Difference between revisions of "User:Rowechen"
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draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));</asy> | draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));</asy> | ||
[[2014 AIME I Problems/Problem 1|Solution]] | [[2014 AIME I Problems/Problem 1|Solution]] | ||
+ | |||
+ | ==Problem 5== | ||
Compute, to the nearest integer, the area of the region enclosed by the graph of | Compute, to the nearest integer, the area of the region enclosed by the graph of |
Revision as of 09:02, 30 May 2020
Here's the AIME compilation I will be doing:
Contents
Problem 1
The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
Problem 5
Compute, to the nearest integer, the area of the region enclosed by the graph of
Problem 6
A flat board has a circular hole with radius and a circular hole with radius
such that the distance between the centers of the two holes is
. Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is
, where
and
are relatively prime positive integers. Find
.
Problem 9
Let and
be real numbers such that
and
. The value of
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 9
Let be the three real roots of the equation
. Find
.
Problem 9
Let be the set of all ordered triple of integers
with
. Each ordered triple in
generates a sequence according to the rule
for all
. Find the number of such sequences for which
for some
.
Problem 12
Suppose that the angles of satisfy
. Two sides of the triangle have lengths 10 and 13. There is a positive integer
so that the maximum possible length for the remaining side of
is
. Find
.
Problem 11
Consider arrangements of the numbers
in a
array. For each such arrangement, let
,
, and
be the medians of the numbers in rows
,
, and
respectively, and let
be the median of
. Let
be the number of arrangements for which
. Find the remainder when
is divided by
.
Problem 10
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path
, which has
steps. Let
be the number of paths with
steps that begin and end at point
Find the remainder when
is divided by
.
Problem 11
Find the least positive integer such that when
is written in base
, its two right-most digits in base
are
.
Problem 14
In ,
,
, and
. Let
,
, and
be points on line
such that
,
, and
. Point
is the midpoint of segment
, and point
is on ray
such that
. Then
, where
and
are relatively prime positive integers. Find
.
Problem 14
For each integer , let
be the area of the region in the coordinate plane defined by the inequalities
and
, where
is the greatest integer not exceeding
. Find the number of values of
with
for which
is an integer.
Problem 13
Let have side lengths
,
, and
. Point
lies in the interior of
, and points
and
are the incenters of
and
, respectively. Find the minimum possible area of
as
varies along
.
Problem 15
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed in a circle with radius
. Let
denote the measure of the acute angle made by the diagonals of quadrilateral
, and define
and
similarly. Suppose that
,
, and
. All three quadrilaterals have the same area
, which can be written in the form
, where
and
are relatively prime positive integers. Find
.
Problem 13
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is where
and
are relatively prime positive integers. Find
.