# 1974 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 or and or , then is equivalent to

## Problem 2

Let and be such that and , . Then equals

## Problem 3

The coefficient of in the polynomial expansion of

is

## Problem 4

What is the remainder when is divided by ?

## Problem 5

Given a quadrilateral inscribed in a circle with side extended beyond to point , if and , find .

## Problem 6

For positive real numbers and define ' then

## Problem 7

A town's population increased by people, and then this new population decreased by . The town now had less people than it did before the increase. What is the original population?

## Problem 8

What is the smallest prime number dividing the sum ?

## Problem 9

The integers greater than one are arranged in five columns as follows:

(Four consecutive integers appear in each row; in the first, third and other odd numbered rows, the integers appear in the last four columns and increase from left to right; in the second, fourth and other even numbered rows, the integers appear in the first four columns and increase from right to left.)

In which column will the number fall?

## Problem 10

What is the smallest integral value of such that

has no real roots?

## Problem 11

If and are two points on the line whose equation is , then the distance between and , in terms of and is

## Problem 12

If and when , then equals

## Problem 13

Which of the following is equivalent to "If P is true, then Q is false."?

## Problem 14

Which statement is correct?

## Problem 15

If , then equals

## Problem 16

A circle of radius is inscribed in a right isosceles triangle, and a circle of radius is circumscribed about the triangle. Then equals

## Problem 17

If , then equals

## Problem 18

If and , then, in terms of and , equals

## Problem 19

In the adjoining figure is a square and is an equilateral triangle. If the area of is one square inch, then the area of in square inches is

## Problem 20

Let

Then

## Problem 21

In a geometric series of positive terms the difference between the fifth and fourth terms is , and the difference between the second and first terms is . What is the sum of the first five terms of this series?

## Problem 22

The minimum of is attained when is

## Problem 23

In the adjoining figure and are parallel tangents to a circle of radius , with and the points of tangency. is a third tangent with as a point of tangency. If and then is

## Problem 24

A fair die is rolled six times. The probability of rolling at least a five at least five times is

## Problem 25

In parallelogram of the accompanying diagram, line is drawn bisecting at and meeting (extended) at . From vertex , line is drawn bisecting side at and meeting (extended) at . Lines and meet at . If the area of parallelogram is , then the area of the triangle is equal to

## Problem 26

The number of distinct positive integral divisors of excluding and is

## Problem 27

If for all real , then the statement: " whenever and and " is true when

## Problem 28

Which of the following is satisfied by all numbers of the form

where is or , is or ,..., is or ?

## Problem 29

For let be the sum of the first terms of the arithmetic progression whose first term is and whose common difference is ; then is

## Problem 30

A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If is the ratio of the lesser part to the greater part, then the value of

is