¶ 1988 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1988 AHSME problems, please refer below:
Problem 1:
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Problem 2: Triangles and are similar, with corresponding to and to . If and , then is
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Problem 3: Four rectangular paper strips of length and width are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?
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Problem 4: The slope of the line is
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Problem 5: If and are constants and
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Problem 6: A figure is an equiangular parallelogram if and only if it is a
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A. rectangle
B. regular polygon
C. rhombus
D. square
E. trapezoid
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Problem 7: Estimate the time it takes to send blocks of data over a communications channel if each block consists of "chunks" and the channel can transmit chunks per second.
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A. seconds
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D. minutes
E. hours
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Problem 8: If and , what is the ratio of to
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Problem 9: An table sits in the corner of a square room, as in Figure below. The owners desire to move the table to the position shown in Figure . The side of the room is feet. What is the smallest integer value of for which the table can be moved as desired without tilting it or taking it apart?
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Problem 10: In an experiment, a scientific constant is determined to be with an error of at most . The experimenter wishes to announce a value for in which every digit is significant. That is, whatever is, the announced value must be the correct result when is rounded to that number of digits. The most accurate value the experimenter can announce for is
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Problem 11: On each horizontal line in the figure below, the five large dots indicate the populations of cities and in the year indicated. Which city had the greatest percentage increase in population from to
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Problem 12: Each integer through is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack's integer and Jill's integer?
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Problem 13: If then what is
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Problem 14: For any real number and positive integer , define
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Problem 15: If and are integers such that is a factor of , then is
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Problem 16: and are equilateral triangles with parallel sides and the same center, as in the figure. The distance between side and side is the altitude of . The ratio of the area of to the area of is
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Problem 17: If and , find .
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Problem 18: At the end of a professional bowling tournament, the top bowlers have a playoff. First bowls . The loser receives prize and the winner bowls in another game. The loser of this game receives prize and the winner bowls . The loser of this game receives prize and the winner bowls . The winner of this game gets prize and the loser gets prize. In how many orders can bowlers through receive the prizes?
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Problem 19: Simplify
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Problem 20: In one of the adjoining figures a square of side is dissected into four pieces so that and are the midpoints of opposite sides and is perpendicular to . These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, , in this rectangle is
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Problem 21: The complex number satisfies . What is
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Problem 22: For how many integers does a triangle with side lengths , and have all its angles acute?
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Problem 23: The six edges of tetrahedron measure , , , , and units. If the length of edge is , then the length of edge is
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Problem 24: An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is , and one of the base angles is . Find the area of the trapezoid.
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Problem 25: , and are pairwise disjoint sets of people. The average ages of people in the sets and are given in the table below.
Find the average age of the people in the set .
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Problem 26: Suppose that and are positive numbers for which
What is the value of
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Problem 27: In the figure, , and is tangent to the circle with center and diameter . In which one of the following cases is the area of an integer?
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A. ,
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Problem 28: An unfair coin has probability of coming up heads on a single toss. Let be the probability that, in independent tosses of this coin, heads come up exactly times. If , then
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A. must be
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C. must be greater than
D. is not uniquely determined
E. there is no value of for which
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Problem 29: You plot weight () against height () for three of your friends and obtain the points , , . If and , which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.
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Problem 30: Let . Given , consider the sequence defined by for all . For how many real numbers will the sequence , , , take on only a finite number of different values?
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E. infinitely many
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The problems on this page are the property of the MAA's American Mathematics Competitions