Mock AIME 6 Pre 2005/Problems
. Find the remainder when
is divided by
.
. Suppose
,
, and
. Given that
and
, where
is a prime, find
.
. Suppose there are
red points and
blue points on a circle. Let
be the probability that a convex polygon whose vertices are among the
points has at least one blue vertex, where
and
are relatively prime. Find
.
. Let
be the circumcircle of
and let
,
, and
be points on sides
,
, and
, respectively, such that
,
,
,
, and
, and
,
, and
are concurrent. Let
be a point on minor arc
. Extend
to
such that
is tangent to the circle at
and
. Find
.
. Let
where
. Find the remainder when
is divided by
.
. Let
and
for
. Find the smallest
such that
is an integer.
. Let
be the region inside the graph
and to the right of the line
. If the area of
is in the form
where
is squarefree, find
.
. If
and
is real, evaluate
.
. You have
boxes and
balls.
,
, and
of these balls are blue, green, and red, respectively. Suppose the boxes are numbered
through
. You place
blue ball,
green balls, and
red balls in box
. Then
blue balls,
green balls, and
red balls in box
. Similarly, you put
blue balls,
green balls, and
red balls in box
for
. Repeat the entire process (from boxes
to
) until you run out of one color of balls. How many red balls are in boxes
,
, and
? (NOTE: After placing the last ball of a certain color in a box, you still place the balls of the other colors in that box. You do not, however, place balls in the following box.)
. There are two ants on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. If the probability that they both return to their starting position after
moves is
, where
and
are relatively prime, find
. (NOTE: They do not stop if they collide.)
. Evaluate
.
. A rigged coin has the property that when it is flipped
times the probability of getting heads
times is equal to the probability of getting heads
times. If
is the probability of getting a two heads in a row, where
and
are relatively prime, find
.
. In
,
,
, and
. Let
be on side
and
be on side
such that
is perpendicular to
and
. If
, find
. (NOTE:
denotes the area of
.)
. Let
be a polynomial of degree
with leading coefficient
such that for
,
. Find the number of
's at the end of
.
. An AIME has
questions,
of each of three difficulties: easy, medium, and hard. Let
denote the number of easy questions up to question
(including question
). Similarly define
and
. Let
be the number of ways to arrange the questions in the AIME such that, for any
,
and if a easy and hard problem are consecutive, the easy always comes first. Find the remainder when
is divided by
.