Matrices and Forgotten Activations

Matrices and Forgotten Activations

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, Regional Stage, Grades 9–11
“Artificial Intelligence” profile, Tour 1, 17 January 2026
Task B

Points for the task: 50
Answer submission format: entering the answer
Number of attempts: 10
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.