MathDB

2012 Chile Junior Math Olympiad

Part of Chile Junior Math Olympiad

Subcontests

(1)
1

2012 Chile NMO Juniors XXIV

p1. Find a natural number NN such that the sum of the digits of NN is 100100 and the sum of the digits of 2N2N is 110110.
[url=https://artofproblemsolving.com/community/c1068820h2935480p26269350]p2. What is the minimum number of movements that a horse must carry out on chess, on an 8×88\times 8 board, to reach the upper right square starting at the lower left? Remember that the horse moves in the usual LL-shaped manner.
[url=https://artofproblemsolving.com/community/c6h2927799p26188318]p3. On a flat and unlimited world, a walker makes a journey obeying the following rules: ∙\bullet He walks only on line segments. ∙\bullet These segments alternate between two types of segments: segments in direction North and segments in any direction other than South direction. That is, it begins traveling a segment in a North direction, then make one in a direction other than the South direction, then another segment in the North direction, and so on alternate. Prove that if the walker manages to return to the starting point then there necessarily must be another point on the plane through which the path passes more than once.
p4. There are three clocks on a wall, all showing twelve o'clock. The first one is delayed 22 minutes per day, the second one is delayed 5 minutes per day, while the third is in custody. If each clock advances at a constant speed, calculate how long it will take before the three clocks return to the same time hour. [hide=original wording]En una pared se encuentran tres relojes, todos marcando las doce en punto. El primero de ellos se retrasa 2 minutos por dıa, el segundo se retrasa 5 minutos por dıa, mientras que el tercero se encuentra detenido. Si cada reloj avanza a velocidad constante, calcule cuanto tiempo transcurrira antes de que los tres relojes vuelvan a marcar la misma hora.
PS. Problem 2 was also proposed as Seniors P1.