Difference between revisions of "2000 AIME I Problems"
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== Problem 9 == | == Problem 9 == | ||
The system of equations | The system of equations | ||
| − | < | + | <cmath>\begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ |
\log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ | \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ | ||
\log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ | \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ | ||
| − | \end{eqnarray*}</ | + | \end{eqnarray*}</cmath> |
has two solutions <math>(x_{1},y_{1},z_{1})</math> and <math>(x_{2},y_{2},z_{2})</math>. Find <math>y_{1} + y_{2}</math>. | has two solutions <math>(x_{1},y_{1},z_{1})</math> and <math>(x_{2},y_{2},z_{2})</math>. Find <math>y_{1} + y_{2}</math>. | ||
Revision as of 19:35, 1 March 2015
| 2000 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
| ||
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.Contents
Problem 1
Find the least positive integer 
such that no matter how 
is expressed as the product of any two positive integers, at least one of these two integers contains the digit 
.
Problem 2
Let 
and 
be integers satisfying 
. Let 
, let 
be the reflection of 
across the line 
, let 
be the reflection of across the y-axis, let 
be the reflection of across the x-axis, and let 
be the reflection of across the y-axis. The area of pentagon 
is 
. Find 
.
Problem 3
In the expansion of 
where 
and 
are relatively prime positive integers, the coefficients of 
and 
are equal. Find 
.
Problem 4
The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
![[asy]draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]](http://latex.artofproblemsolving.com/d/8/1/d8140d628cc0905f333aa8deaface1b89ee6c983.png)
Problem 5
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is 
One marble is taken out of each box randomly. The probability that both marbles are black is 
and the probability that both marbles are white is 
where 
and are relatively prime positive integers. What is 
?
Problem 6
For how many ordered pairs 
of integers is it true that 
and that the arithmetic mean of 
and 
is exactly 
more than the geometric mean of and ?
Problem 7
Suppose that 
and 
are three positive numbers that satisfy the equations 
and 
Then 
where and are relatively prime positive integers. Find .
Problem 8
A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is ![$m - n\sqrt [3]{p},$](http://latex.artofproblemsolving.com/1/5/8/1586f2bca69882cdcb09b653822a61d6a5931db7.png)
where 
and 
are positive integers and is not divisible by the cube of any prime number. Find 
.
Problem 9
The system of equations
has two solutions 
and 
. Find 
.
Problem 10
A sequence of numbers 
has the property that, for every integer 
between 
and 
inclusive, the number 
is less than the sum of the other 
numbers. Given that 
where and are relatively prime positive integers, find .
Problem 11
Let 
be the sum of all numbers of the form 
where and are relatively prime positive divisors of 
What is the greatest integer that does not exceed 
?
Problem 12
Given a function 
for which
holds for all real what is the largest number of different values that can appear in the list 
?
Problem 13
In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at 
miles per hour along the highways and at 
miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is 
square miles, where and are relatively prime positive integers. Find .
Problem 14
In triangle 
it is given that angles and are congruent. Points 
and 
lie on 
and 
respectively, so that 
Angle 
is 
times as large as angle 
where is a positive real number. Find the greatest integer that does not exceed 
.
Problem 15
A stack of 
cards is labelled with the integers from to 
with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: 
In the original stack of cards, how many cards were above the card labeled 1999?
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. 
