Individual Problems and Solutions
For problems and detailed solutions to each of the 1961 AHSME problems, please refer below:
Problem 1: When simplified becomes:
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Problem 2: An automobile travels feet in seconds. If this rate is maintained for 3 minutes, how many yards does it travel in the 3 minutes?
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Problem 3: If the graphs of and are to meet at right angles, the value of is:
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Problem 4: Let the set consisting of the squares of the positive integers be called ; thus is the set . If a certain operation on one more members of the set always yields a member of the set, we say that the set is closed under that operation. Then is closed under:
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A. addition
B. multiplication
C. division
D. extraction of a positive integral root
E. none of these
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Problem 5: Let . Then equals:
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Problem 6: When simplified becomes:
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Problem 7: The third term in the expansion of is, when simplified:
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Problem 8: Let the two base angles of a triangle be and , with larger than . The altitude to the base divides the vertex angle into two parts, and , with adjacent to side . Then:
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Problem 9: Let be the result of doubling both the base and the exponent of . If equals the product of by , then equals:
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Problem 10: Each side of triangle is 12 units. is the foot of the perpendicular dropped from on , and is the midpoint of . The length of , in the same unit, is:
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Problem 11: Tangents and are drawn to a circle from exterior point . Tangent intersects in and in , and touches the circle at . If , then the perimeter of triangle APR is:
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Problem 12: The first three terms of a geometric progression are . The fourth term is:
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Problem 13: The symbol means if is a positive number or zero, and if is a negative number. For all real values of the expression is equal to:
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Problem 14: A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus in terms of , where is the area of the rhombus in square inches:
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Problem 15: If men working hours a day for each of days produce articles, then the number of articles (not necessarily an integer) produced by men working hours a day for each of days is:
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Problem 16: An altitude of a triangle is increased by a length . How much must be taken from the corresponding base so that the area of the new triangle is one-half that of the original triangle?
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Problem 17: In the base ten number system the number 526 means . In the Land of Mathesis, however, numbers are written in the base . Jones purchases an automobile there for 440 monetary units (abbreviated m.u.). He gives the salesman a . bill, and receives, in change, . The base is:
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Problem 18: The yearly changes in the population census of a town for four consecutive years are, respectively, increase, increase, decrease, decrease. The net change over the four years, to the nearest percent, is:
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Problem 19: Consider the graphs of and . We may say that:
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A. They do not intersect
B. They intersect in one point only
C. They intersect in two points only
D. They intersect in a finite number of points but more than two
E. They coincide
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Problem 20: The set of points satisfying the pair of inequalities and is contained entirely in quadrants:
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A. I and II
B. II and III
C. I and III
D. III and IV
E. I and IV
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Problem 21: Medians and of triangle intersect in . The midpoint of is . Let the area of triangle times the area of triangle ABC . Then equals:
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Problem 22: If is divisible by , then it is also divisible by:
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Problem 23: Points and are both in the line and on the same side of the midpoint of line . divides in the ratio and divides in the ratio . If , then the length of is:
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Problem 24: Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by from that of each adjacent book. For the price of the book at the extreme right a customer can buy the middle book and an adjacent one. Then:
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A. The adjacent book referred to is at the left of the middle book
B. The middle book sells for
C. The cheapest book sells for
D. The most expensive book sells for
E. None of these is correct
Solution:
Problem 25: Triangle is isosceles with base . Points and are respectively in and and such that . The number of degrees in angle is:
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Problem 26: For a given arithmetic series the sum of the first 50 terms is 200 , and the sum of the next 50 terms is 2700. The first term of the series is:
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Problem 27: Given two equiangular polygons and with different numbers of sides; each angle of is degrees and each angle of is degrees, where is an integer greater than 1. The number of possibilities for the pair is:
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A. Infinite
B. Finite, but more than two
C. Two
D. One
E. Zero
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Problem 28: If is multiplied out, the units' digit in the final product is:
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Problem 29: Let the roots of be and . The equation with roots and is:
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Problem 30: If and , then equals:
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Problem 31: In triangle the ratio . The bisector of the exterior angle at intersects extended at ( is between and ). The ratio PA:AB is:
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Problem 32: A regular polygon of sides is inscribed in a circle of radius . The area of the polygon is . Then equals:
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Problem 33: The number of solutions of , in which and are integers, is:
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A. Zero
B. One
C. Two
D. Three
E. More than three, but finite
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Problem 34: Let be the set of values assumed by the function when is any member of the interval . If there exists a number such that no number of the set is greater than , then is an upper bound of . If there exists a number such that no number of the set is less than , then is a lower bound of . We may then say:
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A. is in is not in
B. is in is not in
C. both and are in $S
D. neither nor is in
E. does not exist either in or outside
Solution:
Problem 35: The number 695 is to be written with a factorial base of numeration, that is, where are integers such that , and means . Find :
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Problem 36: In triangle the median from is given perpendicular to the median from . If and , then the length of is:
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Problem 37: In racing over a given distance , at uniform speed, can beat by 20 yards, can beat by 10 yards, and can beat by 28 yards. Then , in yards, equals:
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Problem 38: Triangle ABC is inscribed in a semicircle of radius . Base AB coincides with diameter AB. Point does not coincide with either or . Let . Then, for all permissible positions of :
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Problem 39: Any five points are taken inside or on a square of side 1. Let be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than . Then is:
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Problem 40: Find the minimum value of if :
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The problems on this page are the property of the MAA's American Mathematics Competitions