Distant Mathematics Challenge 2010 Aim. The challenge was designed to help select students who will be invited to the Durham Gifted and Talented Summer School in August 2010. For full details of the application process, please, contact Mr Shane Collins (firstname.lastname@example.org). Our wider mission is to promote mathematical thinking, so submissions from everyone in years 9–13 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 Friday 28th May 2010. Please, attach an oﬃcial letter conﬁrming names and year groups of students. Winners in each year group will receive diplomas and prizes by post. Marking. The credit will be given for fully justiﬁed mathematical argu- ments. The initial full mark for each problem is 10. After analysing the raw data, each problem will receive a weight from 1 to 10. Then marks for each problem will be automatically rescaled according to its actual diﬃculty, i.e. the average mark over all submissions. Results will be at http://www.maths.dur.ac.uk/∼dma0vk/challenge.html in June 2010. Problem 1 (Happy numbers). Happy New A number is called a happy number if it can be rep- Year 2010! resented in the form ab + ba for some positive integers a, b, e.g. 57 = 52 + 25 . Is 2010 a happy number? Problem 2 (From a triangle to a rectangle). Show how to cut (along straight segments) any tri- angle into 3 pieces and make a rectangle from them. Problem 3 (Winning strategy). Two players write down n digits from 1 to 5 in turns, i.e. the 1st player chooses a 1st digit, the 2nd player chooses a 2nd digit etc. Can the second player guarantee that the resulting n-digit number is divisible by 9 if n = 2010 or n = 2012? 1 Problem 4 (Maximum area). Each side of a triangle is not greater than 2. √ Prove that its area is not greater than 3. Problem 5 (Integer-valued polynomial). Prove that a quadratic polynomial ax2 + bx + c takes integer values for all integers x if and only if 2a, a + b and c are integers. Problem 6 (Minimum distance). Five arbitrary points are plotted in the interior of the equi- lateral triangle with sides of length 2. Prove that among the plotted points one can always choose two points such that the distance between them is strictly less than 1. Problem 7 (Functional equation). Find all functions f (x) satisfying the equation 2f (1 − x) + 1 = xf (x) for any real x. Problem 8 (Integer vertices). Does there exist an equilateral triangle such that all its vertices have both coordinates integer? Problem 9 (Equation with a parameter). Find all solutions of the equation f (x) = f (a), (x2 − x + 1)3 where f (x) = and a > 2. x2 (x − 1)2 Problem 10 (Monotonous digits). The 10 digits 0, 1, 2, . . . , 9 are written in an arbitrary order. Prove that one can always remove 6 digits such that the remaining 4 digits are ordered monotonically, i.e. they are either increasing or decreasing.