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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Problem 2: The expression is equal to:
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Problem 3: The first three terms of an arithmetic progression are , in the order shown. The value of 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 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 . Then, if all measurements are in degrees, angle equals:
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Problem 8: Given the set of numbers; , of which one is and all the others are . 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
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Problem 10: A man drives miles to the seashore in hours and minutes. He returns from the shore to the starting point in hours and 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 equals:
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A. a number between and
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Problem 15: Given triangle with base fixed in length and position. As the vertex 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
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Problem 16: Given rectangle with one side inches and area square inches. Rectangle with diagonal 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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Problem 18: A regular dodecagon ( 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
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E. no value of
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Problem 23: In triangle is the altitude to and is the altitude to . 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 is an acute angle
C. determined only if is an acute angle
D. determined only if is an acute triangle.
E. none of these is correct
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Problem 24: Three machines , and , working together, can do a job in hours. When working alone needs an additional 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 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?
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A. only
B. only
C. and only
D. and only
E. all three
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Problem 28: The set of -values satisfying the equation consists of:
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A. , only
B. , only
C. , only
D. or $100A, only
E. more than two real numbers
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Problem 29: Which of the following sets of -values satisfy the inequality
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B. or
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Problem 30: Consider the statements:
where and are statements, each of which may be true or false.
How many of these statements 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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Problem 34: For what real values of does have real roots?
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A. none
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E. all
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Problem 35: A man on his way to dinner shortly after p.m. observes that the hands of his watch form an angle of . Returning before 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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Problem 37: is a square with side of unit length. Points and are taken respectively on sides and so that and the quadrilateral 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 , the population was one more than a perfect square. Now, with an additional increase of , 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 inches and 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
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The problems and solutions on this page are the property of the MAA's American Mathematics Competitions