Difference between revisions of "1991 AIME Problems"
Mathcool2009 (talk | contribs) (→Problem 11: I made the diagram more clear than before--the old diagram implies that the centers of the disks are on C, which is false.) |
|||
Line 65: | Line 65: | ||
Twelve congruent disks are placed on a circle <math>C^{}_{}</math> of radius 1 in such a way that the twelve disks cover <math>C^{}_{}</math>, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers and <math>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. | Twelve congruent disks are placed on a circle <math>C^{}_{}</math> of radius 1 in such a way that the twelve disks cover <math>C^{}_{}</math>, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers and <math>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. | ||
− | [[Image:AIME_1991_Problem_11. | + | [[Image:AIME_1991_Problem_11.png|300px]] |
[[1991 AIME Problems/Problem 11|Solution]] | [[1991 AIME Problems/Problem 11|Solution]] |
Revision as of 14:07, 23 June 2014
1991 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Find if and are positive integers such that
Problem 2
Rectangle has sides of length 4 and of length 3. Divide into 168 congruent segments with points , and divide into 168 congruent segments with points . For , draw the segments . Repeat this construction on the sides and , and then draw the diagonal . Find the sum of the lengths of the 335 parallel segments drawn.
Problem 3
Expanding by the binomial theorem and doing no further manipulation gives
where for . For which is the largest?
Problem 4
How many real numbers satisfy the equation ?
Problem 5
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will be the resulting product?
Problem 6
Suppose is a real number for which
Find . (For real , is the greatest integer less than or equal to .)
Problem 7
Find , where is the sum of the absolute values of all roots of the following equation:
Problem 8
For how many real numbers does the quadratic equation have only integer roots for ?
Problem 9
Suppose that and that where is in lowest terms. Find
Problem 10
Two three-letter strings, and , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an when it should have been a , or as a when it should be an . However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let be the three-letter string received when is transmitted and let be the three-letter string received when is transmitted. Let be the probability that comes before in alphabetical order. When is written as a fraction in lowest terms, what is its numerator?
Problem 11
Twelve congruent disks are placed on a circle of radius 1 in such a way that the twelve disks cover , no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from , where are positive integers and is not divisible by the square of any prime. Find .
Problem 12
Rhombus is inscribed in rectangle so that vertices , , , and are interior points on sides , , , and , respectively. It is given that , , , and . Let , in lowest terms, denote the perimeter of . Find .
Problem 13
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
Problem 14
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by , has length 31. Find the sum of the lengths of the three diagonals that can be drawn from .
Problem 15
For positive integer , define to be the minimum value of the sum
where are positive real numbers whose sum is 17. There is a unique positive integer for which is also an integer. Find this .
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.