# 1977 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

## Problem 1

If and , then equals

## Problem 2

Which one of the following statements is false? All equilateral triangles are

## Problem 3

A man has $2.73 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is

## Problem 4

In triangle and . If points , and lie on sides and , respectively, and and , then equals

## Problem 5

The set of all points such that the sum of the (undirected) distances from to two fixed points and equals the distance between and is

## Problem 6

If and are not zero, then equals

## Problem 7

If , then equals

## Problem 8

For every triple of non-zero real numbers, form the number . The set of all numbers formed is

## Problem 9

In the adjoining figure and arc , arc , and arc all have equal length. Find the measure of .

## Problem 10

If , then equals

## Problem 11

For each real number , let be the largest integer not exceeding (i.e., the integer such that ). Which of the following statements is (are) true?

## Problem 12

Al's age is more than the sum of Bob's age and Carl's age, and the square of Al's age is more than the square of the sum of Bob's age and Carl's age. What is the sum of the ages of Al, Bob, and Carl?

## Problem 13

If is a sequence of positive numbers such that for all positive integers , then the sequence is a geometric progression

## Problem 14

How many pairs of integers satisfy the equation ?

## Problem 15

Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is

## Problem 16

If , then the sum equals

## Problem 17

Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one?

## Problem 18

If then

## Problem 19

Let be the point of intersection of the diagonals of convex quadrilateral , and let , and be the centers of the circles circumscribing triangles , and , respectively. Then

## Problem 20

For how many paths consisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram above, is the word CONTEST spelled out as the path is traversed from beginning to end?

## Problem 21

For how many values of the coefficient a do the equations $\begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.) have a common real solution?

## Problem 22

If is a real valued function of the real variable , and is not identically zero, and for all and , then for all and

## Problem 23

If the solutions of the equation are the cubes of the solutions of the equation , then

## Problem 24

Find the sum .

## Problem 25

Determine the largest positive integer n such that is divisible by .

## Problem 26

Let , and be the lengths of sides , and , respectively, of quadrilateral . If is the area of , then

## Problem 27

There are two spherical balls of different sizes lying in two corners of a rectangular room, each touching two walls and the floor. If there is a point on each ball which is inches from each wall which that ball touches and inches from the floor, then the sum of the diameters of the balls is

## Problem 28

Let . What is the remainder when the polynomial is divided by the polynomial ?

## Problem 29

Find the smallest integer such that for all real numbers , and .

## Problem 30

If , and are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon (see adjoining figure), then