Subcontests
(6)P-representable Polynomial
Let P be a polynomial of degree greater than or equal to 4 with integer coefficients. An integer x is called P-representable if there exists integer numbers a and b such that x=P(a)−P(b). Prove that, if for all N≥0, more than half of the integers of the set {0,1,…,N} are P-representable, then all the even integers are P-representable or all the odd integers are P-representable. Carioca Points
A sequence P1,…,Pn of points in the plane (not necessarily diferent) is carioca if there exists a permutation a1,…,an of the numbers 1,…,n for which the segments
Pa1Pa2,Pa2Pa3,…,PanPa1
are all of the same length.Determine the greatest number k such that for any sequence of k points in the plane, 2023−k points can be added so that the sequence of 2023 points is carioca. Weighting options in Brazil
Ann and Beto play with a two pan balance scale. They have 2023 dumbbells labeled with their weights, which are the numbers 1,2,…,2023, with none of them repeating themselves. Each player, in turn, chooses a dumbbell that was not yet placed on the balance scale and places it on the pan with the least weight at the moment. If the scale is balanced, the player places it on any pan. Ana starts the game, and they continue in this way alternately until all the dumbbells are placed. Ana wins if at the end the scale is balanced, otherwise Beto win. Determine which of the players has a winning strategy and describe the strategy.