Durham university problems2011
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1. The Distant Mathematics Challenge 2011 was sponsored by Durham University's Department of Mathematical Sciences to select students for their gifted and talented summer school. Students in years 7-13 were invited to submit solutions to 8 mathematical problems by March 24, 2011. 2. The problems involved topics like geometry, sequences, and divisibility. Students would receive points for justified solutions, and overall scores would be scaled according to problem difficulty. Results would be available in April 2011 on the department's website. 3. One problem asked how many pieces could be cut from a pentagonal cake after 99 straight cuts.
Durham university problems2011
- 1. Distant Mathematics Challenge 2011 sponsored by Department of Mathematical Sciences, Durham University Aim. The challenge is designed to help select students for the Gifted and Talented Summer School at Durham University in August 2011. For full details of the application process, please contact Mr Shane Collins (s.m.collins@dur.ac.uk). Our wider aim is to promote mathematical think- ing. Submissions from all students in year groups 7–13 are welcome. Submissions from the same school are expected to be sent in one pack to Dr Vitaliy Kurlin (Department of Mathematical Sciences, Durham Uni- versity, Durham DH1 3LE) by 5pm on Thursday 24th March 2011. If you have questions on the problems, please e-mail vitaliy.kurlin@durham.ac.uk. Please attach a letter to confirm full names and year groups of students. Marking. The credit is given for justified mathematical arguments. The full mark for each of eight problems is 10. Then marks for each problem will be rescaled according to the actual difficulty of the problem (the average mark over all submissions). The results of the challenge will be available in April 2011 at http://www.maths.dur.ac.uk/∼dma0vk/challenge.html. Problem 1: Attenborough and a bear. David Attenborough walked two miles from his camp in the eastward direction. Then he turned to the north and walked one mile. Suddenly, he spotted a bear and started to run to the south. After running one mile to the south, David reached the camp. What was the colour of the bear? Problem 2: the first spaceship. 50 years ago the first manned spaceship made a few turns (90 min each) around Earth. During the first half of each turn on the sunny side of Earth, the spaceship was getting 5 degrees warmer. During the second half of each turn, the spaceship was becoming 3 degrees cooler. How long into the flight will the original temperature increase by 35 degrees? 1
- 2. 2 Problem 3: share a cake with many friends. You are cutting a pentagonal cake along straight lines. What is the maximum number of pieces that you can get after 99 cuts? Problem 4: count a lot of sequences. An ordered sequence consists of only 1 or 2. The sum of all elements is 15. How many such sequences are there? Problem 5: a long way for a queen. Walking on a chessboard 8×8, a queen passed through the centre of each cell once. The path of the queen turned out to be closed and without self-intersections. How short and how long can this path be? All cells are unit squares. A queen can move in any direction: horizontally, vertically, or along a diagonal. Problem 6: a simple multiple of 2011. Find a number of the form 111 . . . 1 that is divisible by 2011. Problem 7: cockroaches and robots. Two tiny cockroaches were tied to each other by a thread of length 20 cm. Despite this constraint, the cockroaches managed to run from one corner A to the opposite corner B along quite arbitrary paths. Robotic cleaners W and E are round disks of radius 10cm. The robot W is moving from A to B along the path of the first cockroach. The robot E is moving from B to A along the path of the second cockroach. Prove that the robots will inevitably collide in all cases. The cockroaches and the robots can choose any changeable velocities. Problem 8: pick the right apples! There are 2011 ap- ples on several trees: fewer than 262 apples on each tree. Can you pick apples in such a way that (1) there are at least 262 apples left, and (2) all the trees that still have some apples have exactly the same number of apples?