Difference between revisions of "1977 AHSME Problems"
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+ | {{AHSME Problems | ||
+ | |year = 1977 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
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== Problem 3 == | == Problem 3 == | ||
− | A man has | + | A man has \$2.73 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is |
<math>\text{(A)}\ 3 \qquad | <math>\text{(A)}\ 3 \qquad | ||
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If <math>x, y</math> and <math>2x + \frac{y}{2}</math> are not zero, then | If <math>x, y</math> and <math>2x + \frac{y}{2}</math> are not zero, then | ||
− | <math>\left( 2x + \frac{y}{2} \right)\left[(2x)^{-1} + \left( \frac{y}{2} \right)^{-1} \right]</math> | + | <math>\left( 2x + \frac{y}{2} \right)^{-1} \left[(2x)^{-1} + \left( \frac{y}{2} \right)^{-1} \right]</math> |
equals | equals | ||
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<math>\text{(A)}\ \frac{\sqrt{2}}{2} \qquad | <math>\text{(A)}\ \frac{\sqrt{2}}{2} \qquad | ||
\text{(B)}\ -10i\sqrt{2} \qquad | \text{(B)}\ -10i\sqrt{2} \qquad | ||
− | \text{(C)}\ \frac{21\sqrt{2}}{2} | + | \text{(C)}\ \frac{21\sqrt{2}}{2} \qquad\\ |
\text{(D)}\ \frac{\sqrt{2}}{2}(21 - 20i) \qquad | \text{(D)}\ \frac{\sqrt{2}}{2}(21 - 20i) \qquad | ||
\text{(E)}\ \frac{\sqrt{2}}{2}(21 + 20i) </math> | \text{(E)}\ \frac{\sqrt{2}}{2}(21 + 20i) </math> | ||
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== Problem 21 == | == Problem 21 == | ||
− | For how many values of the coefficient a do the equations < | + | For how many values of the coefficient a do the equations <cmath>\begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*}</cmath> have a common real solution? |
<math>\textbf{(A)}\ 0 \qquad | <math>\textbf{(A)}\ 0 \qquad | ||
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<math>\textbf{(A) }f(0)=1\qquad | <math>\textbf{(A) }f(0)=1\qquad | ||
\textbf{(B) }f(-x)=-f(x)\qquad | \textbf{(B) }f(-x)=-f(x)\qquad | ||
− | \textbf{(C) }f(-x)=f(x)\qquad | + | \textbf{(C) }f(-x)=f(x)\qquad \\ |
− | \textbf{(D) }f(x+y)=f(x)+f(y)\qquad\\ | + | \textbf{(D) }f(x+y)=f(x)+f(y) \qquad \\ |
\textbf{(E) }\text{there is a positive real number }T\text{ such that }f(x+T)=f(x) </math> | \textbf{(E) }\text{there is a positive real number }T\text{ such that }f(x+T)=f(x) </math> | ||
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[[1977 AHSME Problems/Problem 30|Solution]] | [[1977 AHSME Problems/Problem 30|Solution]] | ||
+ | |||
+ | == See also == | ||
+ | |||
+ | * [[AMC 12 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{AHSME box|year=1977|before=[[1976 AHSME]]|after=[[1978 AHSME]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 04:09, 27 November 2020
1977 AHSME (Answer Key) Printable versions: • 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 and , then equals
Problem 2
Which one of the following statements is false? All equilateral triangles are
Problem 3
A man has $2.73 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is
Problem 4
In triangle and . If points , and lie on sides and , respectively, and and , then equals
Problem 5
The set of all points such that the sum of the (undirected) distances from to two fixed points and equals the distance between and is
Problem 6
If and are not zero, then equals
Problem 7
If , then equals
Problem 8
For every triple of non-zero real numbers, form the number . The set of all numbers formed is
Problem 9
In the adjoining figure and arc , arc , and arc all have equal length. Find the measure of .
Problem 10
If , then equals
Problem 11
For each real number , let be the largest integer not exceeding (i.e., the integer such that ). Which of the following statements is (are) true?
Problem 12
Al's age is more than the sum of Bob's age and Carl's age, and the square of Al's age is more than the square of the sum of Bob's age and Carl's age. What is the sum of the ages of Al, Bob, and Carl?
Problem 13
If is a sequence of positive numbers such that for all positive integers , then the sequence is a geometric progression
Problem 14
How many pairs of integers satisfy the equation ?
Problem 15
Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is
Problem 16
If , then the sum equals
Problem 17
Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one?
Problem 18
If then
Problem 19
Let be the point of intersection of the diagonals of convex quadrilateral , and let , and be the centers of the circles circumscribing triangles , and , respectively. Then
Problem 20
For how many paths consisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram above, is the word CONTEST spelled out as the path is traversed from beginning to end?
Problem 21
For how many values of the coefficient a do the equations have a common real solution?
Problem 22
If is a real valued function of the real variable , and is not identically zero, and for all and , then for all and
Problem 23
If the solutions of the equation are the cubes of the solutions of the equation , then
Problem 24
Find the sum .
Problem 25
Determine the largest positive integer n such that is divisible by .
Problem 26
Let , and be the lengths of sides , and , respectively, of quadrilateral . If is the area of , then
Problem 27
There are two spherical balls of different sizes lying in two corners of a rectangular room, each touching two walls and the floor. If there is a point on each ball which is inches from each wall which that ball touches and inches from the floor, then the sum of the diameters of the balls is
Problem 28
Let . What is the remainder when the polynomial is divided by the polynomial ?
Problem 29
Find the smallest integer such that for all real numbers , and .
Problem 30
If , and are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon (see adjoining figure), then
See also
1977 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1976 AHSME |
Followed by 1978 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.