Difference between revisions of "Mock AIME I 2015 Problems"
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Contents
Problem 1
David, Justin, Richard, and Palmer are demonstrating a "math magic" concept in front of an audience. There are four boxes, labeled A, B, C, and D, and each one contains a different number. First, David pulls out the numbers in boxes A and B and reports that their product is . Justin then claims that the product of the numbers in boxes B and C is
, and Richard states the product of the numbers in boxes C and D to be
. Finally, Palmer announces the product of the numbers in boxes D and A. If
is the number that Palmer says, what is
?
Problem 2
Suppose that and
are real numbers such that
and
. The value of
can be expressed in the form
where
and
are positive relatively prime integers. Find
.
Problem 3
Let be points in the plane such that
,
, and
. Semicircles with diameters
and
intersect at a point
with
. Find the length of line segment
.
Problem 4
At the AoPS Carnival, there is a "Weighted Dice" game show. This game features two identical looking weighted 6 sided dice. For each integer , Die A has
probability of rolling the number
, while Die B has a
probability of rolling
. During one session, the host randomly chooses a die, rolls it twice, and announces that the sum of the numbers on the two rolls is
. Let
be the probability that the die chosen was Die A. When
is written as a fraction in lowest terms, find the sum of the numerator and denominator.
Problem 5
In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the probability that the two marbles are of opposite color is . Let
be the
smallest possible values for the total number of marbles in the urn. Compute the remainder when
is divided by
.
Problem 6
Find the number of digit numbers using only the digits
such that every pair of adjacent digits is no more than
apart. For instance,
and
are acceptable numbers, while
and
are not.
Problem 7
For all points in the coordinate plane, let
denote the reflection of
across the line
. For example, if
, then
. Define a function
such that for all points
,
denotes the area of the triangle with vertices
,
, and
. Determine the number of lattice points
in the first quadrant such that
.
Problem 8
Let be consecutive terms (in that order) in an arithmetic sequence with common difference
. Suppose
and
are roots of a monic quadratic
with
. Then
for positive relatively prime integers
and
. Find the remainder when
is divided by
.
Problem 9
Compute the number of positive integer triplets with
that satisfy the following properties:
(a) is a perfect square,
(b) is a power of
,
(c) is a multiple of
.
Problem 10
Let be a function defined along the rational numbers such that
for all relatively prime positive integers
and
. The product of all rational numbers
such that
can be written in the form
for positive relatively prime integers
and
. Find
.
Problem 11
Suppose ,
, and
are complex numbers that satisfy the system of equations
If
for positive relatively prime integers
and
, find
.
Problem 12
Alpha and Beta play a game on the number line below.
Both players start at
. Each turn, Alpha has an equal chance of moving
unit in either the positive or negative directions while Beta has a
chance of moving
unit in the positive direction and a
chance of moving
unit in the negative direction. The two alternate turns with Alpha going first. If a player reaches
at any point in the game, he wins; however, if a player reaches
, he loses and the other player wins. If
is the probability that Alpha beats Beta, where
and
are relatively prime positive integers, find
.
Problem 13
Let be a hexagon inscribed inside a circle of radius
. Furthermore, for each positive integer
let
be the midpoint of the segment
, where
. Suppose that hexagon
can also be inscribed inside a circle. If
and
, then
can be written in the form
where
and
are positive relatively prime integers. Find
.
Problem 14
Consider a set of pennies laid out in the formation of an equilateral triangle with "side length"
. You wish to move some of the pennies so that the triangle is flipped upside down. For example, with
, you could take the top penny and move it to the bottom to accomplish this task, as shown:
Let be the minimum number of pennies for which this can be done in terms of
. Find
.
Problem 15
Let be a triangle with
,
, and
. Let
denote its circumcenter and
its orthocenter. The circumcircle of
intersects
and
at
and
respectively. Suppose
where
and
are positive relatively prime integers. Find
.