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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 subset $S$ of the integers is called Saxonian if for every three pairwise different elements $a, b, c \in S$ the number $ab + c$ is the square of an integer. Prove that any Saxonian set is finite. Determine the largest possible number of elements that a Saxonian set can have.

Bob has $n$ coins with integer values $$c_1 \geq c_2 \geq \cdots \geq c_n > 0.$$

He is standing in front of a vending machine that offers $n$ candy bars with positive integer costs $b_1, b_2, \ldots, b_n$. Bob notices that for every $i \in \{1, \ldots, n\}$, it holds that $$b_1 + b_2 + \cdots + b_i \geq c_1 + c_2 + \cdots + c_i.$$

Furthermore, the total value of Bob's coins equals the sum of the costs of all the candy bars. The candy bars can be purchased in any order. In order to buy the $i$-th candy bar, Bob has to insert coins of total value at least $b_i$. However, the machine does not give him back any change.

Prove that Bob can buy at least half of the candy bars.

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?

Let $n$ be a positive integer. In the province of Laplandia there are $100n$ cities, each two connected by a direct road, and each of these roads has a toll station collecting a positive amount of toll revenue. For each road, the revenue of its toll station is split equally between the two cities at the ends of the road (meaning that each of the two cities receives half of the income). For each city, the total toll revenue is given by the sum of the revenues it receives from the $100n - 1$ toll stations on its roads.

According to a new law, the revenues of some of the toll stations will be collected by the federal government instead of by the adjacent cities. The governor of Laplandia is allowed to choose those toll stations. The mayors of the cities demand that for each city, the sum of the remaining revenues it receives from the other toll stations after this change is at least $99\%$ of its former total toll revenue.

Find the largest positive integer $k$, depending on $n$, such that the governor can always choose $k$ toll stations for the federal government to collect the toll revenue, while satisfying the demand of the city mayors.

Odredi najmanju vrijednost koju može poprimiti izraz $$x + |2|x| - 1|$$ za neki realni broj $x$.

Baka Jagoda prodaje trešnje te je uočila da postoji linearna ovisnost između cijene jednog kilograma trešanja i količine prodanih trešanja u danu: svakim povećanjem cijene za 1 € po kilogramu bi u danu prodala 3 kilograma trešanja manje. Najveći iznos od prodaje trešanja bi ostvarila kada bi ih prodavala po cijeni od 3.6 € po kilogramu. Jednog dana unuka Višnja zamijenila je baku na tržnici, sama odredila cijenu kilograma trešanja i prodala trešnje za 18.6 €. Po kojoj je cijeni Višnja mogla prodavati trešnje?

U stožac osnovke polumjera 1 i visine duljine $2\sqrt{2}$ upisan je kvadar takav da jedna strana kvadra pripada osnovki stošca, a vrhovi suprotne strane pripadaju plaštu stošca.

Ako je strana kvadra koja pripada osnovki stošca kvadrat, koliko je najveće oplošje koje takav kvadar može imati?

Odredi najveću moguću površinu pravokutnika upisanog u pravokutni trokut s katetama duljina 5 i 12 tako da se dva vrha pravokutnika nalaze na hipotenuzi, a po jedan vrh na svakoj kateti tog trokuta.

Blok je figura koja se sastoji od šest jediničnih kvadrata kao što je prikazano na slici. Odredi najveći mogući broj blokova koje je moguće postaviti na ploču dimenzija $6 \times 11$ tako da svaki prekriva točno šest polja. Blokovi se mogu rotirati i ne smiju se preklapati.

Tablica dimenzija $2025 \times 2025$ popunjena je tako da se u polju u $i$-tome retku i $j$-tome stupcu nalazi broj $i + j - 1$, za sve $i, j \in \{1, 2, \ldots, 2025\}$. Odabrano je 2025 polja koja se nalaze u različitim retcima i različitim stupcima.

Koja je najmanja moguća vrijednost umnoška brojeva na odabranim poljima?

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.

Za arhipelag od 2026 otoka kažemo da je dobro povezan ako među svakih pet različitih otoka, postoje tri takva da između svaka dva od njih postoji dvosmjerna brodska linija. Odredi najveći prirodni broj $N$ takav da u svakom dobro povezanom arhipelagu postoji niz od barem $N$ različitih otoka takav da su svaka dva uzastopna, te prvi i posljednji otok u nizu povezani brodskom linijom.