2015年AIME I 真题:
Problem 1
The expressions =
and
=
are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers
and
.
Problem 2
The nine delegates to the Economic Cooperation Conference include officials from Mexico,
officials from Canada, and
officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is
, where
and
are relatively prime positive integers. Find
.
Problem 3
There is a prime number such that
is the cube of a positive integer. Find
.
Problem 4
Point lies on line segment
with
and
. Points
and
lie on the same side of line
forming equilateral triangles
and
. Let
be the midpoint of
, and
be the midpoint of
. The area of
is
. Find
.
Problem 5
In a drawer Sandy has pairs of socks, each pair a different color. On Monday, Sandy selects two individual socks at random from the
socks in the drawer. On Tuesday Sandy selects
of the remaining
socks at random, and on Wednesday two of the remaining
socks at random. The probability that Wednesday is the first day Sandy selects matching socks is
, where
and
are relatively prime positive integers. Find
.
Problem 6
Point and
are equally spaced on a minor arc of a circle. Points
and
are equally spaced on a minor arc of a second circle with center
as shown in the figure below. The angle
exceeds
by
. Find the degree measure of
.
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