CELEBRATING A CENTURY OF ADVANCING MATHEMATICS
17th/67th Annual
American Mathematics Competitions
AMC 10/12
TEACHERS’
MANUAL
Instructions and Reporting Forms
for School Contest Managers
Tuesday, February 2, 2016
Wednesday, February 17, 2016
Please read this booklet thoroughly upon receipt.
DATES TO REMEMBER
AMC 10/AMC 12 - Tuesday, February 2, 2016 &/or Wednesday, February 17, 2016
AIME - Thursday, March 3, 2016 or Wednesday, March 16, 2016
USAMO & USAJMO - Tuesday & Wednesday, April 19–20, 2016
A
M
C
10
&
12
American Mathematics Competitions
Dear Contest Managers:
I am very pleased that you will be providing the opportunity for your students to participate in the 67th annual
American Mathematics Contest 12 (AMC 12) or its sister contest, the 17th annual American Mathematics Contest
10 (AMC 10). I believe that you and your students will find these contests to be both interesting and challenging.
Again this year the Committee on the American Mathematics Competitions (CAMC) and panelists under the
leadership of Committee Chairs Silvia Fernandez for the AMC 10 and Jerrold W. Grossman for the AMC 12
developed a total of 4 contests (two AMC 10 and two AMC 12). I extend special thanks to both of them for the
effort involved in developing these excellent contests.
This year, we continue to include in this manual several handouts, included in Section XIV:
* Worksheets for contest preparation (page 32–39)
* A handout for parents on why mathematics is important (page 20)
* Sample Press Release (page 19)
* A Certificate of Participation (pages 22–23)
You may reproduce these pages for your students.
Sincerely,
Mark Saul, Ph.D.
Director, MAA American Mathematics Competitions
American Mathematics Competitions
PO Box 471 | Annapolis Junction, MD 20701
800.527.3690/phone | 240.396.5647/fax | amcinfo@maa.org | maa.org/amc
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ATTENTION CONTEST MANAGERS
IMPORTANT DATES — AMC 10/12 A
By January 19, 2016
Final day to register/order additional contest materials.
By February 1, 2016
Have students pre-fill all name and demographic information on answer sheets.
February 2, 2016
February 3, 2016
Official contest administration date
Return answer sheets to the MAA AMC office the day after contest administration — don’t forget
to use a trackable method. All answer sheets must be received no later than 10 days after contest
administration (February 9, 2016).
IMPORTANT DATES — AMC 10/12 B
By February 3, 2016
Final day to register/order additional contest materials.
By February 16, 2016
Have students pre-fill all name and demographic information on answer sheets.
February 17, 2016
Official contest administration date
February 18, 2016
Return answer sheets to the MAA AMC office the day after contest administration — don’t forget
to use a trackable method. All answer sheets must be received no later than 7 days after contest
administration (February 24, 2016).
• Verify your contest materials immediately. There should be enough answer sheets included to match the number of contests you purchased. If there are missing materials, please contact Customer Service at 800-527-3690 (US), or 301-617-7805
(outside the US) immediately.
• The AMC 10/12 Teachers’ Manual is online this year. Please download and print your copy at maa.org/amc under Teacher
Resources.
• Do not use photocopied answer sheets. Photocopied answer sheets cannot be read by our scanners and cannot be scored,
so please be sure to order enough contests and answer sheets for your students. For information on ordering additional
contests/answer sheets, see the Additional Bundles Form on page 13 in your Teachers’ Manual.
• Check that all answer sheets are complete. Have students fill in the answer sheets, filling in the bubbles completely with
their name, age, date of birth, grade, gender, and address prior to the contest date. Incomplete answer sheets could result in
incorrect or delayed scoring. After the contest administration, please verify that these portions of each answer sheet have
been completed correctly.
• Complete and include the School Identification Form. The correct CEEB (school) code, number of answer sheets returned, and the date the contest was given are crucial to the scoring process. If you are unsure of the correct code, please
contact our office at 800-527-3690 (US), 301-617-7805 (outside the US), or amcinfo@maa.org prior to returning your contest
materials. Errors in completion of this form can lead to inaccurate score reports.
• Complete and include the Certification Form. The form is found on page 11 of the Teachers’ Manual and a copy of the
form is included in your contest package. Please be sure to include all signatures and necessary exceptions.
• Please mail completed answer sheets the day after the contest administration. Answer sheets must be returned within
10 days (for the AMC 10/12 A) and 7 days (for the AMC 10/12 B) of the contest date to receive your contest report and any
awards earned. Those received after this time will be subject to a delay in scoring and will only receive electronic scores
• Returned answer sheets must be shipped via a trackable method. The MAA AMC is not responsible for lost answer
sheet packages.
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Table of Contents
Points to Note.............................................................................................................................................................................. 5
I. Preliminary Instructions for Administering the AMC 10/AMC 12 ..................................................................................... 6
II. Instructions For The Day of The AMC 10/AMC 12............................................................................................................ 6
III. Eligibility........................................................................................................................................................................ 7
International Students & Non-Citizens in USA Schools......................................................................................................................................................................... 7
IV. Team Score Identification ............................................................................................................................................... 7
V. School Results ................................................................................................................................................................ 7
VI. Policy Statements .......................................................................................................................................................... 7
Early Administration........................................................................................................................................................................................................................................ 7
Official Administration.................................................................................................................................................................................................................................... 7
Unofficial Administration .............................................................................................................................................................................................................................. 7
Contest Proctoring........................................................................................................................................................................................................................................... 7
One Contest per Date....................................................................................................................................................................................................................................... 7
Students with Accomodation Plans ............................................................................................................................................................................................................8
Sickness and Other Special Situations ......................................................................................................................................................................................................8
Learning Centers and Home Schools .........................................................................................................................................................................................................8
Questionable Scores........................................................................................................................................................................................................................................8
Follow-up Inquiries and Reexamination.....................................................................................................................................................................................................8
Policy for Changes............................................................................................................................................................................................................................................8
Refund/Credit Policy........................................................................................................................................................................................................................................8
Shipping Address..............................................................................................................................................................................................................................................8
Request for Student Names Policy..............................................................................................................................................................................................................8
VII. AIME Instructions ........................................................................................................................................................... 8
AIME Rules for AMC 10/AMC 12.......................................................................................................................................................................................................................8
AIME School Manager......................................................................................................................................................................................................................................8
Second AIME Testing Date.............................................................................................................................................................................................................................. 9
VIII. USAMO/USAJMO Participant Selection ........................................................................................................................................ 9
IX. The MOSP Program ........................................................................................................................................................10
X. Regions of the AMC 10/AMC 12.......................................................................................................................................10
XI. Intramural and National Awards.....................................................................................................................................10
Intramural Certificates for Students..........................................................................................................................................................................................................10
National School Awards.................................................................................................................................................................................................................................10
XII. Contest Certification......................................................................................................................................................11
Certification by the Contest Manager.........................................................................................................................................................................................................11
Certification by the Principal, official or person with comparable title..........................................................................................................................................11
Service Questionnaire....................................................................................................................................................................................................................................12
XIII. Additional Forms............................................................................................................................................................ 13
Additional Bundles Form ...............................................................................................................................................................................................................................13
Proof of Intent to Pay.....................................................................................................................................................................................................................................13
Rescoring Request Form................................................................................................................................................................................................................................14
Method of Payment.........................................................................................................................................................................................................................................14
Contest Registration Form ...........................................................................................................................................................................................................................15
Publications Order Form................................................................................................................................................................................................................................17
XIV. Classroom Accessories .................................................................................................................................................. 19
Press Release....................................................................................................................................................................................................................................................19
Letter to Parents (for Reproduction)....................................................................................................................................................................................................... 20
Hallway Promotional Flier............................................................................................................................................................................................................................ 21
AMC 10 Participation Certificates.............................................................................................................................................................................................................. 22
AMC 12 Participation Certificates.............................................................................................................................................................................................................. 23
Facsimile of AMC 10 Front Cover................................................................................................................................................................................................................ 24
Facsimile of AMC 12 Front Cover................................................................................................................................................................................................................ 25
Curriculum Inspirations............................................................................................................................................................................................................................... 26
AMC 10/12 Student Practice Questions..................................................................................................................................................................................................... 32
Our Sponsors.............................................................................................................................................................................. 40
Any forms not available here can be downloaded from our web site: maa.org/amc
Points to Note
and we will be happy to grade it. Students may take the contest
booklets home with them the day of the contest.
Important Changes
1. Contest Managers must review all answer sheets for completeness
before returning them for scoring. Full names, school, birth date,
etc. must be completed in their entirety. Incomplete forms will
not be scored.
2. Schools returning packets within the specified time (10 days
for AMC 10/12A and 7 days for AMC 10/12B) will receive their
score reports and any awards earned. Those received after this
time will be subject to a delay in scoring and will only receive
electronic scores.
3. Photocopies of answer sheets will not be accepted. Please be
sure to order the correct number of contests and answer sheets.
Additional orders can be made, in compliance with contest deadlines, using the additional bundles order form in this manual.
Any photocopied answer sheets will be returned to the sender
without scoring.
Answer Forms
The AMC 10 and AMC 12 Answer Forms have been combined to
now include the AMC 10A, AMC 10B, AMC 12A, and AMC 12B
all on one form. Read the instructions on the answer forms and
instruct the students how to properly complete the answer forms
before the official test date.
Solutions & Results
We will send your school’s solution books and results by email (if
available) and first class mail as soon as the answer forms are scored.
After the competition, we will send you a link to a website from
which you can download a copy of the contest pamphlet and a copy
of the solution pamphlet.
If you have not received your results from our office within 30 days
after the contest, please contact us to verify that your answer forms
were in fact received.
All student forms are held for 90 days after the contest date, then
they are recycled.
Format
There are two official dates for the AMC 10 and AMC 12.
The AMC 10 and AMC 12 must be given at the same time
within each participating school on Tuesday, February 2, 2016
(AMC 10-A and AMC 12-A), or Wednesday, February 17,
2016 (AMC 10-B and AMC 12-B) in a convenient 75-minute
interval, preferably in the morning. For time zones outside
the continental United States, the contests must be given after
9 am EST (UTC-5) February 2, 2016 or February 17, 2016. All four
contests consist of 25 questions.
AIME Qualification
Students with sufficiently high scores will qualify to take the 34th
annual American Invitational Mathematics Examination (AIME) on
Thursday, March 3, 2016 or Wednesday, March 16, 2016.
School Flier
NOTE: Each correct answer scores 6 points, a blank scores 1.5
points and an incorrect answer scores 0 points.
We are again providing a flier you can use to promote the contests
within your school. It is included on page 21 of this Manual. It has
space left for you to add school information, such as the location
of the contests, and who to contact locally for more information.
If you would like a color version, visit our web page to download a
pdf version of the flier you need: maa.org/amc
The AMC 10 and AMC 12 have several questions in common. The
students in grades 10 and below should choose between the AMC 10
and AMC 12. Students in grades 11 and 12 may only take the AMC
12. All rules and awards apply to both contests for all schools and
students. Any student who missed the exam may take it unofficially,
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I. Preliminary Instructions for Administering the
AMC 10/AMC 12
1. I nform students far in advance about the dates for the AMC 10/
AMC 12 and obtain a supply of No. 2 lead pencils.
2. Hand out the student Answer Forms and have the students
complete the non-answer sections on the front and back.
Have students use their full legal name, no nicknames or
abbreviations. Have them pay special attention to marking
their name and address accurately. Remind them that student
names listed on the AMC website come from this form. The
AMC Office will not do any editing of the information.
3. Announce that the students may use scratch paper, graph paper,
rulers, compass, protractors, and erasers. CALCULATORS AND
CELL PHONES ARE NOT PERMITTED. No problems on the
contest will require the use of a calculator.
4. Review past tests and solutions or inform your students how
to order copies for themselves. The Publications Order Form is
found on page 17 or it can be printed from the AMC Web page
at maa.org/amc.
5. Encourage participation by students who have not taken these
competitions before, and especially by gifted younger students.
But make sure they know what to expect. For such students,
getting even a few questions correct is an accomplishment.
You might let them know about typical scores of students at
their level in your school in past years, or the national statistics
posted on amc-reg.maa.org/reports/generalreports.aspx. Make
sure they set appropriate goals for themselves, so that taking
the competition is a positive experience for them.
6. Remind students the day before the contest about the time and
place of the AMC 10/AMC 12. Also tell them your plan if the
school should suddenly close. All students must take the contest
at the same time, either in one group or in separate classrooms
and be under the supervision of a proctor.
7. Make sure you have arranged to follow all the rules and
procedures in this manual. EARLY ADMINISTRATION OF
THE CONTESTS IS NEVER PERMITTED, and will lead to
disqualification. These rules and procedures have been instituted
to guarantee valid results for our examination. We must all
observe them carefully.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
II. Instructions for the Day of the AMC 10/AMC 12
If today is not yet Tuesday, February 2, 2016 or February 17, 2016
STOP EVERYTHING. Under no circumstances is the contest to
be given to anyone before the official day, nor should the contest
package(s) be opened before that date. For time zones outside the
continental United States, the contests must be given after 9 am EST
(UTC-5) February 2, 2016 or February 17, 2016.
1. Sign the Certification Form to certify that the package was
opened within an hour before the contest.
2. When the students arrive, seat them so they are separated by
an empty space if possible.
3. Hand out the Answer Forms which have been partially
completed by the students.
4. As you hand out the papers, tell the students not to open the
contest booklet. They should then read the entire front cover.
Give them 5 minutes to do this.
5. Inform the students to, “Carefully read instructions 4 and 5 on
the contest cover.” (see pages 24 and 25 of this manual). The
16.
17.
18.
19.
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AMC 10/AMC 12 has a scoring system which has important
consequences for guessing. Unless you are fairly sure of the
answer, it is better to leave a question unanswered than to guess.
Six points are given for a correct answer, 1.5 points for a blank
answer and 0 points for an incorrect answer. If a student can
reduce the problem to three possible answers, it is advantageous
to guess one of the three possible answers. If a student can only
reduce to 4 possible answers by eliminating 1 of the possibilities,
then it is not advantageous to guess.
Inform the participants they may not talk or ask any questions
during the contest, and they must do their own work.
Remind students they have 75 minutes, then tell them to BEGIN.
(See page 6 for student disabilities policy).
Students who finish the contest early may be dismissed provided
they will be under the supervision of a teacher during the
remainder of the contest period.
You (and other teachers, if there are many participants) should
proctor continually as you would for any important contest.
Students whose eyes wander should be warned; students caught
copying answers or collaborating must be disqualified. Try to
provide as quiet an environment as possible. Please be sure
that students do not have access to calculators, cell phones,
smart watches, or other electronic devices for computation or
communication.
Announce when there are 30 minutes remaining and when
there are 5 minutes remaining.
When time is up, tell the students to STOP and have them sign
their name in the space provided on the Student Answer Form.
Collect the Answer Forms as quickly as possible.
Please do not grade the answer forms. They are to be sent to
the AMC office for grading. Students may circle their answers
on the contest booklet. However, the official answers will be the
ones blackened on the answer form.
Fan the forms, making sure none are stuck together, and place
the School ID Form on top. There is only one ID FORM to be
used with all the AMC 10/AMC 12 answer forms.
YOUR SCHOOL’S CEEB NUMBER IS THE NUMBER
PRINTED ON THE FRONT OF THE AMC 10/AMC 12
REPORT ENVELOPE.
Complete the Certification Form (only one form is needed)
and place it on top of the School ID Form and answer forms,
then place all material in the Report Envelope. Seal and send it
within 24 hours or as soon as possible after all administrations
have taken place. The AMC office requires using a trackable
mail service.
Send your answer forms to 9050 Junction Drive, Annapolis
Junction, MD 20701 using a trackable mail service. AMC cannot
be responsible for grading answer forms that are not sent with
a trackable mail service.
AMC has instituted a firm deadline policy for receipt of answer
forms. Answer forms not received with 7 days of the contest
administration may not be scored.
Please affix the proper postage before mailing.
Please note: After the Answer Forms have been delivered to
the school office to be mailed, you may discuss the contest
with your students under the following conditions which take
into consideration the fact that there will be schools taking the
contest in other locations at different times.
a. Inform the students that the contest may not be discussed
with anyone outside of your school verbally, via email, phone,
text messaging, web, social networking site, copier or media
of any type until after the contest period.
VI. Policy Statements
Early Administration
Administration on an earlier date is NEVER permitted and will lead
to disqualification. Such an administration would jeopardize the
validity of all scores from other participating schools.
b. Students may keep the contest booklets and take them home.
III. Eligibility
Official Administration
AMC 12 Eligibility — A student in a program leading to a high
school diploma, and under 19.5 years of age on the day of the contest.
The AMC 10-A/AMC 12-A will be given officially on Tuesday
February 2, 2016. The AMC 10-B/AMC 12-B will be given officially
on Wednesday, February 17, 2016. Only official participants, their
school and their teacher are eligible for National Awards. In addition, official participants are eligible for all intramural awards and
for participation in the AIME.
AMC 10 Eligibility — A student in a program leading to a high
school diploma, and under 17.5 years of age on the day of the contest,
and not enrolled in grades 11 or 12 or equivalent.
Please note: students in grades 11 & 12 cannot take the AMC 10. However, students in grades 9 & 10 may choose which contest they take.
Unofficial Administration
If you are unable to give Contest A on Tuesday, February 2, 2016
because:
a. your school is closed,
b. your school has an academic conflict,
c. the class periods have been shortened due to an assembly or
other reason,
then you may give the second version of the contests (AMC 10-B/
AMC 12-B) on the second official day, Wednesday, February 17,
2016 (See Contest B Registration Form on page 15). You may still
take either exam unofficially on later dates, but those contests will
not be eligible for state and National Awards and will not be eligible
for participation in the AIME.
International Students & Non-Citizens in USA Schools
Students attending school in the United States and Canada together
with U.S. Citizens residing outside of the U.S. and Canada (with
qualifying scores) are eligible to take the USA(J)MO.
Students learning “English as a Second Language” (ESL) may use
a nontechnical print dictionary between their native language and
English. A student may use the dictionary only the first time that
he/she takes the AMC 10/AMC 12. The dictionary must be given to
the school contest manager to examine and retain for the 24-hour
period preceding the contest. The proctor must announce to other
students that the student(s) has/have been given special permission
to use the dictionary during the contest. No electronic translation
devices are allowed.
It is important to note that the only days eligible for official participation are the two official Contest days: February 2, and February 17,
2016.
IV. Team Score Identification
Contest Proctoring
TO RECEIVE OFFICIAL TEAM STATUS AND AWARDS, A
SCHOOL MUST HAVE AT LEAST THREE PARTICIPANTS ON A
CONTEST DATE. The team score for a school is the sum of its three
highest student scores and will be determined by the AMC Office.
The proctoring of the AMC 10/AMC 12 contest should be by:
1st preference: a math teacher at the school
2nd preference: a teacher or administrator at the school
3rd preference: a college or university teacher of mathematics or
a responsible adult who is a math club or team coach, and not
related to any of the participants
th
4 preference: a responsible adult not related to any of the
participants such as librarian, clergy, etc.
V. School Results
The AMC office will send results by email (if available) and first
class mail as soon as the answer forms are scored. If you have not
received your results from our office within 30 days after the AMC
10/AMC 12 please contact us to verify that your answer forms were
in fact received.
• The proctoring of the contest must take place in a public building,
(i.e., school library, college or university, church, etc.).
• All students must take the contest at the same time, either in one
group or in separate classrooms and be under the supervision of
a proctor.
• The proctor should not be related to any of the participants.
• If a parent wishes to help, they can arrange room reservations
and set-up, help with student sign-up, provide treats, etc. Under
no circumstances may a parent proctor a relative.
If you would like to receive your results by email, and have not previously sent us your email address, send a message, including your
name, school name, address, and CEEB number to:
amcinfo@maa.org
Results are not official until you receive the postal mailed paper copy
of your report, and that should be no longer than 30 days after the
AMC 10/AMC 12.
One Contest per Date
Please retain this report for future reference, either in your files or
with the student counselor’s office. Many students cite these scores
on college applications. AMC cannot look up student scores after
the contest cycle has been completed.
A student may take only one exam on a given day. A student taking
both contests on a single date will have both scores cancelled and
invalidated. A student can participate on both contest dates if the
school registers for both contests. The higher score will be used for
individual awards.
7
The American Mathematics Competitions (AMC) receives requests from educational institutions and organizations for the
names, addresses and grade levels of high scoring AMC 12 (or
AMC 10) students. The optional personal data on ethnic origin and gender is used for recruiting and academic purposes.
Students with Accommodation Plans
(e.g., for Visual or Learning Disabilities)
The AMC 10/AMC 12 time limit set by the CAMC for students who
are visually impaired or learning disabled is 120 minutes. A teacher
or a school administrator may read the questions to the student and
mark the answers as directed by the student. The braille and large
print exams must be ordered no later than three weeks before the
test. It is the responsibility of the school and the contest manager to
make arrangements consistent with the student’s accommodation
plan. Please contact us at amcinfo@maa.org for more information.
Blacken this circle if you give the AMC permission to release
this information to these organizations. (Your score will not
be affected if you do not blacken the circle.)
Receiving information is an “opt-in” decision for each individual
student. The AMC handles requests from institutions and
organizations on a case-by-case basis and evaluates each individually
for appropriateness. We provide legitimate educational institutions
of all levels, both secondary and collegiate/university level, with onetime use of “directory information” (as defined by the Department of
Education) for mailings. We also provide professional and scholarly
organizations such as those listed as contributors to the AMC with
one-time use of “directory information” for mailings, generally for
professional or career information.
Sickness and Other Special Situations
A student who is sick or on a field trip on the first contest day may
take the alternate Contest B on Wednesday, February 17, 2016. YOU
MUST REGISTER FOR CONTEST B if you have not already done
so. (See page 15 for a Registration Form.)
Learning Centers and Home Schools
The AMC Office reserves the right to not register any entity that is
not a school, college, or university.
VII. AIME Instructions
Questionable Scores
The 34 annual American Invitational Mathematics Examination
(AIME) will be held on Thursday, March 3, 2016 with a second
alternate exam given on the alternate date of Wednesday, March 16,
2016. These are the only days the exam may be taken officially. You
may give the exam for practice (unofficially), after the official dates.
We will be pleased to grade it for you but your students will not be
eligible to take the USAMO/USAJMO. The contest is provided free
of charge to all those taking the exam on the first date. However
those taking the exam on the second alternate date will be charged
a processing fee to cover additional handling.
th
If it is clear to the Contest Manager from personal observation
that a student has cheated, then the Manager must disqualify the
student. If the Contest Manager receives an accusation of cheating,
or obtains other indirect evidence of cheating, then the Manager
must hold back the student’s paper and immediately report all the
facts of the situation to the AMC Director, who in conjunction with
the Chair of the CAMC, will determine what to do. UNDER NO
CIRCUMSTANCES may the school decide on its own to accept a
questionable score, nor should a school retest the student in question
before receiving instructions from the AMC Office.
AIME Rules for AMC 10/AMC 12
Qualification parameters for AIME vary with the results of the
AMC 10/12 contests. For AMC 10, invitations are issued to the top
2.5% of all scorers (approximately), or scores of at least 120 points
(approximately). For AMC 12, invitations are issued to the top
5% of all scorers (approximately) or scores of at least 100 points
(approximately). The exact cutoffs will depend on the difficulty of
each contest.
Note that the cutoffs for AMC 10 are higher than those for AMC 12.
This is done to include very good younger students for whom the
AIME would be an appropriate experience, but to exclude students
who will not succeed.
Follow-up Inquiries and Reexamination
The results of this contest help to identify students with unusual
mathematical ability. To ensure that this purpose is served, the
CAMC reserves the right to retest students before deciding whether
to grant official status to individual or team scores. Reexamination
will be requested when, after an inquiry, there is a reasonable basis
to doubt a score. Official status will not be granted if a student or
school does not agree to a requested retesting.
Policy for Changes
The CAMC may, from time to time, change the program rules,
regulations, awards and conditions of participation in whole or
in part. Whenever possible you will be notified of these changes
ahead of time.
AIME School Manager
1. The AMC office will include all materials relating to the
examination (including instructions for the exam) with your
AMC 10/AMC 12 results.
2. The AIME II contests and answer booklets are made available
ONLY in electronic form. The AIME I contest and solution
pamphlets are in paper form.
3. All questions or problems concerning the AIME should be
directed to the AMC office (800-527-3690).
4. The AIME is a three-hour examination. Each of its 15 questions
requires a three digit integer answer and each correct answer will
receive one (1) point.
5. Calculators, smart watches, and cell phones are not allowed.
6. The AIME Answer Forms are sent directly to the AMC office
for grading and processing.
Refund/Credit Policy
If your school is unable to take the contests, please use the materials as practice sets for the next year. Do not return them. WE CAN
NOT GIVE REFUNDS OR CREDITS AFTER THE CONTEST
MATERIALS ARE SHIPPED.
Shipping Address
The AMC Office cannot ship to P.O. Boxes or residential addresses.
Request for Student Names Policy
The following statement appears on the student answer forms for
the AMC 10 and AMC 12:
8
7. E
ach participating school will receive a report of their results
and an AIME solution pamphlet.
8. All AMC 10/AMC 12 procedures for disqualification, follow-up
inquiries and reexamination apply to the AIME as appropriate.
9. If you have students who you feel may qualify for the AIME
please order prior year AIME exams and solutions for practice
now. Thus, you will have these practice materials on hand when
you receive your AMC 10/AMC 12 results.
Olympiad participation to 10th graders and below. Students who
take ONLY the AMC 10 test, whether AMC 10 A or AMC 10 B
or both, will NOT be eligible for the USAMO regardless of their
score on the AMC 10 or the AIME.
5. The approximately 270 individual students with the top AMC 12
based USAMO indices will be invited to take the USAMO. These
indices will be selected from the pool of AMC 12 takers with an
AIME score.
6. The approximately 230 individual students with the top AMC
10 based USAJMO indices will be invited to take the USAJMO.
These indices will be selected from the pool of AMC 10 takers
with an AIME score after removing students who also took an
AMC 12 test and qualified for the USAMO in rule 5. This means
young students MUST take the USAMO if they qualify through
an AMC 12 index.
Second AIME Testing Date
Situations in which a student may take a second version of the AIME
to be held on Wednesday, March 16, 2016:
1. A school is located outside of the United States.
2. School is closed on March 3, 2016 (i.e., spring break, weather).
3. Student is out of school the entire day due to attendance at an
academic/school related event.
4. Student is ill and can not attend school on March 3, 2016.
7. We will also invite the student with the numerically largest index, whether AMC 10 based USAJMO index or AMC 12 based
USAMO index, from each U.S. state not already represented in
either the USAMO or the USAJMO. The student will be invited
to the USAMO if the numerically highest index in the state is
AMC 12 based, and invited to the USAJMO if the index is AMC
10 based.
The processing fee for the second AIME is as follows: 1–10 students
= $25, 11+ students = $50. We will need your payment before the
answer forms can be graded. A payment form will be included with
your AIME material, if you have AIME qualifiers. All AIME answer
forms must arrive in the AMC office by March 21, 2016.
Email requests for the second AIME may be sent to:
More information will be provided in the AIME Teachers’ Manual.
You can also visit our website at maa.org/amc for more information.
amcinfo@maa.org
or, you can call the AMC office at 1-800-527-3690. Please have your
school identification number (CEEB) and charge card information
available before calling. Email requests should include the school’s
CEEB number, and complete mailing address. Under no circumstances can a student take both AIME’s.
Which Contest Should Young Advanced Students Take?
A student in 10th grade or below who wishes to qualify for the
USAMO must take either the AMC 12 A or the AMC 12 B contests
in order to be considered for USAMO selection. We recommend
that such a student take an AMC 10 contest on one date if possible.
A student in 10th grade or below wishing to take the AMC 12
and qualify for the USAMO should have a good problem solving
knowledge in advanced algebra, analytic geometry, function notation, logarithms, trigonometry, and complex numbers in order to
score well on the AMC 12, AIME and USAMO. The student must
also be prepared to compete with 11th and 12th grade students on
an equal basis since USAMO qualification is based on score only
with no consideration for grade in school.
VIII. USAMO/USAJMO Participant Selection
The USA Mathematical Olympiad (USAMO) is a two day, nine-hour,
six-question, essay-proof examination. We select approximately 500
students total for the two Olympiads, split approximately between
270 for the USAMO and 230 for the USAJMO. Selection for the
2016 USA Mathematical Olympiad (USAMO) and 2016 USA Junior
Mathematical Olympiad (USAJMO) will be made according to the
following rules:
The minimum USAMO qualifying index for the last few years has
been:
2015.......... 219.0 ....... 294 students
2014.......... 211.5 ....... 266 students
2013.......... 209.0 ....... 264 students
2012.......... 204.5 ....... 264 students
2011.......... 188.0 ....... 279 students
2010.......... 208.5 ....... 277 students
2009.......... 201.5 ....... 514 students
1. Students attending a school in the United States and Canada
together with U.S. Citizens residing outside the United States and
Canada (with qualifying scores) are eligible to take the USAMO
and USAJMO.
2. Selection to the USAMO will be based on the USAMO index,
defined as AMC 12 Score + 10 * AIME Score. Selection to the
USAJMO will be based on the USAJMO index, defined as AMC
10 Score + 10 * AIME Score.
3. Only AMC 12 A or AMC 12 B takers who are students attending a school in the United States and Canada together with U.S.
Citizens residing outside the United States and Canada will be
eligible for the USAMO.
Exact invitation levels will vary each year depending on the difficulty
of the contests and the pool of participating students. The historical
minimum qualifying scores here are for planning purposes only and
do not necessarily indicate future qualifying scores.
4. Only AMC 10 A or AMC 10 B takers who are students attending a school in the United States and Canada together with U.S.
Citizens residing outside the United States and Canada will be
eligible for the USAJMO. Note that this rule limits Junior Math
9
3. Gold Medal — The Gold Medal is awarded to participants who
place first in their respective school for four consecutive years.
4. Silver and Bronze Medals — These medals are awarded to
participants who place first in their school for three and two
consecutive years, respectively.
5. AMC 10/12 Certificate of Distinction — A Certificate of
Distinction is awarded to those students who qualify to take the
American Invitational Mathematics Examination (AIME).
6. AMC 12 Certificate of Achievement — Awarded to students
in grade 10 and below who score a 90 or above on the AMC 12
contest.
7. AMC 10 Certificate of Achievement — Awarded to students in
grade 8 and below who score a 90 or above on the AMC 10 contest.
8. School Certificate of Honor — Awarded to schools with a team
score of 400 or greater on the AMC 12 only.
9. School Certificate of Merit — Awarded to schools with a team
score between 300 and 399 on the AMC 12 only.
IX. The MOSP Program
The Mathematical Olympiad Summer Program (MOSP) is a threeweek, academic challenge designed to broaden participants’ view of
mathematics while fostering excitement toward further math study.
In 2016 MOSP will be held in June at Carnegie Mellon University.
Invited students include:
1. The top 12 winners (or fewer) of the USAMO from the United
States.
2. Approximately 12 students not currently in 12th grade who make
Honorable Mention on the USAMO.
3. Approximately 12 top scorers on the USAMO who are in 9th and
10th grade (“Approximately” to account for ties).
4. Approximately 12 top scorers (from the United States) on the
USAJMO in the 9th and 10th grade.
5. Approximately 8 to 10 girls with sufficiently high scores from the
USAMO and USAJMO to train for Girls Mathematical Olympiads.
6. Numbers are approximate and subject to space availability and
funding. MOSP invitations are extended only to students who
have attended school in the United States for the year prior to the
program, or to U.S. citizens attending school in another country.
National School Awards
The AMC divides the United States and Canada into eleven regions.
In each of these regions the five schools with the highest team scores
(sum of the highest three scores by participants) are recognized by
Donor or Committee Awards.
1. Charles T. Salkind Memorial Silver Cup — This silver cup, is
awarded to the school with the highest team score (in any region).
2. William H. Fagerstrom Memorial Silver Cup — This silver cup
is awarded to the school with the second highest team score (in
any region).
3. Committee Bronze Cups — In each of the regions in which
a silver cup is not awarded the Committee on the American
Mathematics Competitions provides a Bronze Cup to the school
with the highest team score.
4. Schools and students may be eligible to purchase other awards
from an authorized vendor, at their own expense. The school
report forms will give details for ordering such awards.
X. Regions of the AMC 10/AMC 12
The USA and Canada are partitioned into the following regions.
National Awards are given to a minimum of 10 high scoring students
and 5 schools (based on the team score) in each of these regions.
Region
0 Connecticut, Maine, Massachusetts, New Hampshire, Pennsylvania, Rhode Island, Vermont
1 New Jersey, New York
2 Delaware, District of Columbia, Maryland, North Carolina,
South Carolina, Virginia, West Virginia
3 Alabama, American Embassy and APO/FPO Schools, Florida,
Georgia, Puerto Rico, Virgin Islands
4 Indiana, Michigan, Ohio
5 Arkansas, Iowa, Kansas, Minnesota, Nebraska, North Dakota,
Oklahoma, South Dakota, Wisconsin
6 Illinois, Kentucky, Missouri, Tennessee
7 Louisiana, Mississippi, Texas
8 Alaska, Arizona, Colorado, Guam, Hawaii, Idaho, Montana,
Nevada, New Mexico, Oregon, Utah, Washington, Wyoming
9 California
10Canada
XI. Intramural and National Awards
Intramural Certificates for Students
The AMC Office will send you Intramural Certificates, along with
your school results. An order form for additional intramural awards
will be included with your results.
1. AMC 10 & AMC 12 School Winner Pin — For each school
making a report on three or more students the Mathematics
Award Pin is awarded to the student from that school with the
highest score.
2. AMC 10 & AMC 12 Honor Roll of Distinction Pin — The
Honor Roll of Distinction Pin is awarded to the top 1% of all
participating students.
10
AMC 10
2016
AMC 12
XII. Contest Certification
The AMC 10 and AMC 12 must be administered by a teacher or an adult not related to any of the participants. The administration of
the contest must take place in a public building (e.g. school, library, church). Please send all Answer Forms from your school or group
at one time.
The Contest Manager and the Principal, Vice Principal, or Headmaster must sign this form, which is to be returned with your student
Answer Forms.
Certification by the Contest Manager
I certify that the following statements are true or that, if there are any exceptions, I have checked the box at the bottom of this page and
have listed them on a separate page. I understand that the absence of either signature from this form, and a consideration of the exceptions may result in DISQUALIFICATION of all scores from our school.
1. I certify that the exam package(s) were retained in their sealed condition within an hour of the start of the contest.
2. I accept for our school the rules and procedures described on this page and pages 4–7, and accept that failure to follow these rules
and procedures may result in DISQUALIFICATION from official standing of all scores from our school.
3. The contest was held on Tuesday, February 2, 2016 or Wednesday, February 17, 2016. For time zones outside the continental United
States, the contests must be given after 9 am EST (UTC-5) February 2, 2016 or February 17, 2016.
4. The AMC 10/AMC 12 were given at the same time.
5. The participants were continually monitored during the contest, and they were separated by an empty space, if possible.
6. No aids were permitted other than scratch paper, graph paper, ruler, compass, and protractor. NO CALCULATORS, SMART
WATCHES, OR CELL PHONES WERE USED (see Section I. Item 3).
7. Participants had exactly 75 minutes working time. (See page 6 for Students with Accommodation Plans)
8. No students were permitted to proctor or grade the contest.
9. After the contest, the answer forms were kept secure and no changes were made in the answers.
10. No parent assisted in the administration of the contest.
11. I have followed all the rules as stated in this Teachers’ Manual.
Signature: _______________________________________ E-mail (please print):____________________________________
Time, Day & Date Test was given:_______________________________________________________________________
Name of School:____________________________________________________________________________________
City: ____________________________________________ State/Province: ____________________________________ Telephone: ( ______ )____________________________ School CEEB Number: ___ ___ ___ ___ ___ ___
Certification by the Principal, Official or Person of Comparable Title
I agree that the Contest Manager followed all the rules and procedures listed above.
Signature:_____________________________________ Date:_________________________________________
Title:_________________________________________ PLEASE INDICATE THE EXACT NUMBER
OF AMC 10/AMC 12 ANSWER FORMS RETURNED FOR GRADING. ____________
Are there students who saw the contest booklet, but you are not sending an answer form? YES or NO
If accommodations for a Learning Disability were made, please list student name(s):
11
EXCEPTIONS

Questionnaire to Help Us Serve You Better
Thank you in advance for taking the time to answer these questions for us. Answer to the best of your ability, and if
you don’t know, just give us a “best estimate”. Please fill in your answers in the ovals provided on the back of the school
ID form. Thank you!
1. Rate the difficulty of the AMC 10 and AMC 12 problems 1–15 compared to the difficulty of your mathematics
curriculum (for 9th & 10th grade curricula for the AMC 10, and 11th & 12th grade curricula for the AMC 12)
0)much easier
1)easier
2)about the same
3)harder
4)much harder
5)N/A
2. What level math classes take the AMC contest? Mark all that apply:
0) calculus
1)pre-calculus
2) statistics
3) algebra II
4) algebra I
5) Math Club
3. Would you be interested in a shorter time version of the AMC contest? For example, a 30 minute contest.
0) very interested
1) interested
2) neutral
3) not interested
4) do not want
4. Would you be interested in an AMC 10/12 A contest date in the fall and an AMC 10/12 B contest date in the spring?
0) very interested
1) interested
2) neutral
3) not interested
4) do not want
5. Would you be interested in receiving the AMC 10/12 Teachers’ Manual as a PDF or other electronic document?
0) very interested
1) interested
2) neutral
3) not interested
4) do not want
6. The AMC is considering electronic delivery of the AMC 10/12 competitions. Your school would receive the
contest in PDF format a few days before the date of administration, and you would be responsible for printing out copies for students to take locally. Answer sheets would be sent to you, returned to AMC and scored
centrally as before. Would this procedure be less convenient than the current arrangement?
0) much less convenient
4) much more convenient
1) slightly less convenient
2) about equally convenient
3) slightly more convenient
7. If you order solution pamphlets, how do you use them?
0)give pamphlet to each student
1) discuss solutions in class 2) keep on file 3) I do not order solution pamphlets.
8. Is the curriculum in your school or district adopting or moving toward the Common Core State Standards?
0)yes
1)no
2)not sure
9. Is the AMC 10/12 a good supplement to the Common Core State Standards in Mathematics’ topics?
0)yes
1)somewhat
2)no
3) not sure
10.Which of the following features would you be interested in having access to using your online AMC Contest
Account? Mark all that apply:
0)ability to edit student information
1)prefilled student certificates
2)more payment options
3)ability to buy downloadable AMC Publications, like Math Club Packages, past contests, etc.
4)ability to build custom reports
5)I did not create an AMC Contest Account this year.
12
XIII. Additional Forms used
XIII. Additional Forms
Additional Bundles Form
Please fill in the information below and FAX your order. The administrator or authorized person of the school agrees to pay the American
Mathematics Competitions for the following materials:
School Name ______________________________________________________ CEEB # _________________
Address ___________________________________________________________________________________
City__________________________________________________State______Zip _______________________
Person placing the order _____________________________________ Email ____________________________
Contest date (please circle one): A or B
Contest Bundles of ten . . . . . . . . . . . . #_________.@ $22.50/bundle =. . . . . $ ___________
Solutions Sets of ten (optional). . . . . #_________.@ $ 9/set =. . . . . . . . . . . . $ _____________
AMC 10
AMC 12
Contest Bundles of ten . . . . . . . . . . . #_________.@ $22.50/bundle = . . . . $ ___________
Solutions Sets of ten (optional). . . . . #_________.@ $ 9/set =. . . . . . . . . . . $ ___________
Postage/handling Fee (see below) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ _________
Total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ _________
VISA/MC/AMEX/Discover# : _______________________________ Exp. Date: ___________ Sec. Code: ______
Name (Please Print):
___________________________________________ Phone #: ______________________
Email (needed for email receipt): ____________________________________________________________________
Shipping Charges
1. VISA, MasterCard, American Express, & Discover accepted.
2. Make checks payable to:
Mathematical Association of America
3. PAYMENT IN U.S. FUNDS ONLY.
4. USA: A one time processing fee of $5 plus applicable
domestic shipping charges.
5. OUTSIDE USA: Add $35 to your order, plus a processing
fee of $5. Payment via credit card or wire transfer only.
SCAN and EMAIL to: amcinfo@maa.org
FAX to: 240.396.5647
MAIL your order to:
MAA American Mathematics Competitions
ATTN: AMC 10/12 Additional Bundles
PO Box 471
Annapolis Junction, MD 20701
Orders for Additional Bundles will be accepted up to one week before the contest date.
Proof of Intent to Pay
 Billed
P.O. Number ________________________________________________________________________
Complete billing address: __________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
13
Rescoring Request Form
I would like to have the following student’s answer form rescored. I understand that there is a $25.00 charge for
each student answer form rescored. Rescoring requests are only accepted within 7 days after the school report has
been mailed.
$ 25.00/each
Student Name___________________________________________________________________________ $________________
Circle contest taken: AMC 10-A AMC 10-B AMC12-A AMC12-B AIME I AIME II
Student Name___________________________________________________________________________ $________________
Circle contest taken: AMC 10-A AMC 10-B AMC12-A AMC12-B AIME I AIME II
Grand Total ____________________________________________________________________________ $________________
Teacher’s Name _________________________________________________ CEEB # ____________________________________
School Name _____________________________________________________________________________________________
Address __________________________________________________________________________________________________
City State________________________________________________________________ Zip_____________________________
Email where re-score should be sent ______________
Method of Payment: (Rescores will not be processed until payment is received.)
Check (U.S. funds only) made payable to Mathematical Association of America and mailed with this form to the:
MAA AMERICAN MATHEMATICS COMPETITIONS
PO Box 471
Annapolis Junction, MD 20701
Charge to Visa/Mastercard/AMEX/Discover#: ____________________________________________ Sec. Code: _____
Name on card (print):____________________________________________________________________________
Signature:_____________________________________________________________________________________
Expiration Date:______________________________ Telephone:_________________________________________
FAX to: 240.396.5647
14
17
18
XIV. Classroom Accessories
Press Release
The sample Contest Announcement news release below, should be prepared and distributed to the newspapers, radio and
television stations in your region. To make preparation of the news release easier, visit our web site and download the text
from the on-line copy of the AMC 10/12 Teachers’ Manual. See the AMC website for the 2015 Summary of Results which
includes statistics and figures from the 2015 contest. Statistics and figures for the 2016 Contest will be available on our website
in March 2016.
(School or School District)
FOR IMMEDIATE RELEASE
(School) STUDENTS PARTICIPATE IN
NATIONAL MATHEMATICS COMPETITION
( # ) students at (School) participated in the 67th annual American Mathematics Contest 12, and/or the 17th annual American
Mathematics Contest 10. The contests were held on Tuesday, February 2, 2016 (or Wednesday, February 17, 2016). The
students competed for local, regional and national student and school awards. The contest, which covers high school
mathematics, is given in participating schools. Its purpose is to spur interest in mathematics and develop talent through the
excitement of friendly competition at problem solving in a timed format. In 2015, over 210,000 students from 4,100 schools
participated in the AMC 10 & AMC 12 contest including (_#_) students from (_#_) schools in (State). Top scorers at (school)
were (________, _______, etc.).
According to Dr. Mark Saul, who serves as Director of the American Mathematics Competitions, the AMC 12 (first offered
in 1950), and the AMC 10 (first offered in 2000), are part of a series of contests sponsored each year by The Mathematical
Association of America, through their program, the American Mathematics Competitions. The AMC offers the only math
competition series in the country leading to the United States of America Mathematical Olympiad (USAMO), the United
States of America Junior Mathematical Olympiad (USAJMO) and the Mathematical Olympiad Summer Program (MOSP).
From this group of students, the AMC sends the highly competitive USA Team to the prestigious annual International
Mathematical Olympiad.
The AMC program includes :
American Mathematics Contest 8 (AMC8)
American Mathematics Contest 10 (AMC 10)
American Mathematics Contest 12 (AMC 12)
American Invitational Mathematics Examination (AIME)
USA Mathematical Olympiad (USAMO)
USA Junior Mathematical Olympiad (USAJMO)
Mathematical Olympiad Summer Program (MOSP)
International Mathematical Olympiad (IMO)
Grades 6–8
Grades 10 & below
Grades 12 & below
All who qualify
All who qualify
All who qualify
Qualify thru USAMO
Top six from USAMO, MOSP
November
2 dates in February
2 dates in February
2 dates in March-April
late-April
late-April
June
July
The AMC is located at the Mathematical Association of America and receives support from The Akamai Foundation, American
Mathematical Society, American Statistical Association, Art of Problem Solving, Casualty Actuarial Society, Collaborator’s
Circle, Conference Board of the Mathematical Sciences, The D.E. Shaw Group, Expii, IDEA MATH, Jane Street Capital,
MathWorks, Mu Alpha Theta, National Council of Teachers of Mathematics, Simons Foundation, Society for Industrial and
Applied Mathematics, Star League, Tudor Investment Corporation, and Two Sigma. The Contests are given across the U.S.A.,
Canada, and in many schools abroad.
Details concerning the 2016 AMC 10/12 contests for High School, as well as the rest of AMC’s programs are available on the
AMC web site: maa.org/amc
For further information contact the AMC by email at amcinfo@maa.org or by telephone at 800-527-3690.
19
Letter to Parents (for Reproduction)
February 2016
Dear Parent or Guardian:
On February 2, 2016 and/or February 17, 2016 your son or daughter will participate in the 67th annual American
Mathematics Competitions contest. This contest has grown from a single city-wide competition in New York City
in 1950, organized by the local section of The Mathematical Association of America to a sequence of contests. In
2015, over 210,000 students from over 4,100 U.S. and international schools competed for school, regional, and
national awards in this contest and found it fun and rewarding. Top 10, well-known U.S. universities and colleges,
including internationally recognized U.S. technical institutions, ask for AMC scores on their application forms.
Your students deserve the chance to list these scores on their applications!
Each year the AMC 10 and AMC 12 are on the National Association of Secondary School Principals Advisory
List of Contests and Activities. The AMC Contests are sponsored by the Mathematical Association of America,
and are considered to be such a valuable stimulus to student interest in mathematics that 19 professional societies
and organizations support the contests with financial contributions.
With these contests, there are awards in each school for the student with the highest score, certificates for highscoring students in each school, state-wide awards, regional awards, and even national awards. These contests lead
to other more selective math contests, all the way to the USA team sent to the International Mathematical Olympiad,
the premier international high school level problem solving contest.
But the real rewards come from challenging each student with mathematics that is new, different, and “outside
of the box.” The problems on the contest are hard, but designed to be within reach. Just by participating in the
contest your student should still feel accomplishment, because these problems are meant to be more challenging
than routinely encountered in mathematics courses.
Mathematics is increasingly important in our technological and scientific age. Taking more mathematics in high
school is the gateway to jobs and careers of all kinds, even those that are not explicitly mathematical, scientific,
or technological. We hope that by offering these contests, we can challenge and inspire students to learn more
mathematics. We hope that your son or daughter enjoys the contests, and will continue to take mathematics courses
and competitions in high school and beyond.
Sincerely,
Mark Saul, Ph.D.
Director, MAA American Mathematics Competitions
PO Box 471 | Annapolis Junction, MD 20701 | 800.527.3690 | amcinfo@maa.org | maa.org/amc
American Mathematics Competitions
Participate in the 2016
AMC 10 and the
AMC 12 Contests!
Tuesday, February 2, 2016 and/or
Wednesday, February 17, 2016
Practice Materials
can be found at:
maa.org/amc
For more information, contact:
amcinfo@maa.org
AMC 10 Participation Certificates
American Mathematics Competitions
CERTIFICATE
Awarded to
for participating in the
American Mathematics Contest 10
(AMC 10)
__________________________________
2016
__________________________________
Silvia Fernandez, Chair
AMC 10 Subcommittee
Mark Saul, Director
MAA American Mathematics Competitions
American Mathematics Competitions
CERTIFICATE
Awarded to
for participating in the
American Mathematics Contest 10
(AMC 10)
__________________________________
Mark Saul, Director
MAA American Mathematics Competitions
2016
__________________________________
Silvia Fernandez, Chair
AMC 10 Subcommittee
AMC 12 Participation Certificates
American Mathematics Competitions
CERTIFICATE
Awarded to
for participating in the
American Mathematics Contest 12
(AMC 12)
__________________________________
2016
__________________________________
Mark Saul, Director
MAA American Mathematics Competitions
Jerrold W. Grossman, Chair
AMC 12 Subcommittee
American Mathematics Competitions
CERTIFICATE
Awarded to
for participating in the
American Mathematics Contest 12
(AMC 12)
__________________________________
Mark Saul, Director
MAA American Mathematics Competitions
2016
__________________________________
Jerrold W. Grossman, Chair
AMC 12 Subcommittee
A
ruary 2, 2016
Your School’s Results**
administer this exam is
READ THE MANUAL
erify on the AMC 10
ers’ Manual) that you
e exam.
il to the AMC office no
oblems or solutions for this
cipate seriously jeopardizes
a copier, telephone, email,
tion rules.
ons
es
s
cs
Facsimile of AMC 10 Front Cover
Tuesday, February 2, 2016
CELEBRATING A CENTURY OF ADVANCING MATHEMATICS
American Mathematics Competitions
17th Annual
AMC 10A
American Mathematics Contest 10A
Tuesday, February 2, 2016
INSTRUCTIONS
1. DO NOT OPEN THIS BOOKLET UNTIL YOUR PROCTOR TELLS YOU.
2. This is a 25-question multiple-choice test. Each question is followed by answers
marked A, B, C, D, and E. Only one of these is correct.
3. Mark your answer to each problem on the AMC 10 Answer Form with a #2 pencil.
Check the blackened circles for accuracy and erase errors and stray marks completely.
Only answers properly marked on the answer form will be graded. No copies.
4. SCORING: You will receive 6 points for each correct answer, 1.5 points for each
problem left unanswered, and 0 points for each incorrect answer.
5. No aids are permitted other than scratch paper, graph paper, rulers, compass,
protractors, and erasers. No calculators, smartwatches, or computing devices are
allowed. No problems on the test will require the use of a calculator.
6. Figures are not necessarily drawn to scale.
7. Before beginning the test, your proctor will ask you to record certain information
on the answer form.
8. When your proctor gives the signal, begin working on the problems. You will have
75 minutes to complete the test.
9. When you finish the exam, sign your name in the space provided on the Answer Form.
The Committee on the American Mathematics Competitions (CAMC) reserves the right to re-examine
students before deciding whether to grant official status to their scores. The CAMC also reserves the right to
disqualify all scores from a school if it is determined that the required security procedures were not followed.
Students who score well on this AMC 10 will be invited to take the 34th annual American Invitational Mathematics
Examination (AIME) on Thursday, March 3, 2016 or Wednesday, March 16, 2016. More details about the
AIME are on the back page of this test booklet.
The publication, reproduction or communication of the problems or solutions for this contest during the
period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination
at any time via copier, telephone, email, internet, or media of any type is a violation of the competition rules.
© 2016 Mathematical Association of America
24
Facsimile of AMC 12 Front Cover
Tuesday, February 2, 2016
A
RUARy 2, 2016
Your School’s Results**
o administer this exam is
E READ THE MANUAL
verify on the AMC 12
hers’ Manual) that you
he exam.
ail to the AMC office no
roblems or solutions for this
icipate seriously jeopardizes
ia copier, telephone, email,
tition rules.
tions
nces
ics
tics
CELEBRATING A CENTURY OF ADVANCING MATHEMATICS
American Mathematics Competitions
67th Annual
AMC 12A
American Mathematics Contest 12A
Tuesday, February 2, 2016
INSTRUCTIONS
1. DO NOT OPEN THIS BOOKLET UNTIL YOUR PROCTOR TELLS YOU.
2. This is a 25-question multiple-choice test. Each question is followed by answers marked A,
B, C, D, and E. Only one of these is correct.
3. Mark your answer to each problem on the AMC 12 Answer Form with a #2 pencil. Check
the blackened circles for accuracy and erase errors and stray marks completely. Only answers
properly marked on the answer form will be graded. No copies.
4. SCORING: You will receive 6 points for each correct answer, 1.5 points for each problem left
unanswered, and 0 points for each incorrect answer.
5. No aids are permitted other than scratch paper, graph paper, rulers, compass, protractors,
and erasers. No calculators, smartwatches, or computing devices are allowed. No problems
on the test will require the use of a calculator.
6. Figures are not necessarily drawn to scale.
7. Before beginning the test, your proctor will ask you to record certain information on the
answer form.
8. When your proctor gives the signal, begin working on the problems. You will have 75 minutes
to complete the test.
9. When you finish the exam, sign your name in the space provided on the Answer Form.
The Committee on the American Mathematics Competitions (CAMC) reserves the right to re-examine
students before deciding whether to grant official status to their scores. The CAMC also reserves the right to
disqualify all scores from a school if it is determined that the required security procedures were not followed.
Students who score well on this AMC 12 will be invited to take the 34th annual American Invitational Mathematics
Examination (AIME) on Thursday, March 3, 2016 or Wednesday, March 16, 2016. More details about the
AIME are on the back page of this test booklet.
The publication, reproduction or communication of the problems or solutions for this contest during the
period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination
at any time via copier, telephone, email, internet, or media of any type is a violation of the competition rules.
© 2016 Mathematical Association of America
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Problem Solving Strategy Essay # 1:
Engage in Successful Flailing
James Tanton, PhD, Mathematics, Princeton 1994; MAA Mathematician in Residence
***
Teachers and schools can benefit from the chance to challenge students with interesting
mathematical questions that are aligned with curriculum standards at all levels of difficulty.
This is the first essay in a series to give credence to this claim on the MAA website, www.maa.org/math-competitions. For over
six decades, the MAA AMC has been creating and sharing marvelous stand-alone mathematical tidbits. Take them out of their
competition coverings and see opportunity after opportunity to engage in great conversation with your students. Everyone can
revel in the true creative mathematical experience!
I personally believe that the ultimate goal of the mathematics curriculum is to teach self-reliant thinking, critical questioning
and the confidence to synthesize ideas and to re-evaluate them. Content, of course, is itself important, but content linked to
thinking is the key. Our complex society is demanding of the next generation not only mastery of quantitative skills, but also the
confidence to ask new questions, explore, wonder, flail, innovate and succeed. Welcome to these essays!
We will demonstrate the power of mulling on interesting mathematics and develop the art of asking questions. We will help
foster good problem solving skills, and joyfully reinforce ideas and methods from the mathematics curriculum. We will be
explicit about links with goals of the Common Core State Standards for Mathematics. Above all, we will show that deep and rich
thinking of mathematics can be just plain fun!
So on that note … Let’s get started!
CONTENTS: Here we shall:
a.
b.
c.
d.
e.
Present a problem
Discuss problem-solving methods in the context of solving that problem.
Explicitly connect the problem to the curriculum, the Core Content Standards and the Mathematical Practices.
Take the problem further.
Explore deeper curricular mathematics inspired by the problem.
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AMC 10/12 Student Practice Questions
Guide to Student Practice Questions
Each of the following four large congruent squares is
subdivided into combinations of congruent triangles or
rectangles and is partially shaded. What percent of the total
area is partially shaded?
The original problem
and choices from the
2011 AMC 8 contest
(A) 12
1
2
(B) 20
(C) 25
(D) 33
1
3
(E) 37
1
2
2011 AMC 8, Problem #7—
“Find the shaded portion of each square separately. ”
Problem number
Hint
Solution
Answer (C): The upper left and the lower right squares are each one-fourth shaded, for a total of one-half square.
The shaded portions of the upper right and lower left squares make up one-half square. So the total shaded area is
one full square, which is 25% of the total area.
Difficulty: Medium
SMP-CCSS: 2, 7
CCSS-M: 6G.1, 6RP.3C
Standards for Math Practice
Common Core State Standard
7
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Solution from official solutions
Difficulty, Percent correct
Easy 100-80%
Med Easy 80-60%
Medium
60-40%
Med Hard 40-20%
Hard 20-0%
AMC 10/12 Student Practice Questions continued
33
AMC 10/12 Student Practice Questions continued
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AMC 10/12 Student Practice Questions continued
35
AMC 10/12 Student Practice Questions continued
36
AMC 10/12 Student Practice Questions continued
David drives from his home to the airport to catch a flight. He
drives 35 miles in the first hour, but realizes that he will be 1
hour late if he continues at this speed. He increases his speed
by 15 miles per hour for the rest of the way to the airport and
arrives 30 minutes early. How many miles is the airport from
his home?
(A) 140
(B) 175
(C) 210
(D) 245
(E) 280
2014 AMC 10A, Problem #15—
“What is the remaining distance after one hour of driving, and the
remaining time until his flight?”
Solution
Answer (C): Let d be the remaining distance after one hour of driving, and let t be the remaining time until his
flight. Then d = 35(t + 1), and d = 50(t − 0.5). Solving gives t = 4 and d = 175. The total distance from home
to the airport is 175 + 35 = 210 miles.
OR
Let d be the distance between David’s home and the airport. The time required to drive the entire distance at 35
d
MPH is 35
hours. The time required to drive at 35 MPH for the first 35 miles and 50 MPH for the remaining d − 35
miles is 1 + d−35
50 . The second trip is 1.5 hours quicker than the first, so
d
−
35
1+
d − 35
50
= 1.5.
Solving yields d = 210 miles.
Difficulty: Medium Hard
SMP-CCSS: 1. Make Sense of Problems and Persevere in Solving Them, 2. Reason Abstractly and
Quantitatively.
CCSS-M: A-CED. Create equations that describe numbers or relationships.
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AMC 10/12 Student Practice Questions continued
Susie pays for 4 muffins and 3 bananas. Calvin spends twice
as much paying for 2 muffins and 16 bananas. A muffin is how
many times as expensive as a banana?
(A)
3
2
(B)
5
3
(C)
7
4
(D) 2
(E)
13
4
2014 AMC 10B, Problem #4—
“What is the cost of muffin and a banana?”
Solution
Answer (B): Let a muffin cost m dollars and a banana cost b dollars. Then 2(4m + 3b) = 2m + 16b, and
simplifying gives m = 53 b.
Difficulty: Medium Easy
SMP-CCSS: 1. Make Sense of Problems and Persevere in Solving Them, 2. Reason Abstractly and
Quantitatively.
CCSS-M: A.SSE.B. Write expressions in equivalent forms to solve problems.
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American Mathematics Competitions
The
MAA American Mathematics Competitions
are supported by
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Art of Problem Solving
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Conference Board of the Mathematical Sciences
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Expii
IDEA MATH
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MathWorks
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National Council of Teachers of Mathematics
Simons Foundation
Society for Industrial and Applied Mathematics
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Tudor Investment Corporation
Two Sigma
American Mathematics Competitions
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