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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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 . 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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Problem 13: A circle is inscribed in a triangle with sides of lengths , and . Let the segments of the side of length , made by a point of tangency, be and , with . Then the ratio is:
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Problem 14: A farmer bought sheep. He sold of them for the price paid for the sheep. The remaining sheep were sold at the same price per head as the other . 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 , 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 . Of the three possibilities: parallelogram straight line trapezoid, figure , depending upon the location of the points , and , can be:
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A. only
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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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Problem 19: If and , the numerical value of
is:
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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 so that . What is the ratio of the area of triangle to the area of quadrilateral
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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 miles and First beats Third by 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 , and the common difference is . 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, 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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A. must equal
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Problem 33: is a point interior to rectangle such that inches, inches, and inches. Then , in inches, equals:
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Problem 34: If is a multiple of , the sum , where , equals:
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Problem 35: The sides of a triangle are of lengths , and . The altitudes of the triangle meet at point . If is the altitude to side of length , 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 . The circle is made to roll along the side , remaining tangent to it at a variable point and intersecting sides and in variable points and , respectively.
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Let be the number of degrees in arc . Then , for all permissible positions of the circle:
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A. varies from to
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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 inches and inches. The median is inches. Then , in inches, is:
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Problem 39: The magnitudes of the sides of triangle are , and , 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 P.M. on March . When the watch shows A.M. on March , the positive correction to be added to the time shown by the watch, in minutes, equals:
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The problems and solutions on this page are the property of the MAA's American Mathematics Competitions