Difference between revisions of "2007 AIME I Problems"
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== Problem 2 == | == Problem 2 == | ||
− | A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of | + | A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person. |
[[2007 AIME I Problems/Problem 2|Solution]] | [[2007 AIME I Problems/Problem 2|Solution]] | ||
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== Problem 3 == | == Problem 3 == |
Revision as of 18:27, 26 June 2020
2007 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
How many positive perfect squares less than are multiples of 24?
Problem 2
A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.
Problem 3
The complex number is equal to
, where
is a positive real number and
. Given that the imaginary parts of
and
are the same, what is
equal to?
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
The formula for converting a Fahrenheit temperature to the corresponding Celsius temperature
is
An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.
For how many integer Fahrenheit temperatures between and
inclusive does the original temperature equal the final temperature?
Problem 6
A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of , or to the closest point with a greater integer coordinate that is a multiple of
. A move sequence is a sequence of coordinates which correspond to valid moves, beginning with
, and ending with
. For example,
is a move sequence. How many move sequences are possible for the frog?
Problem 7
Let
Find the remainder when is divided by 1000. (
is the greatest integer less than or equal to
, and
is the least integer greater than or equal to
.)
Problem 8
The polynomial is cubic. What is the largest value of
for which the polynomials
and
are both factors of
?
Problem 9
In right triangle with right angle
,
and
. Its legs
and
are extended beyond
and
. Points
and
lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center
is tangent to the hypotenuse and to the extension of leg
, the circle with center
is tangent to the hypotenuse and to the extension of leg
, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as
, where
and
are relatively prime positive integers. Find
.
Problem 10
In a grid (
rows,
columns),
of the
squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let
be the number of shadings with this property. Find the remainder when
is divided by
.
Problem 11
For each positive integer , let
denote the unique positive integer
such that
. For example,
and
. If
find the remainder when
is divided by 1000.
Problem 12
In isosceles triangle ,
is located at the origin and
is located at (20,0). Point
is in the first quadrant with
and angle
. If triangle
is rotated counterclockwise about point
until the image of
lies on the positive
-axis, the area of the region common to the original and the rotated triangle is in the form
, where
are integers. Find
.
Problem 13
A square pyramid with base and vertex
has eight edges of length 4. A plane passes through the midpoints of
,
, and
. The plane's intersection with the pyramid has an area that can be expressed as
. Find
.
Problem 14
A sequence is defined over non-negative integral indexes in the following way: ,
.
Find the greatest integer that does not exceed
Problem 15
Let be an equilateral triangle, and let
and
be points on sides
and
, respectively, with
and
. Point
lies on side
such that angle
. The area of triangle
is
. The two possible values of the length of side
are
, where
and
are rational, and
is an integer not divisible by the square of a prime. Find
.
See also
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.