Northeastern WOOTers Mock AIME I Problems
Contents
Problem 1
Let ,
,
, and
be digits, non necessarily distinct and non necessarily non-zero. For how many quadruples
is it true that
is an integer? As an example, if
, then we have
, which is not an integer.
Problem 2
It is given that can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.
Problem 3
Let be a triangle with
,
, and
. Let
be the midpoint of
. Let
and
be trisection points on
. That is,
. Let
and
be the points of intersection of
and
with the cevian
, respectively. Find the area of quadrilateral
.
Problem 4
Let the number of ordered tuples of positive odd integers such that
be
. Find the remainder when
is divided by
.
Problem 5
Let ,
, and
be real numbers. Given that
, the maximum value of
can be represented
, where
and
are positive integers, where
and
are relatively prime. Find
.
Problem 6
Let . Two subsets,
and
, of
are chosen randomly with replacement, with
chosen after
. The probability that
is a subset of
can be written as
, for some primes
and
. Find
.
Problem 7
Find the value of
Problem 8
Dai the Luzon bleeding-heart has numbered lillypads, through
. Then, Ryan the alligator eats those lillypads with the intention of eating Dai. Dai starts on a random lillypad and flies around between lillypads randomly every minute. Ryan also eats a random lillypad every minute. If the expected number of minutes left for Dai to live is
, where
and
are relatively prime positive integers, find
.
Problem 9
Let be a regular hexagon of unit side length. Line
is extended to a point
outside of the hexagon such that
. The line
intersects the lines
and
at points
and
, respectively. Let the area of quadrilateral
be
. Then, the value of
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 10
If are complex numbers such that
then find the value of
.
Problem 11
Cody and Toedy play a game. Cody guesses an integer between and
inclusive, and then Toedy guesses a different integer. A 2014-sided die is rolled. After rolling, whoever guessed closest to the number on the die wins. The winning player wins as much money as the number rolled. What number should Cody guess to maximize his earnings?
Problem 12
Let be a triangle with
,
, and
. A point
is placed on the extension of
past
. Let
and
be the circumcenters of
and
respectively. If
, then the ratio
can be written in the form
for relatively prime positive integers
and
. Find
.
Problem 13
Define a T-Polyomino to be a set of 4 cells in a grid that form a T, as shown below. Dai wants to place T-Polyominos onto a grid such that there is no overlap. He continues to place T-Polyominos randomly until he can no longer do so. Let the probability that he will cover the entire board be
. Then,
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Problem 14
Consider three infinite sequences of real numbers: It is known that, for all integers
, the following statement holds:
The elements of
are defined by the relation
. Let
Then,
can be represented as a fraction
, where
and
are relatively prime positive integers. Find
.
Problem 15
Find the sum of all integers such that
where
denotes the number of integers less than or equal to
that are relatively prime to
.