MathDB
2001 El Salvador Correspondence / Qualifying NMO I

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October 15, 2021
algebrageometrycombinatoricsnumber theoryel salvador NMO

Problem Statement

p1. In the accompanying figure, OO is the center of the circumference, OBOB is perpendicular to the diameter ACAC; segment BFBF is 33 cm, segment FOFO is 22 cm, and line BEBE is tangent to the circumference at point BB. Calculate the length of segment DEDE. https://cdn.artofproblemsolving.com/attachments/e/e/e1e062b7794fe9c16192da8f3a332e3ccafe7a.png
p2. Starting with the sequence 1,9,9,31,9,9,3, we construct the sequence 1,9,9,3,2,3,71, 9, 9, 3, 2, 3, 7, where each digit is the last digit from the right of the sum of the previous four digits in the sequence. Determine if the sequence 7,3,6,77,3,6,7, in that order, will appear at some point in the sequence.
p3. SS is a set with nn integers positive none of which is divisible by nn. Prove that a subset of SS exists such that the sum of its elements is divisible by nn.
p4. On a table there are 20012001 boxes containing 1,2,3,4...20011, 2, 3, 4 ... 2001 objects, respectively. You can choose any number of boxes and from each one of them subtract the same number of objects. Determine the minimum number of moves to leave all boxes empty.
p5. If x1,x2,z3,x4x_1, x_2, z_3, x_4 are non negative integers, what is the total of possible solutions of the equation x1+x2+x3+x4=20x_1 + x_2 + x_3 + x_4 = 20?
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