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2017 El Salvador Correspondence / Qualifying NMO XVII

Source:

October 17, 2021
algebrageometrycombinatoricsnumber theoryel salvador NMO

Problem Statement

p1. Determine the number of positive integers from 1 1 to 10201710^{2017} that satisfy that the sum of their digits is exactly 22.
p2. A positive integer is such that both adding 20002000 and subtracting 1717 results in a perfect square. Determine that number.
p3. Each integer number is painted red or blue according to the following rules: ∙\bullet The number 11 is red. ∙\bullet If aa and bb are two red numbers, not necessarily different, then the numbers a−ba-b and a+ba + b have different colors. Determine the color of the number 20172017.
p4. Determine all positive real numbers x,y,zx, y, z that satisfy the system of equations 1x+y+z=x+1y+z=x+y+1x=3\frac{1}{x} + y + z = x + \frac{1}{y} + z = x + y +\frac{1}{x} = 3
p5. Let ABCDABCD be a rectangle and MM a point on segment BCBC. The bisector of angle ∠DAM\angle DAM intersect segment CDCD at NN. If AM=BM+DNAM = BM +DN, prove that ABCDABCD is a square.
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