50 AMC Lectures
Chapter 10 Area And Area Method
BASIC KNOWLEDGE
1. FORMULAS
1. S
1
1
1
aha = bhb = chc
2
2
2
(1)
2. Let ha b sin C, hb c sin A, and hc a sin B. Equation (1) becomes:
S
1
1
1
bc sin A = ac sinB = ab sinC
2
2
2
(2)
3. S s(s a)(s b)(s c)
s=
MATHCOUNTS
2006
Chapter Competition
Answer Key
The appropriate units (or their abbreviations) are
provided in the answer blanks.
Note to coordinators: Answers to the Tiebreaker Round
problems appear in the Tiebreaker Round Booklet.
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2016 State Competition Solutions
Are you wondering how we could have possibly thought that a Mathlete would be able
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The MATHEMATICAL ASSOCIATION of AMERICA
American Mathematics Competitions
9th Annual American Mathematics Contest 10
AMC 10
Contest A
Solutions Pamphlet
Tuesday, February 12, 2008
This Pamphlet gives at least one solution for each problem on this years co
Mathcounts Chapter Competition 2001
2001 CHAPTER COMPETITION
SPRINT ROUND QUESTIONS
2.
3.
4.
Th
5.
40 hours of work @ $30 per hour =
40 $30 = $1200 Answer
The probability that it will not rain
tomorrow = 1 - the probability that it will
rain tomorrow. The
1999 Mathcounts National Sprint Round Solutions
5
.
12
A 3-digit number is divisible by 3 if the sum of its digits is divisible by 3.
The first digit cannot be 0, so we have the following four groups of 3 such that the three
different numbers sum to a mul
MC 11
(1)
A restaurant raised its price for a hamburger 20% to $1.50. What did the
hamburger cost before?
(2)
Sara buys 20 dinosaurs and 30 alligators from a pet store. She then sells
each dinosaur for $21.50 and each alligator for $36.70. If Sara sells a
MC 12
(1)
Twenty 11 17 sheets were printed on both sides so that when the
sheets were stacked and folded down the center, an 8 12 11 booklet with the pages
numbered 1-80 would be formed. When stacked and folded, however, the sheets containing
pages 12 and
2016
State Competition
Answer Key
The appropriate units (or their abbreviations) are provided in
the answer blanks.
Note to coordinators: Answers to the Tiebreaker Round
problems appear in the Tiebreaker Round Booklet.
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N
MATHCOUNTS
2004
Chapter Competition
Sprint Round
Problems 130
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of 30 problems. You will
have 40 minutes to complete the problems. You are not
allowed to use
MATHCOUNTS.
2001
n
Chapter Competition I
Sprint Round
Problems l-30
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of 30 problems.
You will have 40 minutes to complete the problems. You
are not allowed t
2016 Chapter Competition Solutions
Are you wondering how we could have possibly thought that a Mathlete would be able
to answer a particular Sprint Round problem without a calculator?
Are you wondering how we could have possibly thought that a Mathlete wo
2015 Chapter Competition Solutions
Are you wondering how we could have possibly thought that a Mathlete would be able to
answer a particular Sprint Round problem without a calculator?
Are you wondering how we could have possibly thought that a Mathlete wo
1. Raquel has collected $3.80 in nickels and dimes. She has
exactly 48 nickels. How many dimes does she have?
1. _
2. One of the following four-digit numbers is not divisible by 4:
3544, 3554, 3564, 3572, 3576. What is the product of the
units digit and t
2005
Chapter Competition
Sprint Round
Problems 130
Name
School
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO.
This round of the competition consists of 30 problems.
You will have 40 minutes to complete the problems. You
are not allowed to use calculators
1
1. Tami had 1800 baseball cards. She sold 2 of them and then
4
gave away 5 of her remaining ones. How many cards does
Tami have left?
1. _
2. During the 20th century, eighteen different men held the office
of President of the United States. Eleven of th
2016
Chapter Competition
Answer Key
The appropriate units (or their abbreviations) are provided
in the answer blanks.
Note to coordinators: Answers to the Tiebreaker Round
problems appear in the Tiebreaker Round Booklet.
National Sponsors
Raytheon Company
MC 1
(1)
The front and back covers of a book are each 3 mm thick, and a stack of 25
sheets of paper is 4 mm thick. Als book is 38 mm thick, and all of the pages are numbered
front and back starting with page 1. What is the number of the last page facing t
MC 3
(1)
(2)
(3)
If 120 is divided into 3 parts which are proportional to 1, 1/2, 1/6, what is
the middle part?
If b3 = .25, what is the value of b3 ?
Two drivers leave city A at the same time for city B which is 450 miles
away. The first driver travels a
MC 8
(1)
At a new job, Bobs starting monthly salary is $1000, and he gets a $20 raise
each succeeding month. In total, how much does Bob earn in his first year at this job?
(2)
When the digits 2 through 9 are placed in the squares below, one digit per
squ
MC 5
(1)
Alexs uncle paid Alex $5 a week for doing a few minor chores. However, if
Alex failed to do all the chores in any week, he did not get paid for that week and he had
to give his uncle $7.50 for that week. During the first 52 weeks, Alex failed to
MC 2
(1)
Leila is preparing boxed lunches for a group of 150 people. The lunches are
the same except for the sandwiches. The four sandwich options are ham, turkey, roast beef
or vegetarian. From past experience, she knows she will need 30% turkey, 31 as m
MC 4
(1)
Jasmine had 3 paperclips on Monday, then she had 6 on Tuesday, and her
number of paperclips proceeded to double on each subsequent day. On what day of the
week did she first have more than 100 paperclips?
(2)
Wally buys a baseball card for $5.00,
MC 7
(1)
Two hundred fifty eighth-graders took a test. Eighty percent of them
passed. One-fifth of those who did not pass received a score below 60. One-half of those
below 60 scored below 50. How many students scored below 50?
(2)
Let p(x) = 2x 1, where
MC 6
(1)
Of the five points (3, 10), (6, 20), (12, 35), (18, 40) and (20, 50), what is
the sum of the x-coordinates of the points that lie in the region above the line y = 2x + 7
in the coordinate plane?
(2)
On a recent math test, the mean score for Mr. B
MC 10
(1)
A triangle has side lengths of 8, 15 and 17 units. What is the area of the
triangle, in square units?
(2)
Solve for x:
2x =
5 .
2x
(3)
The rectangular plot with largest possible area has a perimeter of 86 feet.
The plot has whole number dimensio
MC 9
(1)
A 10-foot diameter carpet costs $1350. If cost is proportional to surface
area, what is the number of dollars you should expect to pay for a similar 16-foot diameter
carpet?
(2)
The sum of the cubes of two different prime numbers is 5256. What ar