MathDB

2001 Chile Junior Math Olympiad

Part of Chile Junior Math Olympiad

Subcontests

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2001 Chile NMO Juniors XIII

p1. An archipelago is made up of two groups of islands: those of the north and those of the south. The locals have built several bridges for us, and each of them connects a northern island with a southern island. It is known that from any island it is possible to reach any other, crossing one or more bridges, and that the number of bridges that reach each island is a multiple of 33. Juan claims that if the bridge fell "Las Perdices" then it would be impossible to go from the Big Island to the Rocky Island. It is possible that Juan is right?
p2. Two natural numbers are called coprime, if their only common divisor is 1 1 (for example 66 and 2525). Show that if 1515 naturals less than 20012001 are chosen, such that any pair of they are coprimes, so not all chosen naturals can be composite numbers.

p3. On the "Matimpiapolis" track (which is circular and one kilometer long) you will run at the Olympic Grand Prix between the cars led by Jorge and Jacinto. At the starting point there is a traffic light (two colors) that remains green for 30 seconds and red for 30 seconds, alternately. Cars must respect the following rules:
∙\bullet They stop in front of the red traffic light. ∙\bullet They maintain a constant speed and can only change it after they have been stopped in the traffic light in red. ∙\bullet Jacinto only drives at 6060 km / h, while Jorge's speed is always greater than 6060 km / h, but less than 360360 km / h.
Show that for every pair of naturals nn and mm such that n≤m≤3nn\le m \le 3n, Jorge can choose their velocities so right after nn minutes of racing, he has passed Jacinto m times. For this, suppose both cars start from the starting point with the traffic light cycle starting in green. If a car reaches the traffic light exactly the instant it changes color, it continues without stopping, and the cars pick up speed instantly.
p4. Determine an integer that is a multiple of 19991999 and whose representation in system decimal has last four digits 20012001.
[url=https://artofproblemsolving.com/community/c4h1846805p12438213]p5. The segment AB AB measures 99 cm. and the point C C lies on it, such that ACCB=2 \dfrac {AC} {CB} = 2 . The point D D is such that ∠ACD=60o \angle ACD = 60^o and ∠ABD=45o \angle ABD = 45^o. Determine the measures of the angles of the triangles △ACD,△CBD \vartriangle ACD, \vartriangle CBD and △ABD \vartriangle ABD .
[url=https://artofproblemsolving.com/community/c4h2917537p26060887]p6. Given nn points in the plane, not all collinear, prove that there exists a polygon of nn sides, which does not cut itself and which has as its vertices the given points.
p7. Determine 55 consecutive integers, less than 200200, such that the sum of them and the sum of their squares, both are capicuas numbers, that is, written in the decimal system, they are read the same from left to right as from right to left (for example, the number 123454321123454321 is capicuas, but not number 990990).