Tournament of Towns 2007 problems and solutions junior O-level

International mathematics Tournament of Towns problems and solutions:
Fall 2007
Junior O-level problems:
1.Black and white checkers are placed on an 8 x 8 chessboard, with at the most one checker on each cell.What is the maximum level of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?
2.Initially , the number 1 and an non-integral number x are written on a blackboard.In each step we can choose 2 numbers on a blackboard, not necessarily different, and write their sum or their difference on the blackboard.We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard.Is it possible to write x^2 on a blackboard in a finite number of moves?
3.D is the midpoint on the side BC of triangle ABC.E and F are points on CA and AB respectively, such that BE is perpendicular to CA and CF is perpendicular to AB.If DEF is an equilateral triangle, does it follow that ABC is also equilateral?
4.Each cell of a 29x29 table contains one of the integers 1,2,3...29 and each of these integers appears 29 times.The sum of all the numbers above the main diagonal is equal to 3 times the sum of all the numbers below this diagonal.Determine the number in central cell of the table.
5.The audience chooses 2 of 5 cards numbered from 1 to 5 respectively.The assistant of a magician chooses 2 of remaining 3 cards, and asks the member of audience to take them to magician, who is in another room.The 2 cards are presented to magician in arbitrary order.By an arrangement with the assistant beforehand, the magician is able to deduce witch 2 cards the audience has chosen only from the 2 cards he receives.Explain how this may be done.

Solutions