# 1981 AHSME Problems

## 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 , then equals

## Problem 2

Point is on side of square . If has length one and has length two, then the area of the square is

## Problem 3

For , equals

## Problem 4

If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is

## Problem 5

In trapezoid , sides and are parallel, and diagonal and side have equal length. If and , then

## Problem 6

If , then equals

## Problem 7

How many of the first one hundred positive integers are divisible by all of the numbers , , , and ?

## Problem 8

For all positive numbers , , , the product equals

## Problem 9

In the adjoining figure, is a diagonal of the cube. If has length , then the surface area of the cube is

## Problem 10

The lines and are symmetric to each other with respect to the line . If the equation of the line is with and , then the equation of is

## Problem 11

The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length

## Problem 12

If , , and are positive numbers and , then the number obtained by increasing by and decreasing the result by exceeds if and only if

## Problem 13

Suppose that at the end of any year, a unit of money has lost of the value it had at the beginning of that year. Find the smallest integer such that after years, the money will have lost at least of its value (To the nearest thousandth ).

## Problem 14

In a geometric sequence of real numbers, the sum of the first terms is , and the sum of the first terms is . The sum of the first terms is

## Problem 15

If , , and , then is

## Problem 16

The base three representation of is The first digit (on the left) of the base nine representation of is

## Problem 17

The function is not defined for , but, for all non-zero real numbers , . The equation is satisfied by

## Problem 18

The number of real solutions to the equation is

## Problem 19

In , is the midpoint of side , bisects , and . If sides and have lengths and , respectively, then find .

## Problem 20

A ray of light originates from point and travels in a plane, being reflected times between lines and before striking a point (which may be on or ) perpendicularly and retracing its path back to (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for ). If , what is the largest value can have?

## Problem 21

In a triangle with sides of lengths , , and , . The measure of the angle opposite the side length is

## Problem 22

How many lines in a three dimensional rectangular coordiante system pass through four distinct points of the form , where , , and are positive integers not exceeding four?

## Problem 23

Equilateral is inscribed in a circle. A second circle is tangent internally to the circumcircle at and tangent to sides and at points and . If side has length , then segment has length

## Problem 24

If is a constant such that and , then for each positive integer , equals

$\textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ 2\cos n\theta\qquad \textbf{(E)}\ 2^n\cos^n\theta}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 25

In in the adjoining figure, and trisect . The lengths of , and are , , and , respectively. The length of the shortest side of is

## Problem 26

Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is , independent of the outcome of any other toss.)

## Problem 27

In the adjoining figure triangle is inscribed in a circle. Point lies on with , and point lies on with . Side and side each have length equal to the length of chord , and . Chord intersects sides and at and , respectively. The ratio of the area of to the area of is

## Problem 28

Consider the set of all equations , where , , are real constants and for . Let be the largest positive real number which satisfies at least one of these equations. Then

## Problem 29

If , then the sum of the real solutions of

is equal to

## Problem 30

If , , , and are the solutions of the equation , then an equation whose solutions are is

## See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.