Junior MathBattle
9:17 PM
Muhammad Yusuf
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1. Could the altitude, bisector and median drawn from different vertices of a triangle create an equilateral triangle at intersection?
2. There are 300 trees in a park. If one marks any 201 trees, then there are always be an oak, a pine and a maple tree among the marked. How many oaks, pines and maple tree could be in the park (describe all possibilities)?
3. Basil places black, red and white rooks on a chessboard (the same number of each color) so that the rooks of different color do not attack each other. Find the maximal number of rooks that can be placed on the chessboard.
4. Each employee of the firm “Horns and Hoofs” is either a Lair or Truth Teller. They all are well acquainted with each other. Once, while they were sitting at a round table, every employee was asked two questions. On the first question ”‘How many Truth Tellers among your neighbors,” only one person answered ”one”, all the others said ”zero”. On the second question if there are Lairs among his/her neighbors all answered ”yes”. Given this information, is it possible to distinguish the Liars from Truth Tellers?
5. Alex thinks of a number (from 1 to 9), and Boris tries to guess it. If he is mistaken, Alex changes his number: he tries to divide his number by Boris’ number; however, if it does not divide evenly, he doubles it. Is there some strategy for Boris to guess Alex’ number?
6. 300 oranges are distributed into 10 boxes, so that no two boxes contain the same number of fruits. A packer picks up 9 oranges from the box with the largest number of fruits and distributes them into the other boxes, one orange to each box. The packer wants to get the same number of fruits at least in two boxes. Is there a guarantee that he can ever complete the task?
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