Only answers properly marked on the answer form will be graded. 6th AMC 10 B. 2. 1. A scout troop buys candy bars at a price of. The best way to prepare for the AMC 10 is to do lots of practice AMC 10B Problems and Answers · AMC 10A Problems and. The AMC 10 and AMC 12 are both question, minute, multiple choice examinations in high school mathematics . AMC Question & Solutions .

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Search the history of over billion web pages on the Internet. Guy, and Loren C. Kuczma Mathematical Olympiads Klosinski, and Loren C. Problems, Xmc tions, and Commentary, Kiran Sollutions. Box 91 Washington, DC 12 fax: Originally given only to stu- dents in the New York City area, the contests were first offered nationally in under the sponsorship of the MAA and the Society of Actuaries.

The pri- mary objective of the contests was, and still is, to promote the study of mathe- matics by providing high school students with a positive experience in creative problem solving.

A secondary objective of identifying mathematically talented students was introduced inwhen about top scorers were invited to participate in the Amv Mathematical Olympiad USAMOan extremely chal- lenging proof-oriented contest.

Two years later a contest for middle school students was created. The op- posing argument was that a lOth-grader whose previously undiscovered talent is first revealed by a high AMC 10 score should not be denied the opportunity to take the AIME. Since two versions of both the AMC 10 and the AMC 12 have been created every year and offered about two weeks apart.

This has been done for two reasons. The immediate impetus for change was a conflict between the contest date and state holidays in Illinois and Louisiana, which would have precluded participation by students in both states. The new arrangement also addressed a growing concern among CAMC members about contest se- curity. In the past, contests were administered for official credit at any time during a window of a few days. With the advent of contest-focused web sites, it was recognized that contest answers might become public knowledge sooutions hours.

Setting a specific date for each contest greatly reduces the probability that a student will learn the answers before taking the contest. The AMC contests have attracted many new sponsors since the early years. In addition to the MAA and the Society of Actuaries, the list of sponsoring organizations grew to include the high school and two-year college honorary mathematics society Mu Alpha Theta, the National Council of Teachers of Mathematics, and the Casualty Actuarial Society by There are currently more than 21 Contributors and Sponsors of the contests, which demonstrates the wide range of interest in the program.

35 Sets of Previous Official AMC 10 Tests with Answer Keys (PDF files)

The time allotted for work was initially 80 minutes, increasing to 90 minutes in and decreasing to 75 minutes in Although the highest possible score on 10v XI the contests has always beenthere have been numerous changes in the number of questions and the scoring formulas used over the years. The first contests had 50 questions, divided into three sections with increasing levels of difficulty. Beginning inin an effort to discourage random guessing while still encouraging educated guessing, a penalty for incorrect responses was imposed.

Beginning in the contest was no longer partitioned into sections. The contest consisted of 30 questions each worth 5 points, with the scoring formula 5 R — W, where R and W represent the number of questions with correct and incorrect responses.

The latter formula is neutral with respect amf guessing and precludes the possibil- ity of a negative score, a potentially devastating result for an unlucky student! The last change, instituted in part to reduce the probability of ties among award recipients, appears to have discour- aged students from attempting the more difficult problems. This seemingly simple guideline leaves signifi- cant room for debate.


The list of topics soolutions acceptable for the contest has also changed with time. Probability did not make its first appearance on the contest untiland trigonometry not until Although many AMC 12 contestants have stud- ied calculus, the CAMC does not consider it to be a standard high school topic at this time.

That is less the case today, for two reasons. First, in the process of reducing the number of questions on the contest from the original 50 to the present 25, it was desired to maintain a large number of problems whose solutions require creative insight. Consequently, the number of routine problems on each contest has been greatly reduced. Thus, for example, a prob- lem that asks students only to solve an algebraic equation is not appropriate for a calculator-permitted AMC 10 or AMC Many good problems involv- ing trigonometry and logarithms are also rendered trivial by calculators.

The problems included in this volume were all on calculator-permitted contests, but beginning in calculators will not be permitted on either the AMC 10 or the AMC 12 contests.

By the late s it was clear that many students in grades 9 and 10 were unable to find many approachable problems on the AHSME. This situation led directly to the creation of the AMC The scope of that contest is restricted to topics that are typically studied in grades 10 and below.

The majority of the problems on the AMC 10 contests involve topics from algebra, geometry, probability, and elementary counting and number theory. Specifically excluded from AMC 10 contests are problems that involve trigonometry, logarithms, complex numbers, and the concept of functions and properties of their graphs.

Problems that cover more advanced topics in geometry are also excluded. The two forms of the AMC 10 con- test, referred to as the 10A and 10B, are given about two weeks apart. The two forms must be comparable in terms of difficulty and topical balance, yet not so similar as to give an advantage to students taking the later 10B.

Preface xiii This arrangement permits a question to appear on both contests given on the same day. Because of those considerations, the content of each contest is inextricably linked to the content of the other three, so that all the AMC 10 and 12 contests must be constructed together. For the contests given in year n, the process begins early in year n — 2, when a call goes out to a panel of about 50 problem posers.

By early summer a packet of to problems is sent out to a panel of packet reviewers, many of whom are also problem posers.

Each reviewer submits a ballot of fa- vorite problems from the packet and a set of comments, including corrections, alternate solutions, and suggestions for improved wording. A tentative draft of each contest is written in the fall and sent out to a panel of draft reviewers for editing. It is common for many of the solufions and solutions to be significantly rewrit- ten at the meeting.

In addition, the Subcommittee often reorders problems, shifts them from one contest to another, or replaces them entirely.

AMC 10B Problems_百度文库

Following the meeting, another draft of each contest is written. Over the next few months that draft undergoes additional reviews, and the contest galleys are printed at the AMC office and sent out for proofreading. solutiobs

The contests are printed during the summer of year n — 1. Performance statistics over the last several years indicate that we have been reasonably successful in judging the relative difficulty of the A and B forms of each contest. Average scores 20005 the two forms have usually been within a few points of each other. We have not always been as successful in judging the relative difficulty of individual problems. As an extreme example. Problem 22 on the AMC 10A had a higher success rate In spite of our best efforts, ambiguities and errors occasionally creep into the final product.

Each problem that ultimately appears on one of the contests must meet several criteria. The easier problems are often some- what similar to textbook exercises.


However, if a problem or a nearly iden- tical one has been used in another contest or published elsewhere, it is rejected. The higher- numbered problems should be difficult by virtue of the insight required for the solution, not the length of a,c computations.

The distracters should trap students who think carelessly, but not those who simply read carelessly. We occasionally fail to anticipate the mistakes the students will make, however.

2005 AMC 10B problems and solutions

For example, a number of top students missed Problem 3 on the AMC 10A Problem 1 on AMC 12A contest because they read the problem hastily, expressed their answers in dollars instead of cents, and found the answer they obtained as one of the choices. Solutions by programming raise unique issues. For example, if a problem requires the th term of a sequence, has a stu- dent gained a significant advantage by programming a calculator to generate successive terms of the sequence?

Here the answer depends on the nature of Preface xv the problem. In some cases it has been decided that the creativity required to write the program is comparable to that required to solve the problem without a calculator. Acknowledgments This book consists of a collection of problems submitted by a large group of people. Quite often a submitted problem will be edited so that it might actually be unrecognizable to the original author, but it is the germ of the idea for the final problem that is so valuable for those creating the final contests.

We have a complete record of the source of the problems sinceand are able to determine the originator of most of those on the earlier contests, but we apol- ogize in advance for any names that have been inadvertently omitted. Those who have submitted the problems that make these contests possible include: We would like to add a special thanks to Harold Reiter, who gave us per- mission to include the first AMC 10 contest, which he directed, in this volume, and to Steve Dunbar and the staff at AMC Headquarters for making life eas- ier for us in numerous ways.

Finally, we greatly appreciate the work of Carrie Davis, a student assistant at Youngstown State University, for doing much of the editing work on the manuscript. At the end of the second day, 32 remained. How many jellybeans were in the jar originally? Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time.

What is the fixed monthly fee? As P moves along a line that is parallel to side AB, how many of the four quantities listed below change?

The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci ac Given that the number of freshmen and sophomore con- testants was the same, which of the following must be true?

A There are five times as many sophomores as freshmen. B There are twice as many sophomores as freshmen.

C There are as many freshmen as sophomores. D There are twice as many freshmen as sophomores. E There are five times as many freshmen as sophomores. The sides of a triangle with positive area have lengths 4, 6, and x. The sides of a second triangle with positive area have lengths 4, 6, and y. Two different prime numbers between 4 and 1 8 are chosen.

When their sum is subtracted from their product, which of the following numbers could be obtained? Solution 0, 1, 2, and 3 consist of 1, 5, 13, and 25 nonoverlapping unit squares, respectively.