[...]... draft is developed and the process of review and meeting is repeated Usually at this xviii CONTEST PROBLEM BOOKV point the exam is ready to go to press, but occasionally a difficulty surfaces after this point and further changes have to be made After a final proofreading review by one or two people associated with the AMC, but who have not previously been involved with producing this exam, the exam goes... case the very top) Second, the formats of the two exams are completely different Speed and shrewd intuition pay o f on f the AHSME (and they are valuable talents), but they are not what the USAMO is about Conversely, a student who is slower but has a fine sense of logic could do well on the USAMO but might never make it through the AHSME filter So the AMC members asked themselves: is it possible to devise... respectively on the f AHSME and AIME, or even triples of problems to appear on AHSME, AIME and USAMO If such linked problems appeared a regularly, that would be an incentive, once the AHSME w s over, for participants who were going on to the AIME to review the PREFACE xvii AHSME very carefully and try to prepare for generalizations of AHSME problems It proved too difficult to produce such linked problems... the exam each year, so even if the average score was the same in two consecutive years, that suggests only that the exams were equally hard Furthermore, the scoring changed in 1986, and the change in Honor Roll cutoff may not have compensated exactly As for beforethefact judgments, the AMC exams are not standardized tests, l i e the SATs The College Board uses some problems over again (so called “equators”),... integers 1 t rough 999 in order The 19831d digit to the right of the decimal point is (42 (B) 3 (C> 5 (D) 7 (El 8 17 The diagram to the right imaginary axis shows several numbers in the complex plane The circle is the unit circle centered at the origin One of these numbers is the reciprocal of F Which one? (4A (D) D (B) B (El E (C) c A F0 real ' axis 0 B A CONTEST PROBLEM BOOK V 4 18 Let f be a polynomial... which allows them to figure out how to set scores so they have the same meaning each year They can have equators because not all of their tests are made public; sometimes all the test problems are collected and not published later However, the AMC immediately makes all its problems and solutions publicly available - our main goal is to promote mathematical learning While we (meaning all the AMC test... leadiig applied math contest, the COMAP Modeling Contest, is a group contest where students have a whole weekend and must write an essay in which, among other things, they can explain and justify their interpretation of the problem Computation More generally, there has been a trend in mathematics education to downplay mechanical computation and emphasize concepts The AMC exams involve much less computation... work The best problems - fresh, intriguing, instructive, with multiple solution methods - are rare and cherished gems Problem posing, like problem solving, is a special talent But we are sure there are many wonderful problem posers out there, and the AMC would like to know you Thanks to All We are the compilers o this book because the first of us was f Chair of the AIME subcommittee during most of the. .. and other problem- solving activities in the early 19509 Therefore, following my emigration to the United States and the completion of my studies, it was o natural f r me to follow the tradition of Hungarian mathematicians and search for opportunities to make my own contributions to the competition scene I was first invited to join the USAMO Subcommittee in 1976,became a member of the AHSME Panel in the. .. about the quality of precollege mathematics education, many organizations that had not been directly involved at this level decided that it was important to become involved As one part of this change, the AMC received two new sponsors: The American Statistical Association in 1985, The American Mathematical h i a t i o n of Two-Year Colleges in 1986 Since 1988, further sponsors have come on board: The . there is plenty of room for improvement. However, the fundamental goal of the mathematics community in providing these exams is to pique interest in mathematics and develop talent. Problems. at the heart of mathematics, and experience greatly sharpens problem- solving skills. We believe that some of the problems on the AHSME and AIME are intriguing in themselves, and/or they. Problems 277 vi Preface This is the fifth book of problems from the American High School Mathematics Examination (AHSME), covering the six examinations from 1983-88. It is also the