# 1962 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 Problem 31
- 32 Problem 32
- 33 Problem 33
- 34 Problem 34
- 35 Problem 35
- 36 Problem 36
- 37 Problem 37
- 38 Problem 38
- 39 Problem 39
- 40 Problem 40
- 41 See also

## Problem 1

The expression is equal to:

## Problem 2

The expression $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}$ (Error making remote request. No response to HTTP request) is equal to:

## Problem 3

The first three terms of an arithmetic progression are , in the order shown. The value of is:

## Problem 4

If , then equals:

## Problem 5

If the radius of a circle is increased by unit, the ratio of the new circumference to the new diameter is:

## Problem 6

A square and an equilateral triangle have equal perimeters. The area of the triangle is square inches. Expressed in inches the diagonal of the square is:

## Problem 7

Let the bisectors of the exterior angles at and of meet at . Then, if all measurements are in degrees, equals:

## Problem 8

Given the set of numbers; , of which one is and all the others are . The arithmetic mean of the numbers is:

## Problem 9

When is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is:

## Problem 10

A man drives miles to the seashore in hours and minutes. He returns from the shore to the starting point in hours and minutes. Let be the average rate for the entire trip. Then the average rate for the trip going exceeds in miles per hour, by:

## Problem 11

The difference between the larger root and the smaller root of x^2 - px + (p^2 - 1)/4 = 0 is:

## Problem 12

When \left ( 1 - \frac{1}{a} \right ) ^6 is expanded the sum of the last three coefficients is:

## Problem 13

varies directly as and inverse as . When and . Find when and .

## Problem 14

Let be the limiting sum of the geometric series , as the number of terms increases without bound. Then equals:

## Problem 15

Given with base fixed in length and position. As the vertex moves on a straight line, the intersection point of the three medians moves on:

## Problem 16

Given rectangle with one side inches and area square inches. Rectangle with diagonal inches is similar to . Expressed in square inches the area of is:

## Problem 17

If and , then , in terms of , is:

## Problem 18

A regular dodecagon ( sides) is inscribed in a circle with radius inches. The area of the dodecagon, in square inches, is:

## Problem 19

If the parabola passes through the points , and , the value of is:

## Problem 20

The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be:

## Problem 21

It is given that one root of , with and real numbers, is . The value of is:

## Problem 22

The number 121_b, written in the integral base b, is the square of an integer, for

## Problem 23

In , is the altitude to and is the altitude to . If the lengths of , and are known, the length of is:

## Problem 24

Three machines working together, can do a job in hours. When working alone, needs an additional hours to do the job; , one additional hour; and , additional hours. The value of is:

## Problem 25

Given square with side feet. A circle is drawn through vertices and and tangent to side . The radius of the circle, in feet, is:

## Problem 26

For any real value of the maximum value of is:

## Problem 27

Let represent the operation on two numbers, and , which selects the larger of the two numbers, with . Let represent the operator which selects the smaller of the two numbers, with . Which of the following three rules is (are) correct?

## Problem 28

The set of -values satisfying the equation consists of:

## Problem 29

Which of the following sets of -values satisfy the inequality ?

## Problem 30

Consider the statements:

How many of these imply the negative of the statement " and are both true?"

## Problem 31

The ratio of the interior angles of two regular polygons with sides of unit length is . How many such pairs are there?

## Problem 32

If for and , find .

## Problem 33

The set of -values satisfying the inequality is:

## Problem 34

For what real values of does have real roots?

## Problem 35

A man on his way to dinner short after 6: 00 p.m. observes that the hands of his watch form an angle of . Returning before 7: 00 p.m. he notices that again the hands of his watch form an angle of . The number of minutes that he has been away is:

## Problem 36

If both and are both integers, how many pairs of solutions are there of the equation ?

## Problem 37

is a square with side of unit length. Points and are taken respectively on sides and so that and the quadrilateral has maximum area. In square units this maximum area is:

## Problem 38

The population of Nosuch Junction at one time was a perfect square. Later, with an increase of , the population was one more than a perfect square. Now, with an additional increase of , the population is again a perfect square.

The original population is a multiple of:

## Problem 39

Two medians of a triangle with unequal sides are inches and inches. Its area is square inches. The length of the third median in inches, is:

## Problem 40

The limiting sum of the infinite series, whose th term is is:

## See also

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