¶ 1977 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1977 AHSME problems, please refer below:
Problem 1: If and , then equals
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Problem 2: Which one of the following statements is false? All equilateral triangles are
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A. equiangular
B. isosceles
C. regular polygons
D. congruent to each other
E. similar to each other
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Problem 3: A man has 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
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Problem 4: In triangle and . If points and lie on sides , and , respectively, and and , then equals
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E. none of these
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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
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A. the line segment from to
B. the line passing through and
C. the perpendicular bisector of the line segment from to
D. an ellipse having positive area
E. a parabola
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Problem 6: If and are not zero, then equals
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E. none of these
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Problem 7: If , then equals
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Problem 8: For every triple of non-zero real numbers, form the number
The set of all numbers formed is
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E. none of these
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Problem 9: In the adjoining figure and and all have equal length. Find the measure of .
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Problem 10: If , then equals
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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?
I. for all
II. for all and
III. for all and
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A. none
B. I only
C. I and II only
D. III only
E. all
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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. The sum of the ages of Al, Bob and Carl is
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Problem 13: If is a sequence of positive numbers such that for all positive integers , then the sequence is a geometric progression
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A. for all positive values of and
B. if and only if
C. if and only if
D. if and only if
E. if and only if
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Problem 14: How many pairs of integers satisfy the equation
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E. more than
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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
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Problem 16: If , then the sum equals
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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?
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Problem 18: If then
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E. none of these answers
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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
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A. is a parallelogram
B. is a parallelogram if and only if is a rhombus
C. is a parallelogram if and only if is a rectangle
D. is a parallelogram if and only if is a parallelogram
E. none of the above are true
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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 below, is the word spelled out as the path is traversed from beginning to end?
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E. none of these answers
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Problem 21: For how many values of the coefficient do the equations
and
have a common real solution?
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E. infinitely many
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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
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E. there is a positive number such that
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Problem 23: If the solutions of the equation are the cubes of the solutions of the equation , then
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E. none of these answers
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Problem 24: Find the sum
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Problem 25: Determine the largest positive integer such that is divisible by .
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E. none of these answers
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Problem 26: Let and be the lengths of sides and , respectively, of quadrilateral . If is the area of , then
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A. if and only if is convex
B. if and only if is a rectangle
C. if and only if is a rectangle
D. if and only if is a parallelogram
E. if and only if is a parallelogram
Solution:
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
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A. inches
B. inches
C. inches
D. inches
E. not determined by the given information
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Problem 28: Let . What is the remainder when the polynomial is divided by the polynomial
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Problem 29: Find the smallest integer such that for all real numbers and .
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E. There is no such integer
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Problem 30: If and are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon, then
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The problems and solutions on this page are the property of the MAA's American Mathematics Competitions