MathDB

2006 Chile Classification NMO Seniors

Part of Chile Classification NMO

Subcontests

(1)
1

2006 Chile Classification / Qualifying NMO Seniors XVIII

p1. In a certain city, the bus system has 9393 lines that pass, among all, through 20062006 stops. This system allows you to travel by bus from each stop to each of the others, perhaps making transshipments. For two lines A,BA, B there is at least one stop of A A that is not of B B, and vice versa. Could it happen in this system: a) if 5858 lines are removed, preserving all stops, so that it is still possible to reach from each stop to each other? b) If 5959 lines are removed, it doesn't matter which ones, then condition (a) is no longer true?
p2. On the blackboard there was a trapezoid ABCDABCD with bases ABAB and CDCD, in which four points were marked points: EE and FF are the midpoints of the non-parallel sides ADAD and BCBC, OO the point of intersection of its diagonals and PP an arbitrary point on the line ABAB. The entire figure was erased, except for the four dots. Describe a procedure to reconstruct the trapezium ABCDABCD.
p3. Isabel has a candlestick with nn equal candles. She likes to turn it on on Sundays with a curious system. The first Sunday light a candle for one hour, the second day she lights two candles for one hour and so on, until the nn-th day she lights all the candles for one hour. For what values of nn can Isabel get all the candles to be equally worn after of the last Sunday? In this case, show in what order they should be turned on.
p4. Three cars a,b,ca, b, c they leave at 66 am from three different towns A,B,CA, B, C around three different peoples (and different from the first three) D,E,FD, E, F, and they travel straight paths ADAD, BEBE, CFCF that are cut in pairs. Each car travels at a constant speed (can vary from car to car) such that each pair of cars reaches the intersection of their respective roads at the same time. The cars a,ba, b intersect at 77 am, the cars a,ca, c intersect at 99 am and cars b,cb, c intersect at 1010 am. Exactly at noon (at 12 12 am) the three cars arrive at their destinations. Distance ACAC is 9999 Km. Determine the distance EFEF.
p5. A few days ago, in order to reestablish contact after separating their university companions, the Zweinstein-Curie couple organized a meeting at their home where they a total of 55 couples participated. During the greetings, very affectionate by the way, there was a great number of handshakes. Since the four marriages did not necessarily know each other. Then, Alberto, the host, decided to find out how many of them had greeted each other. To do this, once the greetings were over, Alberto asked each person, including his wife Marta, how many hands he had shaken. It is understood that no one shakes their own hand or that of their spouse; and that no one shakes another person's hand more than once. Much to Alberto's surprise, everyone has answered his question differently. How many hands did Marta shake?
p6. 123123 digits are ordered in a circular way. When reading the digits hourly from some point, a number with 123123 digits is obtained. Show that if this number is divisible by 2727, then it does not matter since at which point we begin to read, all the resulting 123123-digit numbers will still be divisible by 127127.