Tournament of Towns 2007 problems and solutions junior A-level

International mathematics Tournament of Towns problems and solutions:
Fall 2007
Junior A-level problems:
1.Let ABCD be a rhombus.Let K be a point on the line CD, other than C or D, such that AB = BK.Let P be a point of intersection of BD with the perpendicular bisector of BC.Prove that A, K and P are collinear.
2. (a) Each of Peter and Basil thinks of 3 positive integers.For each pair of his numbers, Peter writes down the greatest common divisor of the 2 numbers.For each pair of his numbers, Basil writes down the least common multiple of the 2 numbers.If both Peter and Basil write down the same 3 numbers prove that these 3 numbers are equal to each other.
(b) Can the analogous result be proved if each of Peter and Basil thinks of 4 positive integers instead?
3.Michael is at the centre of a circle of radius 100 meters. Each minute he will announce the direction in which he will be moving.Catharine can leave it as is or change it to the opposite direction. Then Michael moves exactly 1 metre in direction determined by Catherine.Does Michael have a strategy which guarantees that he can get out of the circle, even though Catherine will try to stop him?
4.Two players take turns entering a symbol in an empty cell of a 1 x n chessboard where n is an integer greater that 1. Aaron always enters the symbol x and Betty always enters the symbol O.
Two identical symbols may not occupy adjacent cells.A player without a move loses a game.If Aaron goes first, which player has a winning strategy?
5.Attached to each of a number of objects is a tag which states the correct mass of the object.The tags have fallen of and have been replaced on the objects at random.We wish to determine if by fact all tags are in fact correct.We may use exactly once a horizontal lever which is supported at its middle. The objects can be hung from the lever at any point on either side of support. The lever either stays horizontal or tilts to one side. Is this task always possible?
6.The audience arranges n coins in a row.The sequence of heads and tails is chosen arbitrary. The audience also chooses a number between 1 and n inclusive. Then the assistant turns one of the coins over and magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by audience.
(a) Prove that if this is possible for some n then it is also possible for 2n.
(b) Determine all n for which this is possible.
7.For each letter in the English alphabet, William assigns an English word which contains that letter.His first document consists only of the word assigned to the letter A.In each subsequent document, he replaces each letter of the preceding document by its assigned word. The fortieth document begins with "`Till whatsoever start that guides my moving".Prove that this sentence reappears later in this document.
Solutions