2001 Chile Classification / Qualifying NMO Seniors XIII
Source:
October 12, 2021
algebrageometrynumber theorycombinatoricschilean NMO
Problem Statement
p1. All positive fractions less than one are considered, whose denominator is and whose numerator is a number that has no common divisors with . Calculate the sum of these fractions.
p2. Triangle is right isosceles. The figure below shows two basic ways to inscribe a square in it. Prove that the square has a greater area than the square .
https://cdn.artofproblemsolving.com/attachments/9/3/84d0cda6e109aaf01fb2f3de4d1933f0f16f82.jpg
p3. Given people, show that there exists a value of such that with people you can form groups of a , so that each pair of people is in exactly one of these groups and show a corresponding conformation of these groups. If the same number of groups must be formed, but of people each and with the condition that each pair is in exactly groups, determine if there is a value of that makes it possible for the problem to have a solution and, if affirmative, display a corresponding formation.
p4. Let be a parallelogram. Side is extended to a point , such that and the side is extended to a point , such that .
Prove that points , , and are collinear.
Prove that the perpendicular to line at point , the perpendicular to line at point , the bisector of the angle and the perpendicular to the diagonal by the vertex are all concurrent at a point .
https://cdn.artofproblemsolving.com/attachments/c/1/e0d585f30c4c9ca274e1ce1599128e3e21a08c.png
p5. Consider two positive integers and that satisfy the relation .
Prove that the numbers , , are three perfect squares.
p6. Let be an quadrilateral inscribed in a circle of radius r, and let E be the point where its diagonals intersect.
Prove if the diagonals are perpendicular to each other, then .
If the previous relation is fulfilled, are the diagonals of the quadrilateral necessarily perpendicular?
p7 On a rectangular board with rows and n columns, place in each of the squares a or a , so that the numbers in each add the same amount and those in each column, the same amount . Show that a necessary and sufficient condition for this assignment to be possible is that .PS. Seniors p2, part of p6, were posted also as [url=https://artofproblemsolving.com/community/c4h2689068p23335606]Juniors variation p2, p6.