ΒΆ 1964 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1964 AHSME problems, please refer below:
Problem 1: What is the value of ?
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Problem 2: The graph of is:
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A. a parabola
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D. a point
E. none of these
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Problem 3: When a positive integer is divided by a positive integer , the quotient is and the remainder is and integers. What is the remainder when is divided by ?
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Problem 4: The expression , where and , is equivalent to:
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Problem 5: If varies directly as and if when , the value of when is:
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Problem 6: If are in geometric progression, the fourth term is:
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Problem 7: Let be the number of real values of for which the roots of are equal. Then equals:
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E. an infinitely large number
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Problem 8: The smaller root of the equation is:
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Problem 9: A jobber buys an article at " less ". He then wishes to sell the article at a gain of of his cost after allowing a discount on his marked price. At what price, in dollars, should the article be marked?
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Problem 10: Given a square with side of length s. On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is:
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Problem 11: Given and ; the value of is:
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Problem 12: Which of the following is the negation of the statement: For all of a certain set, ?
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A. For all
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E. For some
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Problem 13: A circle is inscribed in a triangle with sides of lengths 8,13, and 17. Let the segments of the side of length 8, made by a point of tangency, be and , with . Then the ratio is:
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Problem 14: A farmer bought 749 sheep. He sold 700 of them for the price paid for the 749 sheep. The remaining 49 sheep were sold at the same price per head as the other 700. Based on the cost, the percent gain on the entire transaction is:
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Problem 15: A line through the point ( ) cuts from the second quadrant a triangular region with area . The equation of the line is:
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Problem 16: Let the expression have a remainder of zero when divided by 6, and let be the set of integers . The number of members of satisfying the given condition is:
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Problem 17: Given the distinct points and . Line segments are drawn connecting these points to each other and to the origin 0. Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure OPRQ, depending upon the location of the points , and , can be:
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A. (1) only
B. (2) only
C. (3) only
D. (1) or (2) only
E. all three
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Problem 18: Let be the number of pairs of values of and such that and have the same graph. Then is:
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D. finite but more than 2
E. greater than any finite number
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Problem 19: If and , the numerical value of
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Problem 20: The sum of the numerical coefficients of all the terms in the expansion of is:
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Problem 21: If , then equals:
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Problem 22: Given parallelogram with the midpoint of diagonal . Point is connected to a point in DA so that . What is the ratio of the area of triangle DFE to the area of quadrilateral ABEF?
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Problem 23: Two numbers are such that their difference, their sum, and their product are to one another as . The product of the two numbers is:
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Problem 24: Let constants. For what value of is a minimum ?
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Problem 25: The set of values of for which has two factors, with integer coefficients, which are linear in and , is precisely:
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Problem 26: In a ten-mile race First beats Second by 2 miles and First beats Third by 4 miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third?
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Problem 27: If is a real number and where a , then:
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Problem 28: The sum of terms of an arithmetic progression is 153 , and the common difference is 2. If the first term is an integer, and , then the number of possible values for is:
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Problem 29: In this figure inches, 6 inches, inches, inches. The length of , in inches is:
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Problem 30: If , the larger root minus the smaller root is:
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Problem 31: Let . Then , expressed in terms of , equals:
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Problem 32: If , then:
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Problem 33: is a point interior to rectangle and such that inches, inches, and inches. Then , in inches, equals:
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Problem 34: If is a multiple of 4 , the sum , where , equals:
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Problem 35: The sides of a triangle are of lengths 13,14 , and 15 . The altitudes of the triangle meet at point . If is the altitude to side of length 14 , the ratio is :
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Problem 36: In this figure the radius of the circle is equal to the altitude of the equilateral triangle ABC . The circle is made to roll along the side AB , remaining tangent to it at a variable point T and intersecting sides AC and in variable points and , respectively.
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Let be the number of degrees in arc MTN. Then , for all permissible positions of the circle:
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A. varies from 30 to 90
B. varies from 30 to 60
C. varies from 60 to 90
D. remains constant at 30
E. remains constant at 60
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Problem 37: Given two positive numbers such that . Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than:
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Problem 38: The sides and of triangle are respectively of lengths 4 inches and 7 inches. The median is inches. Then , in inches, is:
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Problem 39: The magnitudes of the sides of triangle ABC are , and c , as shown, with . Through interior point and the vertices lines are drawn meeting the opposite sides in , respectively. Let . Then, for all positions of point is less than:
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Problem 40: A watch loses minutes per day. It is set right at 1 P.M. on March 15. When the watch shows 9 A.M. on March 21, the positive correction to be added to the time shown by the watch, in minutes, equals:
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The problems on this page are the property of the MAA's American Mathematics Competitions