2007 Chile NMO Juniors XIX
Source:
October 19, 2021
algebrageometrycombinatoricsnumber theorychilean NMO
Problem Statement
p1. Pedro and Juan are still at opposite points in a roundabout. If Pedro turns times the roundabout and Juan goes around times in the same time, but in the opposite direction, determine how many times they cross.
[url=https://artofproblemsolving.com/community/c6h1845563p12424471]p2. From a triangle , we build the triangle whose vertices they are the midpoints of the sides of . The triangle is constructed from in a way analogue. We build the triangles . Prove that the center of gravity of the triangle is inside the triangle .
[url=https://artofproblemsolving.com/community/c6h2927784p26188277]p3. Consider a non-convex polygon with sides and a line that does not pass through any of the vertices of the polygon. Prove that the line cannot cut exactly sides of the polygon.
[url=https://artofproblemsolving.com/community/c4h2917775p26063722]p4. Determine whether it is possible to draw a finite number of circles inside of a square with side 1 so that its interiors do not intersect and the sum of its radii is greater than .
p5. Seven guests to a restaurant sit in chairs equally spaced around a round table, but they have not seen that in the points there are cards with the names of the guests. Assume they have sat down with such bad luck that none is in the place where they should.
Show that it is possible to get at least two people in their correct position, without anyone getting up from their seat by turning the table.
Show a configuration where exactly a guest is in his assigned place and where there is exactly no way that the table is turned to be possible to get at least two to fit.
[hide=original wording]Siete invitados a una esta se sientan en sillas igualmente espaciadas alrededor de una mesa redonda, pero no se han jado que en los puntos hay tarjetas con los nombres de los invitados.
Suponiendo que se han sentado con tan mala suerte que ninguno se encuentra en el lugar que le corresponde, muestre que es posible lograr que al menos dos personas queden en su puesto correcto, sin que nadie se pare de su asiento haciendo girar la mesa.
Muestre una con figuracion donde exactamente un invitado esta en su lugar asignado y donde de ninguna forma que se gire la mesa es posible lograr que al menos dos queden bien.
p6. A -page magazine has one page ripped off and has lined pages. The product of the numbers of the hatched and missing pages is . Determine the missing pages.