Junior Mathematical Olympiad 2010The Junior Mathematical Olympiad (JMO) has long aimed to help introduce able students to (and to encourage in) the art of problem-solving and proof The problems are the product of the imaginations of a small number of volunteers !riting for the JMO problems group "fter the JMO# model solutions for each problem are published in the solutions boo$let and the %&MT 'earboo$(o!ever# these neatly printed solutions usually represent only one approach (often !here several are feasible) and convey none of the sense of investigation# rough-!or$ing and false-starts that usually precede the finished article )n mar$ing JMO scripts# mar$ers !ant to encourage complete (and concise) proofs in section * but# in doing so# are content to accept less than perfectly !ritten ans!ers# as long as it is clear !here the proof is going + !e appreciate that there is a limited time available# and that to polish and to present ta$es the place of considering another problem ,evertheless# clarity and insight are !hat are loo$ed for-e have collected belo! a small number of solutions submitted for the JMO in 2010# !ith the aim that future candidates can see !hat some 'ear . students (and younger ones) do achieve and that they might aspire to emulate it )n many !ays they are ordinary solutions# not brilliantly and startlingly original + but mathematically they are to be commended# in particular# for the logical progression from one pointto the ne/t# and for clear presentation)t is not easy to generalise about !hat ma$es a good solution or ho! a candidate can achieve success in the JMO# but there are a fe! points to ponder on0 using trial and improvement to a large degree is generally considered not very mathematical + it !ill lend little insight into the structure of the problem# and even if you get an ans!er that!or$s# there should al!ays be a concern that it is not the only ans!er1 if you do have to resort to calculating your !ay through a large number of cases# then the calculations should be sho!n as part the solution + the reader should not have to ma$e suppositions about !hat you have tried (or not tried)1 diagrams should be large enough to contain subse2uent details + ho!ever# dra!ing 3ust one diagram can convey very little of the order of the proof# especially in geometrical 2uestions1 avoid long sentences# particularly those !hich fre2uently use if and could be the use of algebra ma$es it possible to e/press connections in simple !ays# !here a huge number of !ords !ould be other!ise necessary1 if algebra is used# variables should be fully declared + for e/ample# in *4 belo!# saying merely 56et Jac$ be x7 led to much consternation for both candidates and mar$ers over !hether speed or time !as being referred to1 attempt as many 2uestions as you can do really !ell in + it is better# in terms not only of scoring mar$s but also of honest satisfaction# to spend your time concentrating on a fe! 2uestions and providing full# clear and accurate solutions# rather than to have a go at everything you can# and to achieve not very much in any of them Of the 1000 students or so !ho ta$e part in the JMO# generally only a handful achieve full mar$s in all si/ section * 2uestions )f this all sounds rather negative# it has# in balance# to be said that JMO candidates produce an astonishing amount of !or$ that is !orthy of high praise -e hope that the !or$ belo! !ill give future candidates a flavour of !hat is to be encouraged# in style# method and presentation*1 )n a se2uence of si/ numbers# every term after the second term is the sum of the previous t!o terms "lso# the last term is four times the first term# and the sum of all si/ terms is 14-hat is the first term8*2 The eight-digit number 5ppppqqqq7# !here p and q are digits# is a multiple of 9: -hat are the possible values of p8*4 Jac$ and Jill !ent up a hill They started at the same time# but Jac$ arrived at the top one-and-a-half hours before Jill On the !ay do!n# Jill calculated that# if she had !al$ed :0; faster and Jac$ had !al$ed :0; slo!er# then they !ould have arrived at the top of the hill at the same time (o! long did Jill actually ta$e to !al$ up to the top of the hill8 *9 The solution to each clue of this crossnumber is a t!o-digit number# not beginning !ith