IMC training 2016 (4)

Problem 1. The number 16 is placed in the top left corner square of a  table. The remaining 15 squares are to be filled in using exactly once each of the number 1,2,…,15, so that the sum of the four number in each row, each column and each diagonal is the same. Find the maximum value of the sum of the six numbers in the shaded squares shown in the diagram below.


Problem 2. All but one of the numbers from 1 to 21 are put into the squares of a  4\times 5 table, one number in each square, such that the sum of all the numbers in each row is constant, and the sum of all the numbers in each column is also constant. Find the number which is left out.

Problem 3. The diagram below show ten circles in a triangular array. Place each of the numbers 0 to 9 in a different circles so that for each of the six right-side up triangles marked with plus signs, the sum of the numbers in the three circles at its vertices is the same.

tgProblem 4. Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers a,b,c,d are replaced by a-b,b-c,c-d,d-a.  Is it possible after 2016 such to have numbers a,b,c,d such the numbers |bc-ad|, |ca-bd|, |ab-cd|  are primes?

Problem 5. Assume an  8\times 8 chessboard with the usual coloring. You may repaint all squares

1 – Of a row or column;

2 – Of a 2\times 2 square.

The goal is to attain just one black square. Can you reach the goal?

Problem 6. A rectangular floor is covered by  2\times 2 and 1\times 4  tiles. One tile got smashed. There is a tile of the other kind available. Show that the floor cannot be covered by rearranging the tiles.

Problem 7. A beetle sits on each square of a  9\times 9 chessboard. At a signal each beetle crawls diagonally onto a neighboring square. Then it may happen that several beetles will sit on some squares and none on others. Find the minimal possible number of free squares.

Problem 8. 10\times 10 chessboard cannot be covered by 25 T-tetrominoes.

Problem 9. On an infinite chessboard, a solitaire game is played as follows: at the start, we have  n^2 pieces occupying a square of side n. The only allowed move is to jump over an occupied square to an unoccupied one, and the piece which has been jumped over is removed. For which n can the game end with only one piece remaining on the board?

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