Difference between revisions of "Mock AIME 1 2013 Problems"
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== Problem 2 == | == Problem 2 == | ||
− | Find the number of ordered positive integer | + | Find the number of ordered positive integer triplets <math>(a,b,c)</math> such that <math>a</math> evenly divides <math>b</math>, <math>b+1</math> evenly divides <math>c</math>, and <math>c-a=10</math>. |
[[2013 Mock AIME I Problems/Problem 2|Solution]] | [[2013 Mock AIME I Problems/Problem 2|Solution]] |
Revision as of 22:58, 23 July 2020
Contents
Problem 1
Two circles and
, each of unit radius, have centers
and
such that
. Let
be the midpoint of
and let
be a circle externally tangent to both
and
.
and
have a common tangent that passes through
. If this tangent is also a common tangent to
and
, find the radius of circle
.
Problem 2
Find the number of ordered positive integer triplets such that
evenly divides
,
evenly divides
, and
.
Problem 3
Let be the greatest integer less than or equal to
, and let
. If
, compute
.
Problem 4
Compute the number of ways to fill in the following magic square such that:
1. the product of all rows, columns, and diagonals are equal (the sum condition is waived),
2. all entries are nonnegative integers less than or equal to ten, and
3. entries CAN repeat in a column, row, or diagonal.
Problem 5
In quadrilateral ,
. Also,
, and
. The perimeter of
can be expressed in the form
where
and
are relatively prime, and
is not divisible by the square of any prime number. Find
.
Problem 6
Find the number of integer values can have such that the equation
has a solution.
Problem 7
Let be the set of all
th primitive roots of unity with imaginary part greater than
. Let
be the set of all
th primitive roots of unity with imaginary part greater than
. (A primitive
th root of unity is a
th root of unity that is not a
th root of unity for any
.)Let
. The absolute value of the real part of
can be expressed in the form
where
and
are relatively prime numbers. Find
.
Problem 8
Let and
be two perpendicular vectors in the
plane. If there are
vectors
for
in the same plane having projections of
and
along
and
respectively, then find
(Note:
and
are unit vectors such that
and
, and the projection of a vector
onto
is the length of the vector that is formed by the origin and the foot of the perpendicular of
onto
.)
Problem 9
In a magic circuit, there are six lights in a series, and if one of the lights short circuit, then all lights after it will short circuit as well, without affecting the lights before it. Once a turn, a random light that isn’t already short circuited is short circuited. If is the expected number of turns it takes to short circuit all of the lights, find
.
Problem 10
Let denote the
th triangular number, i.e.
. Let
and
be relatively prime positive integers so that
Find
.
Problem 11
Let and
be the roots of the equation
, and let
and
be the two possible values of
Find
.
Problem 12
In acute triangle , the orthocenter
lies on the line connecting the midpoint of segment
to the midpoint of segment
. If
, and the altitude from
has length
, find
.
Problem 13
In acute ,
is the orthocenter,
is the centroid, and
is the midpoint of
. It is obvious that
, but
does not always hold. If
,
, then the value of
which produces the smallest value of
such that
can be expressed in the form
, for
squarefree. Compute
.
Problem 14
Let If
are its roots, then compute the remainder when
is divided by 997.
Problem 15
Let be the set of integers
such that
for all integers
. Compute the remainder when the sum of the elements in
is divided by
.