MathDB

2007 Chile Junior Math Olympiad

Part of Chile Junior Math Olympiad

Subcontests

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2007 Chile NMO Juniors XIX

p1. Pedro and Juan are still at opposite points in a roundabout. If Pedro turns 917917 times the roundabout and Juan goes around 10901090 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 T=△ABCT = \vartriangle ABC, we build the triangle T1=△A1B1C1T_1 = \vartriangle A_1B_1C_1 whose vertices they are the midpoints of the sides of TT. The triangle T2=△A2B2C2T_2 = \triangle A_2B_2C_2 is constructed from T1T_1 in a way analogue. We build the triangles T3,T4,...,T2007T_3, T_4,..., T_{2007}. Prove that the center of gravity GG of the triangle TT is inside the triangle T2007T_{2007}.
[url=https://artofproblemsolving.com/community/c6h2927784p26188277]p3. Consider a non-convex polygon with 10,00010,000 sides and a line that does not pass through any of the vertices of the polygon. Prove that the line cannot cut exactly 20072007 sides of the polygon.
[url=https://artofproblemsolving.com/community/c4h2917775p26063722]p4. Determine whether it is possible to draw a finite number of circles S1,S2,...,SnS_1, S_2, ..., S_n inside of a square with side 1 so that its interiors do not intersect and the sum of its radii is greater than 20072007.
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. ∙\bullet 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. ∙\bullet 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. ∙\bullet 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. ∙\bullet 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 3232-page magazine has one page ripped off and has 22 lined pages. The product of the numbers of the hatched and missing pages is 61446144. Determine the missing pages.