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2018 Chile Classification / Qualifying NMO Seniors XXX

Source:

October 11, 2021
algebrageometrynumber theorycombinatoricschilean NMO

Problem Statement

p1. From a 10001000-page book, a quantity has been ripped of consecutive of leaves. It is known that the sum of the numbers of the torn pages is 20182018. Determine the numbering of the ripped pages.
p2. A square with side 8 8 cm is divided into 6464 squares of 1 1 cm2^2. 77 little squares are colored black and the rest white. Find the maximum area of a rectangle composed only of small white squares independent of the distribution of the little black squares.
p3. Let n ∈N\in N. We want to find a partition of Ω={1,2,...,n}\Omega = \{1, 2,..., n\} in MM disjoint subsets S1,S2,...,SMS_1, S_2,..., S_M, such that the sum of the elements of each SiS_i is the same. What is the maximum value possible of MM?
p4. In the hypotenuse BCBC of a right isosceles triangle ABCABC are chosen four points P1P_1, P2P_2, P3P_3, P4P_4 such that ∣BP1∣=∣P1P2∣=∣P2P3∣=∣P3P4∣=∣P4C∣|BP_1| = |P_1P_2| = |P_2P_3| = |P_3P_4| = |P_4C|. Choose a point DD on leg ABAB such that 5∣AD∣=∣AB∣5|AD| = |AB|. Calculate the sum of the four angles ∠APiD\angle AP_iD, i=1,...,4i = 1,...,4.
PS. Seniors p1, p2 were posted as [url=https://artofproblemsolving.com/community/c4h2690917p23356802]Juniors p3,p2 respectively.
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