ΒΆ 1969 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1969 AHSME problems, please refer below:
Problem 1: When is added to both the numerator and the denominator of the fraction , the value of the fraction is changed to . Then equals:
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E.
Solution:
Problem 2: If an item is sold for dollars, there is a loss of based on the cost. If, however, the same item is sold for dollars, there is a profit of based on the cost. The ratio is:
Answer Choices:
A.
B.
C.
D. dependent upon the cost
E. none of these.
Solution:
Problem 3: If , written in base , is , the integer immediately preceding , written in base , is:
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E.
Solution:
Problem 4: Let a binary operation on ordered pairs of integers be defined as . Then, if and represent identical pairs, equals:
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Solution:
Problem 5: If a number , diminished by four times its reciprocal, equals a given real constant , then, for this given , the sum of all such possible values of is:
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Solution:
Problem 6: The area of the ring between two concentric circles is square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is:
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Solution:
Problem 7: If the points and lie on the graph of , and , then equals:
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Solution:
Problem 8: Triangle is inscribed in a circle. The measures of the nonoverlapping minor , and are, respectively, , . Then one interior angle of the triangle, in degrees, is:
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Problem 9: The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning with , is:
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Solution:
Problem 10: The number of points equidistant from a circle and two parallel tangents to the circle, is:
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A.
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D.
E. infinite
Solution:
Problem 11: Given points and in the -plane; point is taken so that is a minimum. Then equals:
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D.
E. either or .
Solution:
Problem 12: Let be the square of an expression which is linear in . Then has a particular value between:
Answer Choices:
A. and
B. and
C. and
D. and
E. and
Solution:
Problem 13: A circle with radius is contained within the region bounded by a circle with radius . The area bounded by the larger circle is times the area of the region outside the smaller circle and inside the larger circle. Then equals:
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Solution:
Problem 14: The complete set of -values satisfying the inequality is the set of all such that:
Answer Choices:
A. or or
B. or
C. or
D. or
E. is any real number except or
Solution:
Problem 15: In a circle with center at and radius , chord is drawn with length equal to (units). From a perpendicular to meets at . From a perpendicular to meets at . In terms of the area of triangle , in appropriate square units, is:
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Problem 16: When , is expanded by the binomial theorem, it is found that, when , where is a positive integer, the sum of the second and third terms is zero. Then equals:
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Solution:
Problem 17: The equation is satisfied by:
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E. None of these
Solution:
Problem 18: The number of points common to the graphs of and is:
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B.
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D.
E. infinite
Solution:
Problem 19: The number of distinct ordered pairs where and have positive integral values satisfying the equation is:
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B.
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D.
E. infinite
Solution:
Problem 20: Let equal the product of and . The number of digits in is:
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Solution:
Problem 21: If the graph of is tangent to the graph of , then:
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A. must equal
B. must equal
C. must equal
D. must equal
E. may be any non-negative real number
Solution:
Problem 22: Let be the measure of the area bounded by the -axis, the line , and the curve defined by when when . Then is:
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D.
E. less than but arbitrarily close to it
Solution:
Problem 23: For the number of prime numbers greater than and less than , is: ; thus:
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B.
C.
D.
E. for even, for odd
Solution:
Problem 24: When the natural numbers and , with , are divided by the natural number , the remainders are and , respectively. When and are divided by , the remainders are and , respectively. Then:
Answer Choices:
A. always
B. always
C. sometimes and sometimes
D. sometimes and sometimes
E. always
Solution:
Problem 25: If it is known that and that is a maximum, then the least value that can be taken on by is:
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B.
C.
D.
E. none of these
Solution:
Problem 26: A parabolic arch has a height of inches and a span of inches. The height, in inches, of the arch at a point inches from the center , is:
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Solution:
Problem 27: A particle moves so that its speed for the second and subsequent miles - varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in hours, then the time, in hours, needed to traverse the nth mile is:
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Problem 28: Let n be the number of points interior to the region bounded by a circle with radius , such that the sum of the squares of the distances from to the endpoints of a given diameter is . Then n is:
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B.
C.
D.
E. infinite
Solution:
Problem 29: If and , a relation between and is:
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B.
C.
D.
E. none of these
Solution:
Problem 30: Let be a point of hypotenuse (or its extension) of isosceles right triangle . Let . Then:
Answer Choices:
A. for a finite number of positions of
B. for an infinite number of positions of
C. only if is the midpoint of or an endpoint of
D. always
E. if is a trisection point of
Solution:
Problem 31: Let be a unit square in the -plane with and . Let and be a transformation of the -plane into the -plane. The transform (or image) of the square is:
Answer Choices:
A.
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Solution:
Problem 32: Let a sequence be defined by the relation , , and . If is expressed as a polynomial in , the algebraic sum of its coefficients is:
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Solution:
Problem 33: Let and be the respective sums of the first terms of two arithmetic series. If for all , the ratio of the eleventh term of the first series to the eleventh term of the second series, is:
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D.
E. undetermined
Solution:
Problem 34: The remainder obtained by dividing by is a polynomial of degree less than . Then may be written as:
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Solution:
Problem 35: Let be the -coordinate of the left end point of the intersection of the graphs of and where . Let . Then, as is made arbitrarily close to zero, the value of is:
Answer Choices:
A. arbitrarily close to zero
B. arbitrarily close to
C. arbitrarily close to
D. arbitrarily large
E. undetermined
Solution:
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions