Difference between revisions of "1997 AIME Problems"
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== Problem 14 == | == Problem 14 == | ||
− | Let <math>v</math> and <math>w</math> be | + | Let <math>v</math> and <math>w</math> be distinct, randomly chosen roots of the equation <math>z^{1997}-1=0</math>. Let <math>m/n</math> be the probability that <math>\sqrt{2+\sqrt{3}}\le |v+w|</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
[[1997 AIME Problems/Problem 14|Solution]] | [[1997 AIME Problems/Problem 14|Solution]] |
Revision as of 11:40, 1 January 2019
1997 AIME (Answer Key) | AoPS Contest Collections | ||
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 of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?
Problem 2
The nine horizontal and nine vertical lines on an checkerboard form rectangles, of which are squares. The number can be written in the form where and are relatively prime positive integers. Find
Problem 3
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?
Problem 4
Circles of radii 5, 5, 8, and are mutually externally tangent, where and are relatively prime positive integers. Find
Problem 5
The number can be expressed as a four-place decimal where and represent digits, any of which could be zero. It is desired to approximate by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to is What is the number of possible values for ?
Problem 6
Point is in the exterior of the regular -sided polygon , and is an equilateral triangle. What is the largest value of for which , , and are consecutive vertices of a regular polygon?
Problem 7
A car travels due east at miles per minute on a long, straight road. At the same time, a circular storm, whose radius is miles, moves southeast at mile per minute. At time , the center of the storm is miles due north of the car. At time minutes, the car enters the storm circle, and at time minutes, the car leaves the storm circle. Find .
Problem 8
How many different arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?
Problem 9
Given a nonnegative real number , let denote the fractional part of ; that is, , where denotes the greatest integer less than or equal to . Suppose that is positive, , and . Find the value of .
Problem 10
Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:
i. Either each of the three cards has a different shape or all three of the card have the same shape.
ii. Either each of the three cards has a different color or all three of the cards have the same color.
iii. Either each of the three cards has a different shade or all three of the cards have the same shade.
How many different complementary three-card sets are there?
Problem 11
Let . What is the greatest integer that does not exceed ?
Problem 12
The function defined by , where ,, and are nonzero real numbers, has the properties , and for all values except . Find the unique number that is not in the range of .
Problem 13
Let be the set of points in the Cartesian plane that satisfy
If a model of were built from wire of negligible thickness, then the total length of wire required would be , where and are positive integers and is not divisible by the square of any prime number. Find .
Problem 14
Let and be distinct, randomly chosen roots of the equation . Let be the probability that , where and are relatively prime positive integers. Find .
Problem 15
The sides of rectangle have lengths and . An equilateral triangle is drawn so that no point of the triangle lies outside . The maximum possible area of such a triangle can be written in the form , where , , and are positive integers, and is not divisible by the square of any prime number. Find .
See also
1997 AIME (Problems • Answer Key • Resources) | ||
Preceded by 1996 AIME Problems |
Followed by 1998 AIME Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- 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.