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2011 AMC 12A Problems
1. A cell phone plan costs plus
messages and talked for
dollars each month, plus cents per text message sent,
hours. In January Michelle sent
text
hours. How much did she have to pay?
2. There are coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?
cents for each minute used over
3. A small bottle of shampoo can hold bottle can hold
milliliters of shampoo, whereas a large
milliliters of shampoo. Jasmine wants to buy the minimum
number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
4. At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of
,
, and
minutes per day, respectively. There are twice as
many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
5. Last summer were herons, and were geese?
6. The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one
1
of the birds living on Town Lake were geese, were swans,
were ducks. What percent of the birds that were not swans
more than their number of successful two-point shots. The team's total score was points. How many free throws did they make?
7. A majority of the
students in Ms. Demeanor's class bought pencils at the
school bookstore. Each of these students bought the same number of pencils, and this number was greater than . The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was
. What was the cost of a pencil in cents?
8. In the eight term sequence
,
,
,
,
,
,
,
, the value of
?
9. At a twins and triplets convention, there were sets of twins and sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?
10. A pair of standard -sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
is and
the sum of any three consecutive terms is . What is
11. Circles tangency. Circle area inside circle
and
each have radius 1. Circles
and circle
and
share one point of
What is the
has a point of tangency with the midpoint of but outside circle
2
12. A power boat and a raft both left dock
on a river and headed downstream. The
downriver, then
raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock 9 hours after leaving dock to
13. Triangle
has side-lengths
parallel to
14. Suppose and are single-digit positive integers chosen independently and at random. What is the probability that the point
?
lies above the parabola
and
intersects
at
and
The line
at
immediately turned and traveled back upriver. It eventually met the raft on the river
How many hours did it take the power boat to go from
through the incenter of What is the perimeter of
15. The circular base of a hemisphere of radius rests on the base of a square pyramid of height . The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid?
16. Each vertex of convex polygon
How many different colorings are possible?
3
is to be assigned a color. There are
colors to choose from, and the ends of each diagonal must have different colors.
17. Circles with radii , , and are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
18. Suppose that
?
19. At a competition with equal to
players, the number of players given elite status is . Suppose that
players are given elite status. What
?
20. Let
,
,
, where , , and are integers. Suppose that
,
for
. What is the maximum possible value of
is the sum of the two smallest possible values of
some integer . What is ?
21. Let
, and for integers
, let
. If
is nonempty, the domain of
is the largest value of for which the domain of is
. What is
?
22. Let
be a square region and
an integer. A point rays emanating from
in the interior or that divide
into -ray
is
called n-ray partitional if there are partitional?
triangles of equal area. How many points are -ray partitional but not
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23. Let Suppose that
and and
, where and are complex numbers. for all for which
is defined. What is
?
the difference between the largest and smallest possible values of
24. Consider all quadrilaterals and
such that
,
,
,
. What is the radius of the largest possible circle that fits inside or on
the boundary of such a quadrilateral?
25. Triangle Let
, , and
has
,
,
, and
,
.
be the orthocenter, incenter, and circumcenter of ?
repsectively. Assume that the area of pentagon What is
is the maximum possible.
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