MathDB

2022 Pan-American Girls' Math Olympiad

Part of PAGMO

Subcontests

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Two girls play the longest game ever

Ana and Bety play a game alternating turns. Initially, Ana chooses an odd possitive integer and composite nn such that 2j<n<2j+12^j<n<2^{j+1} with 2<j2<j. In her first turn Bety chooses an odd composite integer n1n_1 such that n1≤1n+2n+⋯+(n−1)n2(n−1)n−1.n_1\leq \frac{1^n+2^n+\dots+(n-1)^n}{2(n-1)^{n-1}}. Then, on her other turn, Ana chooses a prime number p1p_1 that divides n1n_1. If the prime that Ana chooses is 33, 55 or 77, the Ana wins; otherwise Bety chooses an odd composite positive integer n2n_2 such that n2≤1p1+2p1+⋯+(p1−1)p12(p1−1)p1−1.n_2\leq \frac{1^{p_1}+2^{p_1}+\dots+(p_1-1)^{p_1}}{2(p_1-1)^{p_1-1}}. After that, on her turn, Ana chooses a prime p2p_2 that divides n2,n_2,, if p2p_2 is 33, 55, or 77, Ana wins, otherwise the process repeats. Also, Ana wins if at any time Bety cannot choose an odd composite positive integer in the corresponding range. Bety wins if she manages to play at least j−1j-1 turns. Find which of the two players has a winning strategy.
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Leticia's friendly squares

Leticia has a 9×99\times 9 board. She says that two squares are friends is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has 44 friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:
- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.
Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.