MathDB

2002 Chile Classification NMO Juniors

Part of Chile Classification NMO Juniors

Subcontests

(1)
1

2002 Chile Classification / Qualifying NMO Juniors XIV

p1. You know a character who always lies from Monday to Wednesday and always tells the truth from Thursday to Sunday. What day of the week could he have said "yesterday I lied" and "in two more days I shall tell the truth "? Justify your answer.
p2. Sebastian plays the following solitaire number with numbers: start with aa, bb and cc, then choose one of them, delete it and replace it with the sum of the other two remaining minus 1 1. For example, if we eliminate the number bb, is replaced by the number a+c−1 + c -1. After several moves we have the numbers 20012001, 20032003 and 20052005. Say if it is possible that Sebastian started with the triple (a,b,c)=(2,2,2)(a, b, c) = (2, 2, 2).
p3. The following figure represents a river of constant horizontal width, and two houses A A and B B located on opposite sides of the river. There is a bridge of width equal to the width of the river that must be placed horizontally and in such a way that the distance from AA to B B is minimal. Determine the location of the bridge. https://cdn.artofproblemsolving.com/attachments/f/a/53e2af5e8dd4df3522014dc66556a276d90475.jpg
p4. Consider all 66-digit natural numbers of the form abcpqr‾\overline{abcpqr}, where the digits pqr‾\overline{pqr} corresponds to a permutation of abc‾\overline{abc}. Let AA be the sum of all the numbers mentioned above. Prove that AA is divisible by 1313.
p5. In the constellation Ponce there is an odd number of planets. Observations made in the La Silla observatory determined that all of them are located at different distances in pairs. Also in, that on each planet only the planet that is closest to it is observed. Prove there is a planet in the constellation of Ponce that is not observed by any other planet of the constellation of Ponce.
p6. David and Claudio get together to play the rectangle game. The game consists of coloring integer coordinate points of the plane. David chooses a point and colors it blue, then Claudio chooses uncolored 20022002 points and colors them red. Next, David chooses an uncolored point and paints it blue, and so on. David wins if he manages to make a rectangle with his 44 blue vertices, and Claudio wins if he manages to prevent it. Prove that David has a way of winning the game.