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2016 AMC12 A
Problem 1
What is the value of
?
Solution
Problem 2
For what value of
does
?
Solution
Problem
3
The remainder can be defined for all
real numbers
and
with
by
where
What is
the value of
?
denotes the
greatest integer less than or equal to
.
Solution
Problem
4
The mean, median, and mode of the
data values
to
.
What is the value of
?
Solution
Problem
5
Goldbach's conjecture states that
every even integer greater than 2 can be written
as the sum
of two prime numbers (for
example,
). So far, no one has been
able to
prove that the conjecture is
true, and no one has found a counterexample to
show that the
conjecture is false. What
would a counterexample consist of?
are
all equal
Solution
Problem 6
A triangular array
of
coins has
coin in the
first row,
coins in the second row,
coins
in the third row, and
so on up to
coins in the
th
row. What is the sum of the digits of
?
Solution
Problem 7
Which of these
describes the graph of
?
Solution
Problem
8
What is the area of the shaded region
of the given
rectangle?
Solution
Problem 9
The five small
shaded squares inside this unit square are
congruent and have disjoint interiors.
The midpoint of each side of the middle
square coincides with one of the vertices of the
other
four small squares as shown. The
common side length is
integers. What is
?
, where
and
are positive
Solution
Problem
10
Five friends sat in a movie theater
in a row containing
seats, numbered
to
from left to
right. (The directions
seats.) During the movie Ada went to
the lobby to get some popcorn. When she returned,
she
found that Bea had moved two seats
to the right, Ceci had moved one seat to the left,
and
Dee and Edie had switched seats,
leaving an end seat for Ada. In which seat had Ada
been
sitting before she got up?
Solution
Problem 11
Each of the
students in a certain summer camp can
either sing, dance, or act. Some
students have more than one talent, but
no student has all three talents. There
are
students who cannot
sing,
students who cannot dance, and
students who
cannot act.
How many students have two of these talents?
Solution
Problem 12
In
,
,
, and
and
bisects
. Point
lies on
intersect at
. What
is the ratio
:
?
. Point
, and
lies on
bisects
,
. The bisectors
Solution
Problem
13
Let
be a positive
multiple of
. One red ball and
green balls are arranged in a line in
random order. Let
be the
probability that at least
of the green
balls are on the same
and that
approaches
as
?
grows large.
side of the
red ball. Observe that
What is the sum
of the digits of the least value of
such that
Solution
Problem
14
Each vertex of a cube is to be
labeled with an integer from
through
, with each integer
being
used once, in such a way that the sum of the four
numbers on the vertices of a face is
the same for each face. Arrangements
that can be obtained from each other through
rotations
of the cube are considered to
be the same. How many different arrangements are
possible?
Solution
Problem
15
Circles with centers
and
, having radii
and
, respectively, lie on the same side of
line
and are tangent to
at
and
,
respectively, with
between
and
. The