2010 AMC 12A Problems and Solution
Problem 1
What is
?
Solution
.
Problem 2
A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?
Solution
It is easy to see that the ferry boat takes trips total. The total number of people taken to the island is
Problem 3
Rectangle Square
, pictured below, shares shares
of its area with square
. What is
?
.
of its area with rectangle
Solution Solution 1
Let
, let
, and let
.
Solution 2
The answer does not change if we shift horizontal lines to divide
to coincide with
, and add new
into five equal parts:
This helps us to see that
.
and
, where
. Hence
Problem 4
If
, then which of the following must be positive?
Solution
is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use
.
Obviously only
is positive.
Problem 5
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next shots are bullseyes she will be guaranteed victory. What is the minimum value for ?
Solution
Let be the number of points Chelsea currently has. In order to guarantee victory, we must consider the possibility that the opponent scores the maximum amount of points by getting only bullseyes.
The lowest integer value that satisfies the inequality is
.
Problem 6
A
, such as 83438, is a number that remains the same when its digits
are three-digit and four-digit palindromes,
are reversed. The numbers and
respectively. What is the sum of the digits of ?
Solution
is at most
must be
It follows that is
, so
.
, so the sum of the digits is
.
is at most
. The minimum value of and
is
is
.
However, the only palindrome between , which means that
Problem 7
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
Solution
The water tower holds
times more water than Logan's
times shorter .
miniature. Therefore, Logan should make his tower than the actual tower. This is Also, the fact that
meters high, or choice
doesn't matter since only the ratios are important.
Problem 8
Triangle such that
and suppose that
has
. Let . Let
and
be on
and ?
, respectively,
and
,
be the intersection of segments
is equilateral. What is
Solution
Let
.
Since , triangle is a triangle, so
Problem 9
A solid cube has side length inches. A -inch by -inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
Solution Solution 1
Imagine making the cuts one at a time. The first cut removes a box second cut removes two boxes, each of dimensions cuts is
.
. . The
, and the third cut
does the same as the second cut, on the last two faces. Hence the total volume of all
Therefore the volume of the rest of the cube is