1988 AHSME Problems
1988 AHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
Problem 2
Triangles and are similar, with corresponding to and to . If , and , then is:
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?
Problem 4
The slope of the line is
Problem 5
If and are constants and , then is
Problem 6
A figure is an equiangular parallelogram if and only if it is a
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.
Problem 8
If and , what is the ratio of to ?
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?
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
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 ?
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?
Problem 13
If then what is ?
Problem 14
For any real number a and positive integer k, define
What is
?
Problem 15
If and are integers such that is a factor of , then is
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
Problem 17
If and , find
Problem 18
At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First #5 bowls #4. The loser receives th prize and the winner bowls #3 in another game. The loser of this game receives th prize and the winner bowls #2. The loser of this game receives rd prize and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #5 receive the prizes?
Problem 19
Simplify
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
Problem 21
The complex number satisfies . What is ? Note: if , then .
Problem 22
For how many integers does a triangle with side lengths and have all its angles acute?
Problem 23
The six edges of a tetrahedron measure and units. If the length of edge is , then the length of edge is
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.
Problem 25
and are pairwise disjoint sets of people. The average ages of people in the sets and are and respectively. Find the average age of the people in set .
Problem 26
Suppose that and are positive numbers for which
What is the value of ?
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?
Problem 28
An unfair coin has probability of coming up heads on a single toss. Let be the probability that, in independent toss of this coin, heads come up exactly times. If , then
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.
Problem 30
Let . Give , consider the sequence defined by for all . For how many real numbers will the sequence take on only a finite number of different values?
See also
1988 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1987 AHSME |
Followed by 1989 AHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.