A rectangle PQRS with $PQ=45$ and $PS=100$ is cut into 4500 squares of side 1. T is a point on QR such that $QT=60$. Of these 4500 squares, how many are cut by lines PT and TS? (A square does not cut by line if the line only passing through its vertices.)
In a mathematical competition consisting of 25 problems, 8 marks are given for each correct response, 0 marks for each incorrect response and each no response is awarded 3 marks. Tom scored 121 marks in this competition. What is the smallest number of incorrect responses he could have?
The integers 1, 2, 3, …, 2005 are written on the board. What is the smallest number of these integers that can be wiped off so that the product of the remaining integers ends in 8?
In a soccer tournament eight teams play each other once, with two points awarded for a win, one point for a draw and zero for a loss. How many points must a team score to that it is in the top three (i.e. has more points than at least five other teams)?
Two players take it in turns to choose from 25 numbered counters, each labeled with a different odd number from 1 to 49. When one player chooses a counter labeled X, the next player must choose the counter whose label is the greatest odd factor of $99-X$. No matter what the first counter be taken, what is the least number of counters will remain when the game ends?
