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On an infinite square grid, on which some unit squares are coloured red, a ruby rook is a piece which, in one move, can travel any number of squares in one direction parallel to one of the grid lines (either vertically or horizontally), while remaining on red squares at all times throughout the move.

Starting with an uncoloured infinite square grid, Alice performs the following procedure: First, she colours at most 2025 of the unit squares red. Afterwards, she places some ruby rooks on distinct red unit squares, such that the following two rules are satisfied:

• No ruby rook can reach another ruby rook in one move.
• Every ruby rook can reach every other ruby rook in two moves.

Find the maximum possible number of ruby rooks that Alice can place during this procedure.

A snake in an $n \times n$ grid is a path composed of straight line segments between centres of adjacent cells, going through the centres of all the $n^2$ grid cells, which visits each cell exactly once. Here two grid cells are considered to be adjacent if they share an edge. Note that all pieces of the snake path are parallel to grid lines. The figure shows an example of a snake in a $4 \times 4$ grid. This snake makes nine $90^\circ$ turns, marked by small black squares.

Let us now consider a snake through the 2025 cells of a $45 \times 45$ grid. What is the maximum possible number of $90^\circ$ turns that such a snake can make?

Iz ploče dimenzija $2025 \times 2025$ uklonjen je kvadrat dimenzija $7 \times 7$, a preostali dio ploče prekriva se pločicama dimenzija $1 \times 4$ (tako da svaka pločica prekriva točno četiri polja).

(a) Ako uklonimo središnji $7 \times 7$ kvadrat, dokaži da je preostali dio ploče moguće pokriti pločicama dimenzija $1 \times 4$.

(b) Ako uklonimo $7 \times 7$ kvadrat koji sadrži jedan ugao ploče, dokaži da preostali dio ploče nije moguće pokriti pločicama dimenzija $1 \times 4$.

U svako polje pravokutne ploče s 3 stupca i 14 redaka upisan je simbol $X$ ili $O$. Za ploču kažemo da je balansirana ako su zadovoljeni sljedeći uvjeti:

• svaki $3 \times 3$ kvadrat sadržava najviše 5 simbola $X$ i najviše 5 simbola $O$
• u svakom $3 \times 3$ kvadrati nijedna dijagonala ni redak ni stupac ne sadržavaju tri ista simbola.

Za balansiranu ploču $P$, centar od $P$ je ploča s 3 stupca i 12 redaka dobivena uklanjanjem prvoga i posljednjega retka iz $P$.

Među svim balansiranim pločama koliko postoji različitih centara?

Dana je ploča dimenzija $9 \times 9$ čija su sva polja bijela. Odredi najveći broj polja koja je moguće obojiti u crveno tako da svaki dio ploče dimenzija $2 \times 2$ sadržava najviše dva crvena polja.