MathDB

5

Part of 2008 Tournament Of Towns

Problems(7)

TT2008 Junior O-Level - P5

Source:

9/4/2010
On a straight track are several runners, each running at a di fferent constant speed. They start at one end of the track at the same time. When a runner reaches any end of the track, he immediately turns around and runs back with the same speed (then he reaches the other end and turns back again, and so on). Some time after the start, all runners meet at the same point. Prove that this will happen again.
geometryperimetercombinatorics proposedcombinatorics
TT2008 Junior A-Level - P5

Source:

9/4/2010
Let a1,a2,⋯ ,ana_1,a_2,\cdots,a_n be a sequence of positive numbers, so that a1+a2+⋯+an≤12a_1 + a_2 +\cdots + a_n \leq \frac 12. Prove that (1+a1)(1+a2)⋯(1+an)<2.(1 + a_1)(1 + a_2) \cdots (1 + a_n) < 2.
[hide="Remark"]Remark. I think this problem was posted before, but I can't find the link now.
inequalitiesinequalities proposed
TT2008 Senior O-Level - P5

Source:

9/4/2010
On the infinite chessboard several rectangular pieces are placed whose sides run along the grid lines. Each two have no squares in common, and each consists of an odd number of squares. Prove that these pieces can be painted in four colours such that two pieces painted in the same colour do not share any boundary points.
geometryrectangleanalytic geometryalgorithmcombinatorics unsolvedcombinatorics
2008 ToT Spring Junior O P5 good domines in a 3colored 10x10 board

Source:

3/7/2020
Each cell of a 10×1010 \times 10 board is painted red, blue or white, with exactly twenty of them red. No two adjacent cells are painted in the same colour. A domino consists of two adjacent cells, and it is said to be good if one cell is blue and the other is white. (a) Prove that it is always possible to cut out 3030 good dominoes from such a board. (b) Give an example of such a board from which it is possible to cut out 4040 good dominoes. (c) Give an example of such a board from which it is not possible to cut out more than 3030 good dominoes.
combinatoricsColoring
2008 ToT Spring Junior A P5 99 girls in a circle, each with a candy

Source:

3/7/2020
Standing in a circle are 9999 girls, each with a candy. In each move, each girl gives her candy to either neighbour. If a girl receives two candies in the same move, she eats one of them. What is the minimum number of moves after which only one candy remains?
combinatorics
2008 ToT Spring Senior O P5 no of permutations of rows and columns

Source:

3/7/2020
We may permute the rows and the columns of the table below. How may different tables can we generate?
1 2 3 4 5 6 7 7 1 2 3 4 5 6 6 7 1 2 3 4 5 5 6 7 1 2 3 4 4 5 6 7 1 2 3 3 4 5 6 7 1 2 2 3 4 5 6 7 1
tablecombinatorics
2008 ToT Spring Senior A P5 positive integers arranged in a row

Source:

3/7/2020
The positive integers are arranged in a row in some order, each occuring exactly once. Does there always exist an adjacent block of at least two numbers somewhere in this row such that the sum of the numbers in the block is a prime number?
combinatoricsprimeIntegers