# Matrices and Forgotten Activations

*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 B

**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

Vasya is learning to draw pictures with a neural network: he wants to map each point of the plane to a pixel colour in three channels (R, G, B).

The network receives as input the vector

$$
x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix},
$$

and then performs the same operation several times in a row: it multiplies the current column vector by a weight matrix and obtains a new column vector. The sizes of the vectors obtained one after another inside the network are:

$$
2 \to 256 \to 128 \to 64 \to 3,
$$

where $2$ is the size of the input vector and $3$ is the size of the output vector.

Through carelessness, Vasya forgot to add everything “interesting” to the network: both the addition of constants and the non-linear functions. Therefore, all the work of the network up to the last step consists only of successive multiplications by matrices. Denote by

$$
T(x_1, x_2)
$$

the three-dimensional vector obtained at the very last layer before the final processing.

Vasya obtains the actual pixel colour after clipping the result coordinate-wise to the range from $0$ to $255$:

$$
\operatorname{clip}(y)_i =
\begin{cases}
0, & y_i < 0,\\
y_i, & 0 \le y_i \le 255,\\
255, & y_i > 255,
\end{cases}
$$

and, finally,

$$
f(x_1, x_2) = \operatorname{clip}(T(x_1, x_2)).
$$

In an experiment with the network weights already trained, it turned out that

$$
f(-2, 3) = \begin{pmatrix} 60 \\ 40 \\ 100 \end{pmatrix},
\qquad
f(1, 3) = \begin{pmatrix} 200 \\ 50 \\ 80 \end{pmatrix}.
$$

Find the vector $f(-7, 6)$.

## Output format

As your answer, output three numbers separated by spaces. If a number turns out to be non-integer, output it with a precision of 6 decimal places. If there is no answer, or if several answers are possible, output -1.

## Scoring

An exact match of the answer: 50 points.

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