New Sample

New Sample

English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email [email protected].

All-Russian School Olympiad in Informatics 2025–2026, Final Stage
“Artificial Intelligence” profile, Tour 1, Moscow, 23 March 2026
Task E

A model predicts a real number from a real number $x$ by the method of linear regression, that is, by the formula $f(x) = ax + b$, where $a, b \in \mathbb{R}$ are the parameters of the model $f$.

A training sample $(x_1, y_1), (x_2, y_2), \ldots (x_n, y_n)$ is given, where $x_1, \ldots, x_n, y_1, \ldots y_n \in \mathbb{R}$. The model $f$ is trained on these data: the parameters $a$ and $b$ are chosen by the least-squares method, that is, so that the value of the expression

$$\sum_{i=1}^{n} (y_i - f(x_i))^2$$

is minimal.

We define the coefficient of determination $R_0^2$, computed on this sample, as follows:

$$R_0^2 = 1 - \frac{\sum\limits_{i=1}^{n} (y_i - f(x_i))^2}{\sum\limits_{i=1}^{n} (y_i - \bar{y})^2}, \qquad \text{where} \quad \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i.$$

We construct a new sample by adding the $n$ objects $(x_i, f(x_i))$, $i = 1, 2, \ldots, n$, to the original one. Let a model $g(x) = cx + d$ be trained on the resulting sample of $2n$ objects, also by the least-squares method. Denote by $R_1^2$ the coefficient of determination for the model $g$, computed on the sample of all $2n$ objects.

It is assumed that the parameters of both models could be determined by the least-squares method uniquely.

Express $R_1^2$ in terms of $R_0^2$.