Difference between revisions of "1981 AHSME Problems"
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==Problem 1== | ==Problem 1== | ||
If <math>\sqrt{x+2}=2</math>, then <math>(x+2)^{2}</math> equals | If <math>\sqrt{x+2}=2</math>, then <math>(x+2)^{2}</math> equals | ||
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− | label(rotate(A--B)*"$14$", A--B, dir(A--B)*dir(90)); | + | label(rotate(angle(dir(A--B)))*"$14$", A--B, dir(A--B)*dir(90)); |
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If <math> \theta</math> is a constant such that <math> 0 < \theta < \pi</math> and <math> x + \dfrac{1}{x} = 2\cos{\theta}</math>, then for each positive integer <math> n</math>, <math> x^n + \dfrac{1}{x^n}</math> equals | If <math> \theta</math> is a constant such that <math> 0 < \theta < \pi</math> and <math> x + \dfrac{1}{x} = 2\cos{\theta}</math>, then for each positive integer <math> n</math>, <math> x^n + \dfrac{1}{x^n}</math> equals | ||
− | <math> \textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ 2\cos n\theta\qquad \textbf{(E)}\ 2^n\cos^n\theta | + | <math> \textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ 2\cos n\theta\qquad \textbf{(E)}\ 2^n\cos^n\theta</math> |
[[1981 AHSME Problems/Problem 24|Solution]] | [[1981 AHSME Problems/Problem 24|Solution]] | ||
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[[1981 AHSME Problems/Problem 30|Solution]] | [[1981 AHSME Problems/Problem 30|Solution]] | ||
− | ==See | + | == See also == |
− | |||
− | *[[ | + | * [[AMC 12 Problems and Solutions]] |
+ | * [[Mathematics competition resources]] | ||
− | + | {{AHSME box|year=1981|before=[[1980 AHSME]]|after=[[1982 AHSME]]}} | |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:01, 19 February 2020
1981 AHSME (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
If , then equals
Problem 2
Point is on side of square . If has length one and has length two, then the area of the square is
Problem 3
For , equals
Problem 4
If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is
Problem 5
In trapezoid , sides and are parallel, and diagonal and side have equal length. If and , then
Problem 6
If , then equals
Problem 7
How many of the first one hundred positive integers are divisible by all of the numbers , , , and ?
Problem 8
For all positive numbers , , , the product equals
Problem 9
In the adjoining figure, is a diagonal of the cube. If has length , then the surface area of the cube is
Problem 10
The lines and are symmetric to each other with respect to the line . If the equation of the line is with and , then the equation of is
Problem 11
The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length
Problem 12
If , , and are positive numbers and , then the number obtained by increasing by and decreasing the result by exceeds if and only if
Problem 13
Suppose that at the end of any year, a unit of money has lost of the value it had at the beginning of that year. Find the smallest integer such that after years, the money will have lost at least of its value (To the nearest thousandth ).
Problem 14
In a geometric sequence of real numbers, the sum of the first terms is , and the sum of the first terms is . The sum of the first terms is
Problem 15
If , , and , then is
Problem 16
The base three representation of is The first digit (on the left) of the base nine representation of is
Problem 17
The function is not defined for , but, for all non-zero real numbers , . The equation is satisfied by
Problem 18
The number of real solutions to the equation is
Problem 19
In , is the midpoint of side , bisects , and . If sides and have lengths and , respectively, then find .
Problem 20
A ray of light originates from point and travels in a plane, being reflected times between lines and before striking a point (which may be on or ) perpendicularly and retracing its path back to (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for ). If , what is the largest value can have?
Problem 21
In a triangle with sides of lengths , , and , . The measure of the angle opposite the side length is
Problem 22
How many lines in a three dimensional rectangular coordiante system pass through four distinct points of the form , where , , and are positive integers not exceeding four?
Problem 23
Equilateral is inscribed in a circle. A second circle is tangent internally to the circumcircle at and tangent to sides and at points and . If side has length , then segment has length
Problem 24
If is a constant such that and , then for each positive integer , equals
Problem 25
In in the adjoining figure, and trisect . The lengths of , and are , , and , respectively. The length of the shortest side of is
Problem 26
Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is , independent of the outcome of any other toss.)
Problem 27
In the adjoining figure triangle is inscribed in a circle. Point lies on with , and point lies on with . Side and side each have length equal to the length of chord , and . Chord intersects sides and at and , respectively. The ratio of the area of to the area of is
Problem 28
Consider the set of all equations , where , , are real constants and for . Let be the largest positive real number which satisfies at least one of these equations. Then
Problem 29
If , then the sum of the real solutions of
is equal to
Problem 30
If , , , and are the solutions of the equation , then an equation whose solutions are is
See also
1981 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1980 AHSME |
Followed by 1982 AHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.