Difference between revisions of "User:Rowechen"
(→Problem 5) |
|||
Line 10: | Line 10: | ||
[[2007 AIME I Problems/Problem 4|Solution]] | [[2007 AIME I Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | + | If | |
<cmath>\frac{1}{0!10!}+\frac{1}{1!9!}+\frac{1}{2!8!}+\frac{1}{3!7!}+\frac{1}{4!6!}+\frac{1}{5!5!}</cmath> | <cmath>\frac{1}{0!10!}+\frac{1}{1!9!}+\frac{1}{2!8!}+\frac{1}{3!7!}+\frac{1}{4!6!}+\frac{1}{5!5!}</cmath> |
Revision as of 15:04, 31 May 2020
Here's the AIME compilation I will be doing:
Contents
Problem 3
A triangle has vertices ,
, and
. The probability that a randomly chosen point inside the triangle is closer to vertex
than to either vertex
or vertex
can be written as
, where
and
are relatively prime positive integers. Find
.
Problem 4
Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are ,
, and
years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all be collinear again?
Problem 5
If
is written as a common fraction reduced to lowest terms, the result is . Compute the sum of the prime divisors of
plus the sum of the prime divisors of
.
Problem 9
Let , and for each integer
let
. Find the least
such that
is a multiple of
.
Problem 8
Two real numbers and
are chosen independently and uniformly at random from the interval
. Let
and
be two points on the plane with
. Let
and
be on the same side of line
such that the degree measures of
and
are
and
respectively, and
and
are both right angles. The probability that
is equal to
, where
and
are relatively prime positive integers. Find
.
Problem 7
Triangle has side lengths
,
, and
. Points
are on segment
with
between
and
for
, and points
are on segment
with
between
and
for
. Furthermore, each segment
,
, is parallel to
. The segments cut the triangle into
regions, consisting of
trapezoids and
triangle. Each of the
regions has the same area. Find the number of segments
,
, that have rational length.
Problem 10
Find the number of functions from
to
that satisfy
for all
in
.
Problem 11
Find the number of permutations of such that for each
with
, at least one of the first
terms of the permutation is greater than
.
Problem 14
The incircle of triangle
is tangent to
at
. Let
be the other intersection of
with
. Points
and
lie on
and
, respectively, so that
is tangent to
at
. Assume that
,
,
, and
, where
and
are relatively prime positive integers. Find
.
Problem 10
Four lighthouses are located at points ,
,
, and
. The lighthouse at
is
kilometers from the lighthouse at
, the lighthouse at
is
kilometers from the lighthouse at
, and the lighthouse at
is
kilometers from the lighthouse at
. To an observer at
, the angle determined by the lights at
and
and the angle determined by the lights at
and
are equal. To an observer at
, the angle determined by the lights at
and
and the angle determined by the lights at
and
are equal. The number of kilometers from
to
is given by
, where
,
, and
are relatively prime positive integers, and
is not divisible by the square of any prime. Find
.
Problem 11
lines and
circles divide the plane into at most
disjoint regions. Compute
.
Problem 15
Find the number of functions from
to the integers such that
,
, and
for all and
in
.
Problem 14
The sequence satisfies
and
for
. Find the greatest integer less than or equal to
.
Problem 15
Let be a diameter of a circle with diameter
. Let
and
be points on one of the semicircular arcs determined by
such that
is the midpoint of the semicircle and
. Point
lies on the other semicircular arc. Let
be the length of the line segment whose endpoints are the intersections of diameter
with the chords
and
. The largest possible value of
can be written in the form
, where
,
, and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 14
Let and
be real numbers satisfying
and
. Evaluate
.