statement

Let $x_1, \dots, x_{10} \in [0, 1]$, and let the following conditions hold:

$$
|x_i - x_j| \ge 0.01 \quad \text{for all } i \ne j,
$$

and every subsegment $[a, a + 0.25] \subset [0, 1]$ contains at least one point of the set $\{x_1, \dots, x_{10}\}$.

Sort the numbers:

$$
x_{(1)} \le x_{(2)} \le \dots \le x_{(10)}.
$$

Denote by

$$
\overline{x} = \frac{1}{10} \sum_{k=1}^{10} x_{(k)}, \qquad
m = \frac{x_{(5)} + x_{(6)}}{2}
$$

the mean and the median (taken as the mean of the two middle numbers in order).

By how much at most can the mean $\overline{x}$ differ from the median $m$? That is, find

$$
\max |\overline{x} - m|
$$

under the conditions described.

## Output format

Round the answer to 6 decimal places, using a point as the decimal separator.

## Scoring

An exact match of the answer: 50 points.

Test results are **not available** during the tour.