Difference between revisions of "1982 AHSME Problems"
m (→Problem 8) |
m (→Problem 21) |
||
Line 328: | Line 328: | ||
[[1982 AHSME Problems/Problem 21|Solution]] | [[1982 AHSME Problems/Problem 21|Solution]] | ||
− | + | ||
== Problem 22 == | == Problem 22 == | ||
Revision as of 11:29, 19 October 2020
1982 AHSME (Answer Key) Printable version: | 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
When the polynomial is divided by the polynomial , the remainder is
Problem 2
If a number eight times as large as is increased by two, then one fourth of the result equals
Problem 3
Evaluate at .
Problem 4
The perimeter of a semicircular region, measured in centimeters, is numerically equal to its area, measured in square centimeters. The radius of the semicircle, measured in centimeters, is
Problem 5
Two positive numbers and are in the ratio where . If , then the smaller of and is
Problem 6
The sum of all but one of the interior angles of a convex polygon equals . The remaining angle is
Problem 7
If the operation is defined by , then which one of the following is FALSE?
Problem 8
By definition, and , where are positive integers and . If form an arithmetic progression with , then equals
Problem 9
A vertical line divides the triangle with vertices , and in the into two regions of equal area. The equation of the line is
Problem 10
In the adjoining diagram, bisects , bisects , and is parallel to . If , and , then the perimeter of is
Problem 11
How many integers with four different digits are there between and such that the absolute value of the difference between the first digit and the last digit is ?
Problem 12
Let , where and are constants. If , the equals
Problem 13
If , and , then equals
Problem 14
In the adjoining figure, points and lie on line segment , and , and are diameters of circle , and , respectively. Circles , and all have radius and the line is tangent to circle at . If intersects circle at points and , then chord has length
Problem 15
Let denote the greatest integer not exceeding . Let and satisfy the simultaneous equations
If is not an integer, then is
Problem 16
A wooden cube has edges of length meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is
Problem 17
How many real numbers satisfy the equation ?
Problem 18
In the adjoining figure of a rectangular solid, and . Find the cosine of .
Problem 19
Let for . The sum of the largest and smallest values of is
Problem 20
The number of pairs of positive integers which satisfy the equation is
Problem 21
In the adjoining figure, the triangle is a right triangle with . Median is perpendicular to median , and side . The length of is
Problem 22
In a narrow alley of width a ladder of length a is placed with its foot at point P between the walls. Resting against one wall at , the distance k above the ground makes a angle with the ground. Resting against the other wall at , a distance h above the ground, the ladder makes a angle with the ground. The width is equal to
Problem 23
The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. The cosine of the smallest angle is
Problem 24
In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If , and , then equals
Problem 25
The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection to intersection , always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability whether to go east or south. Find the probability that through any given morning, he goes through .
Problem 26
If the base representation of a perfect square is , where , then equals
Problem 27
Suppose is a solution of the polynomial equation , where , and are real constants and . Which of the following must also be a solution?
Problem 28
A set of consecutive positive integers beginning with is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is . What number was erased?
Problem 29
Let , and be three positive real numbers whose sum is . If no one of these numbers is more than twice any other, then the minimum possible value of the product is
Problem 30
Find the units digit of the decimal expansion of
.
See also
1982 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1981 AHSME |
Followed by 1983 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.