ΒΆ 1962 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1962 AHSME problems, please refer below:
Problem 1: The expression is equal to:
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Solution:
Problem 2: The expression is equal to:
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Solution:
Problem 3: The first three terms of an arithmetic progression are , in the order shown. The value of x is:
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E. undetermined
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Problem 4: If , then equals:
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Problem 5: If the radius of a circle is increased by 1 unit, the ratio of the new circumference to the new diameter is:
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Problem 6: A square and an equilateral triangle have equal perimeters. The area of the triangle is square inches. Expressed in inches the diagonal of the square is:
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E. none of these
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Problem 7: Let the bisectors of the exterior angles at and of triangle meet at D. Then, if all measurements are in degrees, angle BDC equals:
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Problem 8: Given the set of numbers; , of which one is and all the others are 1. The arithmetic mean of the numbers is:
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Problem 9: When is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is:
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A. more than 5
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D.
E.
Solution:
Problem 10: A man drives 150 miles to the seashore in 3 hours and 20 minutes. He returns from the shore to the starting point in 4 hours and 10 minutes. Let be the average rate for the entire trip. Then the average rate for the trip going exceeds , in miles per hour, by:
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Problem 11: The difference between the larger root and the smaller root of is:
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Problem 12: When is expanded the sum of the last three coefficients is:
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Problem 13: varies directly as and inversely as . When and . Find when and .
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Problem 14: Let be the limiting sum of the geometric series , as the number of terms increases without bound. Then s equals:
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A. a number between 0 and 1
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D.
E.
Solution:
Problem 15: Given triangle with base fixed in length and position. As the vertex C moves on a straight line, the intersection point of the three medians moves on:
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A. a circle
B. a parabola
C. an ellipse
D. a straight line
E. a curve here not listed
Solution:
Problem 16: Given rectangle with one side 2 inches and area 12 square inches. Rectangle with diagonal 15 inches is similar to . Expressed in square inches the area of is:
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Problem 17: If and , then , in terms of , is:
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E. 2 b
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Problem 18: A regular dodecagon ( 12 sides) is inscribed in a circle with radius inches. The area of the dodecagon, in square inches, is:
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Problem 19: If the parabola passes through the points , and , the value of is:
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Problem 20: The angles of a pentagon are in arithmetic progression. One of the angles, in degrees, must be:
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Problem 21: It is given that one root of , with and real numbers, is . The value of is:
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A. undetermined
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Problem 22: The number , written in the integral base , is the square of an integer, for
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A. , only
B. and , only
C.
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E. no value of
Solution:
Problem 23: In triangle is the altitude to AB and AE is the altitude to BC. If the lengths of , and are known, the length of is:
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A. not determined by the information given
B. determined only if A is an acute angle
C. determined only if B is an acute angle
D. determined only if is an acute triangle.
E. none of these is correct
Solution:
Problem 24: Three machines , and , working together, can do a job in hours. When working alone needs an additional 6 hours to do the job; , one additional hour; and additional hours. The value of is:
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Problem 25: Given square with side 8 feet. A circle is drawn through vertices and and tangent to side . The radius of the circle, in feet, is:
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Problem 26: For any real value of the maximum value of is:
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Problem 27: Let represent the operation on two numbers, and , which selects the larger of the two numbers, with . Let represent the operation which selects the smaller of the two numbers, with .
Which of the following three rules is (are) correct?
(1)
(2)
(3)
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A. (1) only
B. (2) only
C. (1) and (2) only
D. (1) and (3) only
E. all three
Solution:
Problem 28: The set of -values satisfying the equation consists of:
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A. , only
B. 10 , only
C. 100 , only
D. 10 or 100 , only
E. more than two real numbers
Solution:
Problem 29: Which of the following sets of -values satisfy the inequality ?
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B. or
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Solution:
Problem 30: Consider the statements: (1) (2) (4) , where and are statements each of which may be true or false. How many of these imply the truth of ?
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Problem 31: The ratio of the interior angles of two regular polygons with sides of unit length is . How many such pairs are there?
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E. infinitely many
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Problem 32: If for and , find .
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Problem 33: The set of -values satisfying the inequality is:
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A. or
B. or
C. or
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Problem 34: For what real values of does have real roots?
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A. none
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D. or
E. all
Solution:
Problem 35: A man on his way to dinner shortly after 6:00 p.m. observes that the hands of his watch form an angle of . Returning before 7:00 p.m. he notices that again the hands of his watch form an angle of . The number of minutes that he has been away is:
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Problem 36: If both and are integers, how many pairs of solutions are there of the equation ?
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E. more than 3
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Problem 37: ABCD is a square with side of unit length. Points and are taken respectively on sides and so that and the quadrilateral CDFE has maximum area. In square units this maximum area is:
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Problem 38: The population of Nosuch Junction at one time was a perfect square. Later, with an increase of 100 , the population was one more than a perfect square. Now, with an additional increase of 100 , the population is again a perfect square. The original population is a multiple of:
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Problem 39: Two medians of a triangle with unequal sides are 3 inches and 6 inches. Its area is square inches. The length of the third median, in inches, is:
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Problem 40: The limiting sum of the infinite series, whose th term is is:
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E. larger than any finite quantity
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions