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Brand new. We distribute directly for the publisher. This is the seventh book of problems and solutions from the Mathematics Competitions (published by the MAA.)Contest Problem
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Book VIIchronicles 275 problems from the American Mathematics Contests (AMC 12 and AMC 10 for the years 1995 through 2000, including the 50th Anniversary AHSME issued in 1999. Twentythree additional problems with solutions are included. A Problem Index classifies the 275 problems in to the following subject areas: Algebra, Complex Numbers, Discrete Mathematics (including Counting Problems), Logic, and Discrete Probability, Geometry (including Three Dimensional Geometry), Number Theory (including Divisibility, Representation, and Modular Arithmetic), Statistics, and Trigonometry.For over 50 years many excellent exams have been prepared by individuals throughout our mathematical community in the hope that all secondary school students will have an opportunity to participate in these problem solving and enriching mathematics experiences. The American Mathematics Contests are intended for everyone from the average student at a typical school who enjoys mathematics to the very best student at the most special school.
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Table of Contents
Preface ix
46th AHSME, 1995
47th AHSME, 1996
48th AHSME, 1997
49th AHSME, 1998
50th AHSME, 1999
Sample AMC 10, 1999
51st AMC 12, 2000
1st AMC 10, 2000
50th Anniversary AHSME
46th AHSME solutions, 1995
47th AHSME solutions, 1996
48th AHSME solutions, 1997
49th AHSME solutions, 1998
50th AHSME solutions, 1999
Sample AMC 10 solutions, 1999
51st AMC 12 solutions, 2000
1st AMC 10 solutions, 2000
Additional Problems
Solutions to Additional Problems
Classification
About the Editor
Preface
Before 1992, the scoring of the exam was done locally, in some states by the teachermanagers themselves, and in other states by the volunteer state director. Beginning in 1992, all the scoring was done at the AMC office in Lincoln, Nebraska. Beginning in 1994, students were asked to indicate their sex on the answer form. The following table shows the degree of participation and average scores among females versus that for males. Year & Females & Mean & Males & Mean & Unspecified & Mean
1994 & 104,471 & 68.8 & 120,058 & 76.0 & 6,530 & 70.6
1995 & 115,567 & 72.3 & 133,523 & 78.5 & 6,877 & 73.7
1996 & 124,491 & 65.8 & 142,750 & 71.2 & 6,659 & 67.8
1997 & 120,649 & 63.8 & 140,359 & 69.8 & 7,944 & 65.5
1998 & 108,386 & 66.5 & 128,172 & 71.9 & 7,438 & 67.8
1999 & 105,705 & 66.1 & 126,992 & 71.1 & 8,200 & 67.5
2000(12) & 71,272 & 61.0 & 89,965 & 67.9 & 5671 & 64.3
2000(10) & 49,288 & 60.8 & 52,836 & 67.5 & 4870 & 63.6 Related Exams
Until the introduction of the AIME in 1983, the AHSME was used for several purposes. It was introduced in order to promote interest in problem solving and mathematics among high school students. It was also used to select participants in the United States of America Mathematical Olympiad (USAMO), the six question, nine hour exam given each May to honor and reward the top high school problem solvers in America. The USAMO was used to pick the sixstudent United States Mathematical Olympiad team for the International Mathematical Olympiad competition held each July. With the introduction of the AIME, which was given the primary role of selecting USAMO participants, the AHSME question writing committee began to focus on the primary objective: providing students with an enjoyable problemsolving adventure. The AHSME became accessible to a much larger body of students. Some 7th and 8th graders, encouraged by their successes on the AJHSME, began participating. Calculators
In 1994, calculators were allowed for the first time. At that time, the AMC established the policy that every problem had to have a solution without a calculator that was no harder than a calculator solution. In 1996, this rule was modified to read `every problem can be solved without the aid of a calculator'. Of course the availability of the graphing calculator, and now calculators with computer algebra systems (CAS) capabilities has changed the types of questions that can be asked. Allowing the calculator has had the effect of limiting the use of certain computational problems. Referring to the Special Fiftieth Anniversary AHSME, problems [195438], [19615], [196929], [197420], [197630], [198018], [198124], and [199214] would all have to be eliminated, either because of the graphing calculator's ``solve and graphing'' capabilities or because of the symbolic algebra capabilities of some recent calculators. But the AMC has felt, like NCTM, that students must learn when not to use the calculator as well as when and how to use it. Thus questions which become more difficult when the calculator is used indiscriminately are becoming increasingly popular with the committee. For example, consider [199921] below: how many solutions does $\cos(\log x)=0$ have on the interval $(0,1)$? Students whose first inclination is to construct the graph of the function will be led to the answer 2 since in each viewing window, the function appears to have just two intercepts. However, the composite function has infinitely many $x$intercepts. Scoring
The number of problems and the scoring system has changed over the history of the exam. In the first years of the AHSME, there were a total of 50 questions with point values of 1, 2, or 3. In 1960, the number of questions was reduced from 50 to 40 and, in 1967, was further reduced from 40 to 35. The exam was reduced to 30 questions in 1974. In 1978, the scoring system was changed to the formula $30 + 4{\rm R}  {\rm W}$, where $\rm R$ is the number of correct answers and $\rm W$ is the number of wrong answers. In 1985, the scoring formula was $30 + 4 {\rm R} {\rm W}$, for a 30problem contest. In 1986, the scoring formula changed to $5 {\rm R} +2 {\rm B}$, where $\rm B$ is the number of blanks, again for a 30question test. The formula and number of questions remained unchanged until the year 2000. Beginning with the 2000 exams, some important changes took place. In order to accommodate school systems, the number of questions was reduced from 30 to 25 and the time allowed was reduced from 90 minutes to 75 minutes. The committee sought to continue to make the first five problems straightforward and the last five very challenging. The intension was to remove one question from what had been 610, one from what had been 1115, and three from those in the range 1625. The committee was also concerned that a bad experience with the AHSME might discourage capable younger students. The solution was a new exam, the AMC 10, specifically designed for students in grades 10 and below. The decision to include only topics that younger students might have seen was made. No trigonometry, logarithms, or complex numbers would appear on the AMC~10. Also in the year 2000, the scoring formula changed to $6 {\rm R}+2 {\rm B}$ for the 25question test. In 2002, the formula changed again to $6 {\rm R} +2.5 {\rm B}$ for a 25question test. Students qualify for the AIME in much the same way as before with one exception. Because the exam was much harder in some years, the AMC decided to guarantee that at least $ 5\%$ of the AMC 12 participants qualify for the AIME. The `top $5\%$' rule was instituted in the year 2001. Since 2001, invitation to the AIME has been limited to students who `scored 100 or placed in the top $5\%$ among all scores on the AMC 12'. On the AMC 10 from 2000 to 2003, invitation was limited to the top $1\%$. In 2004 and 2005 the invitation to the AIME for AMC 10 participants was based upon a score of 120, or placement in the top $1\%$ of scores on the AMC 10.