Solving BAMO Problems


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1 Solving BAMO Problems Tom Dvis Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only the first section is specific to BAMO; the rest of the document concerns generlpurpose mthemticl problem solving techniques. 1 BAMO BAMO (the By Are Mthemticl Olympid) is n olympidstyle contest consisting of five problems to be solved in four hours. The term olympid style mens tht ech problem requires written solution, generlly in the form of mthemticl proof. The people who compose the exm try to rrnge the problems roughly in order of difficulty, so most people should be ble to solve problem 1, nd lmost nobody should be ble to solve problem 5. All problems re of equl vlue (7 points possible on ech), nd most of the scores on ny prticulr problem will be 0, 1, 6, or 7; middle scores re rre. Obviously you should try to work the problems in order; if you re hving trouble solving problem 2, it s unlikely tht you will mke much progress on problem 5. But of course you should look t ll the problems. For exmple, if you re extremely good t geometry nd the third problem is geometric, it my well be tht problem 3 is esier for you thn problem 2. Generlly, however, the rrngement of the problems firly ccurtely reflects their difficulty. If you do solve problem, rther thn beginning work immeditely on the next it is lmost certinly good ide to check over your solution nd mke certin it is clerly written, tht you didn t leve nything out, nd tht it in fct solves the problem you re trying to solve. It is shme to get problems 2 nd 3 correct nd to get zero on problem 1 since it seemed so esy tht you mde some silly mistke on it. Remember tht 4 hours is long time, nd it is often better to spend 5 extr minutes on problem 1 to mke sure tht it is perfect thn to spend tht 5 minutes mking zero progress on problem 5. 2 Writing Solutions Remember tht there will be humns grding your work. They re trying to be s fir s possible, but if your writing is difficult to red, or the solution is disorgnized or if the sentences re bdly written nd difficult to understnd, you will mke it hrd for the reder to understnd your solution, nd will thus be less likely to get high mrks. When you re finished with ech problem, tke look t it nd pretend tht you re the person trying to grde it. How would you like to grde it? Here re some ides for how to write proof or essy tht is esy to understnd: 1
2 1. First nd foremost, remember everything you lerned in your English writing clsses. Orgnize your thoughts, use complete sentences, et ceter. 2. Write n outline before you begin, where outline simply mens sentence or two explining how your proof works. For exmple, you might write something like this: The proof will be done by induction on, the number of sides of the polygon. We will show it is true for tringle ( ), the smllest polygon nd then we will induct on. For lrger thn 3, the proof will be divided into two cses, depending on whether is odd or even. Then write your proof in three prts, idelly with short heder in front of ech, like: Cse, Cse, odd, nd finlly, Cse, n even. 3. If you write something tht you lter decide you don t need or is incorrect, be sure to cross it out completely so tht the reder will understnd clerly tht it is not prt of your solution. 4. If your solution covers multiple pges, mke sure you number them in n obvious wy: Problem 3, pge 2, for exmple. 5. This ws sid bove, but it is so importnt tht it s worth repeting: when you finish writing solution, tke few minutes to rered wht you hve written to mke certin it sys wht you think it does. 3 How to Get Strted Mthemtics must be written into the mind, not red into it. No hed for mthemtics nerly lwys mens Will not use pencil. Arthur Lthm Bker Do not spend lot of time just string t blnk sheet of pper. Do something! Try to serch for pttern, drw picture, work out some simple cses, try to find simpler relted problem nd work tht, chnge the nottion, et ceter. Here re some ides of ctivities you cn usully do, even if you hve no ide t ll how to pproch the problem: 1. Serch for pttern: Imgine you re sked to find the sum of the first few odd numbers: Work out the vlues for smll vlues of k:! " All the nswers re perfect squres! With clue like this, it will probbly be much esier to find out why. 2
3 0 = = 2. Drw picture: For geometry problem this should be obvious, but you cn often drw pictures for other problems s well. For exmple, suppose you wnt to show tht #$%$ & & & $ ')( '+* ',$ #  %/. Here is picture tht might help: The sum is like tringle, so is obviously relted to the re of tht tringle. 3. Check some simple cses: In the exmple bove, check the first few vlues '2(43 ' of. When you check vlues, be sure to try the esiest ones first. In other words, don t check until you ve checked ' ( 5 '( # '( % ' (26,,, nd. Remember to try zero. If you re supposed to show something bout generl tringle, try it on few tringles tht you cn clculte with esily, such s n equilterl tringle or right tringle. 4. Solve simpler relted problem: For exmple, if the problem sks bout the rrngement of queens on chessbord, try to solve the problem with bords tht re smller thn n 9;: 9 chessbord: look t the # : # bord, the % : % bord, nd so on. 5. Chnge the nottion: If your problem involves, for exmple, binomil coefficients, replce them by the fctoril equivlents: < '= > ( '+??* ')7 6. Think bout similr problems: If this problem reminds you of one you hve solved previously, how did you solve tht one? ? 4 Generl Techniques You cn usully pply the techniques in the previous section even if you hve no ide how to strt. The techniques listed here re more specific, but it s worth keeping them ll in mind when you pproch new problem. Remember tht sometimes there re mny techniques tht will work; to get top score, ll you need to do is to find one of them. 1. Divide into Cses. If you know how to solve the problem under certin conditions, perhps you cn divide it into cses. Also, be certin to be sure tht you hve proved it in ll cses; for exmple, if the problem concerns two prllel lines, be sure tht your proof works if the lines re prllel. If it doesn t, you my hve to prove tht s specil cse. 3
4 2. Look for Symmetry. Symmetry cn be geometric or lgebric. For exmple, if you hve to multiply out the A;B)CB D E F, nd fter some struggle, you find tht the coefficient ofahg C G D is 30, then so will be the coefficient ofaig C D G nd ofa C G D G since the originl expression ws symmetric ina,c, nd D. 3. Use Induction. If you cn ssign n integer size to ech version of problem nd it looks like the problem for lrger size cn be solved in terms of similr problems of smller size, perhps induction will work. Induction does not hve to be used on lgebric problems. As n exmple, suppose you wnt to show tht ny polygon (convex or not) cn be cut into tringles using digonls tht lie within the polygon. Surely the smllest polygon ( tringle) cn be so divided. If you cn then show tht ny polygon cn be split into two smller polygons with digonl, you cn use induction to prove the desired result. The Towers of Hnoi problem is nother good exmple. 4. Work Bckwrds. Lots of gmes work this wy. For exmple, suppose you ply gme where you begin with pile of 50 sticks, nd move consists of tking 1, 2, 3, or 4 sticks from the pile. You lternte moves with your opponent, nd the first person unble to mke move loses. If you move first, do you hve strtegy tht will gurntee win? 50 is pretty big number, but work bckwrds. Who wins if the strting pile hs zero sticks? 1 stick? 2 sticks? Work bckwrds to see which positions re sfe to leve n opponent. Zero sticks is clerly sfe, nd piles with 1, 2, 3, or 4 sticks re unsfe. 5 is sfe becuse ny move your opponent mkes leves him in n unsfe position, nd so on. 5. Consider Prity. Sometimes problems hve n oddeven condition. Given polygon with 101 sides tht hs n xis of symmetry, show tht the xis psses through vertex. This is esy if you pir ech vertex with the symmetric vertex cross the xis. 6. Use the Pigeon Hole Principle. If you plce more thn J things into J boxes, t lest one box will hve more thn one thing in it. In group of 13 or more students, t lest two will hve birthdy in the sme month. 7. Use Proof by Contrdiction. If you cn t prove something, ssume it is flse nd see wht you cn conclude from tht. If you cn conclude something tht is obviously flse beginning with tht ssumption, then your ssumption must be wrong nd therefore the originl sttement is true. Prove tht there is no lrgest prime number. Assume there is lrgest, sykml, wherek+n,k G,...,KOL is the list of ll the primes. Then multiply them ll together nd dd 1:PRQ K+N K G+S S S KOL"BT.P cn t be multiple of ny of thekmu, since if you dividep bykou it leves reminder of 1. So eitherp is prime or it is the product of primes not in the list. In either cse the originl ssumption tht there were only finite number of primes leds to nonsensicl result, so there must be n infinite number of primes. 8. Look for Invrints. Sometimes these is property of your problem tht is preserved no mtter wht opertions re performed. Here s good exmple. Suppose you begin with chocolte br tht is 8 squres by 5 squres nd ply the following gme. If it is your turn to move, you select piece (t the beginning, of course, you hve only the originl piece), nd you brek it long one of the lines between the squres. For exmple, the first move might be to brek the br into V1WYX nd XWYX piece. If you cn t brek piece, you lose. 4
5 Here s the invrint to consider: fter ech move, there is one more piece, nd the gme ends when there re 40. Thus, no mtter wht the moves re, the gme is over in exctly 39 moves, so it is not relly gme t ll. 9. Fctor Into Primes. Mny problems bout divisibility cn be solved by reclling tht every integer hs unique fctoriztion into prime numbers. Show tht between ny pir of twin primes except 3 nd 5, the number between them is multiple of 6. (Twin primes re two prime numbers tht differ by 2.) Any set of three successive numbers includes one tht is multiple of three. Since, (except in the cse of 3 nd 5) neither prime cn be multiple of three, the number between them must be. Every pir of twin primes consists of two odd numbers so the number between is multiple of 2. Any number tht is multiple of both 2 nd 3 is multiple of 6. 5 Smple Problems Here is list of smple problems shmelessly copied from vrious contests. These problems re not for solution; insted, for ech one think of s mny pproches s you cn tht might work to solve it, nd think of pictures or digrms you might drw. 1. The yer Z [ \ [;]2[1^ Z _1^ Z `. Compute the next greter yer tht cn be written s the product of three positive integers in rithmetic progression, given tht the sum of those integers is Compute the vlue of: Z [ [ b c de Z b b b c de [ [ b c d Z [ [ b c Z b b b c [ [ b cgf 3. Ifh1ij]2k,j+ik] l,kily]2h, nd b is positive integer, compute the gretest possible vlue for h1ij+ik+il. 4. A chord of constnt length slides round in semicircle. The midpoint of the chord nd the projections of its ends upon the bse form the vertices of tringle. Prove tht the tringle is isosceles nd ll possible such tringles re similr. 5. In how mny wys cn 10 be expressed s sum of 5 nonnegtive integers when order is tken into ccount? In other words, bi_imibin is different from bibi_imin. 6. There re 100 soldiers in detchment, nd every evening three of them re on duty. Cn it hppen tht fter certin period of time ech soldier hs shred duty with every other soldier exctly once? 7. The prime numberso ndp nd the nturl numberq stisfy the following eqution: Find the numbers. Z o i Z p i Z oip ] Z q f 8. There re 7 glsses on tble ll stnding upside down. One move consists of turning over ny 4 of them. Is it possible to rech sitution where ll the glsses re right side up? 9. Prove tht if two qudrilterls hve the sme midpoints for ll of their sides, then their res re equl. 5
6 { x ~ 10. For wht vlues ofr does the system of equtions:shtuv t w s,y t{ v t u r z 11. Show tht: } ~ { { } ~ { { } u } ~ ~ ~ y1 hve exctly zero, one, two, three, nd four solutions, respectively? { { ƒ 12. Using strightedge nd compss, construct trpeziod given the lengths of ll of its sides. 13. On every squre of bord is written either 1 or. For ech rowˆ, let Š be the product of Œ the Ž Ž numbers in tht row. Similrly, let Š be the product of the numbers in column ˆ. Show tht Š x Š Š z is never equl to zero. 14. The sequence u r I I is defined s follows:r is positive rtionl number smller thn, nd ifr I š for some reltively prime integers ndš, then u t {œ r š ƒ Show thtr, for ll. 15. Mr. nd Mrs. Adms recently ttended prty t which there were three other couples. Vrious hndshkes took plce. No one shook hnds with his/her own spouse, no one shook hnds with the sme person twice, nd of course, no one shook his/her own hnd. After ll the hndshking ws finished, Mr. Adms sked ech person, including his wife, how mny hnds he or she hd shken. To his surprise, ech gve different nswer. How mny hnds did Mrs. Adms shke? 6 Bibliogrphy Here is short list of books on mthemticl problem solving strtegies. 1. Arthur Engel. ProblemSolving Strtegies. Springer, New York, Dmitri Fomin, Sergey Genkin, Ili Itenberg. Mthemticl Circles (Russin Experience). Americn Mthemticl Society, Providence, Loren C. Lrson. ProblemSolving Through Problems. SpringerVerlg, New York, George Póly. How to Solve It. Doubledy, second edition, Pul Zeitz. The Art nd Crft of Problem Solving. John Wiley & Sons, Inc., New York,
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