1991 AHSME Problems
1991 AHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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
If for any three distinct numbers , , and we define , then is
Problem 2
Problem 3
Problem 4
Which of the following triangles cannot exist?
An acute isosceles triangle
An isosceles right triangle
An obtuse right triangle
A scalene right triangle
A scalene obtuse triangle
Problem 5
In the arrow-shaped polygon [see figure], the angles at vertices and are right angles, , and . The area of the polygon is closest to
Problem 6
If , then
Problem 7
If , and , then
Problem 8
Liquid does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film cm thick. A rectangular box measuring cm by cm by cm is filled with liquid . Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?
Problem 9
From time to time a population increased by , and from time to time the population increased by . Therefore, from time to time the population increased by
Problem 10
Point is units from the center of a circle of radius . How many different chords of the circle contain and have integer lengths?
Problem 11
Jack and Jill run 10 km. They start at the same point, run 5 km up a hill, and return to the starting point by the same route. Jack has a 10 minute head start and runs at the rate of 15 km/hr uphill and 20 km/hr downhill. Jill runs 16 km/hr uphill and 22 km/hr downhill. How far from the top of the hill are they when they pass each other going in opposite directions (in km)?
Problem 12
The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of integers. Let be the measure of the largest interior angle of the hexagon. The largest possible value of , in degrees, is
Problem 13
Horses and are entered in a three-horse race in which ties are not possible. The odds against winning are and the odds against winning are , what are the odds against winning? (By "odds against winning are " we mean the probability of winning the race is .)
Problem 14
If is the cube of a positive integer and is the number of positive integers that are divisors of , then could be
Problem 15
A circular table has 60 chairs around it. There are people seated at this table in such a way that the next person seated must sit next to someone. What is the smallest possible value for ?
Problem 16
One hundred students at Century High School participated in the AHSME last year, and their mean score was 100. The number of non-seniors taking the AHSME was more than the number of seniors, and the mean score of the seniors was higher than that of the non-seniors. What was the mean score of the seniors?
Problem 17
A positive integer is a palindrome if the integer obtained by reversing the sequence of digits of is equal to . The year 1991 is the only year in the current century with the following 2 properties:
(a) It is a palindrome (b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome.
How many years in the millenium between 1000 and 2000 have properties (a) and (b)?
Problem 18
If is the set of points in the complex plane such that is a real number, then is a
right triangle circle hyperbola line parabola
Problem 19
Triangle has a right angle at and . Triangle has a right angle at and . Points and are on opposite sides of . The line through parallel to meets extended at . If where and are relatively prime positive integers, then is
Problem 20
The sum of all real such that is
Problem 21
For all real numbers except and the function is defined by . Suppose . What is the value of ?
Problem 22
Two circles are externally tangent. Lines and are common tangents with and on the smaller circle and on the larger circle. If , then the area of the smaller circle is
Problem 23
If is a square, is the midpoint of , is the midpoint of , and intersect at , and and intersect at , then the area of quadrilateral is
Problem 24
The graph, of is rotated counter-clockwise about the origin to obtain a new graph . Which of the following is an equation for ?
Problem 25
If and for then is closest to which of the following numbers?
Problem 26
An -digit positive integer is cute if its digits are an arrangement of the set and its first digits form an integer that is divisible by , for . For example, is a cute -digit integer because divides , divides , and divides . How many cute -digit integers are there?
Problem 27
If then
Problem 28
Initially an urn contains 100 white and 100 black marbles. Repeatedly 3 marbles are removed (at random) from the urn and replaced with some marbles from a pile outside the urn as follows: 3 blacks are replaced with 1 black, or 2 blacks and 1 white are replaced with a white and a black, or 1 black and 2 whites are replaced with 2 whites, or 3 whites are replaced with a black and a white. Which of the following could be the contents of the urn after repeated applications of this procedure?
2 black 2 white 1 black 1 black and 1 white 1 white
Problem 29
Equilateral triangle has on and on . The triangle is folded along so that vertex now rests at on side . If and then the length of the crease is
Problem 30
For any set , let denote the number of elements in , and let be the number of subsets of , including the empty set and the set itself. If , , and are sets for which and , then what is the minimum possible value of ?
See also
1991 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1990 AHSME |
Followed by 1992 AHSME | |
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All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.