# Difference between revisions of "1951 AHSME Problems"

m |
|||

Line 352: | Line 352: | ||

== Problem 42 == | == Problem 42 == | ||

+ | |||

+ | If <math> x =\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}} </math>, then: | ||

+ | |||

+ | <math> \textbf{(A)}\ x = 1\qquad\textbf{(B)}\ 0 < x < 1\qquad\textbf{(C)}\ 1 < x < 2\qquad\textbf{(D)}\ x\text{ is infinite} </math> | ||

+ | <math> \textbf{(E)}\ x > 2\text{ but finite} </math> | ||

[[1951 AHSME Problems/Problem 42|Solution]] | [[1951 AHSME Problems/Problem 42|Solution]] | ||

== Problem 43 == | == Problem 43 == | ||

+ | |||

+ | Of the following statements, the only one that is incorrect is: | ||

+ | |||

+ | <math> \textbf{(A)}\ An inequality will remain true after each side is increased, decreased, multiplied or divided (zero excluded) by the same positive quantity. </math> | ||

+ | <math> \textbf{(B)}\ The arithmetic mean of two unequal positive quantities is greater than their geometric mean. </math> | ||

+ | <math> \textbf{(C)}\ If the sum of two positive quantities is given, ther product is largest when they are equal. </math> | ||

+ | <math> \textbf{(D)}\ If </math>a<math> and </math>b<math> are positive and unequal, \frac{1}{2}(a^{2}+b^{2}) is greater than [\frac{1}{2}(a+b)]^{2}. </math> | ||

+ | <math> \textbf{(E)}\ If the product of two positive quantities is given, their sum is greatest when they are equal. </math> | ||

[[1951 AHSME Problems/Problem 43|Solution]] | [[1951 AHSME Problems/Problem 43|Solution]] | ||

== Problem 44 == | == Problem 44 == | ||

+ | |||

+ | If <math> \frac{xy}{x+y}= a,\frac{xz}{x+z}= b,\frac{yz}{y+z}= c </math>, where <math> a, b, c </math> are other than zero, then <math>x</math> equals: | ||

+ | |||

+ | <math> \textbf{(A)}\ \frac{abc}{ab+ac+bc}\qquad\textbf{(B)}\ \frac{2abc}{ab+bc+ac}\qquad\textbf{(C)}\ \frac{2abc}{ab+ac-bc} </math> | ||

+ | <math> \textbf{(D)}\ \frac{2abc}{ab+bc-ac}\qquad\textbf{(E)}\ \frac{2abc}{ac+bc-ab} </math> | ||

[[1951 AHSME Problems/Problem 44|Solution]] | [[1951 AHSME Problems/Problem 44|Solution]] | ||

== Problem 45 == | == Problem 45 == | ||

+ | |||

+ | If you are given <math> \log 8\approx .9031 </math> and <math> \log 9\approx .9542 </math>, then the only logarithm that cannot be found without the use of tables is: | ||

+ | |||

+ | <math> \textbf{(A)}\ \log 17\qquad\textbf{(B)}\ \log\frac{5}{4}\qquad\textbf{(C)}\ \log 15\qquad\textbf{(D)}\ \log 600\qquad\textbf{(E)}\ \log .4 </math> | ||

[[1951 AHSME Problems/Problem 45|Solution]] | [[1951 AHSME Problems/Problem 45|Solution]] | ||

== Problem 46 == | == Problem 46 == | ||

+ | |||

+ | <math>AB</math> is a fixed diameter of a circle whose center is <math>O</math>. From <math>C</math>, any point on the circle, a chord CD is drawn perpendicular to AB. Then, as moves over a semicircle, the bisector of angle <math>OCD</math> cuts the circle in a point that always: | ||

+ | |||

+ | <math> \textbf{(A)}\ \text{bisects the arc }AB\qquad\textbf{(B)}\ \text{trisects the arc }AB\qquad\textbf{(C)}\ \text{varies} </math> | ||

+ | <math> \textbf{(D)}\ \text{is as far from }AB\text{ as from }D\qquad\textbf{(E)}\ \text{is equidistant from }B\text{ and }C </math> | ||

[[1951 AHSME Problems/Problem 46|Solution]] | [[1951 AHSME Problems/Problem 46|Solution]] | ||

== Problem 47 == | == Problem 47 == | ||

+ | |||

+ | If <math>r</math> and <math>s</math> are the roots of the equation <math>ax^2+bx+c=0</math>, the value of <math> \frac{1}{r^{2}}+\frac{1}{s^{2}} </math> is: | ||

+ | |||

+ | <math> \textbf{(A)}\ b^{2}-4ac\qquad\textbf{(B)}\ \frac{b^{2}-4ac}{2a}\qquad\textbf{(C)}\ \frac{b^{2}-4ac}{c^{2}}\qquad\textbf{(D)}\ \frac{b^{2}-2ac}{c^{2}} </math> | ||

+ | <math> \textbf{(E)}\ \text{none of these} </math> | ||

[[1951 AHSME Problems/Problem 47|Solution]] | [[1951 AHSME Problems/Problem 47|Solution]] | ||

== Problem 48 == | == Problem 48 == | ||

+ | |||

+ | The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as: | ||

+ | |||

+ | <math> \textbf{(A)}\ 1: 2\qquad\textbf{(B)}\ 2: 3\qquad\textbf{(C)}\ 2: 5\qquad\textbf{(D)}\ 3: 4\qquad\textbf{(E)}\ 3: 5 </math> | ||

[[1951 AHSME Problems/Problem 48|Solution]] | [[1951 AHSME Problems/Problem 48|Solution]] |

## Revision as of 09:42, 29 August 2011

## 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 Problem 41
- 42 Problem 42
- 43 Problem 43
- 44 Problem 44
- 45 Problem 45
- 46 Problem 46
- 47 Problem 47
- 48 Problem 48
- 49 Problem 49
- 50 Problem 50
- 51 See also

## Problem 1

The percent that is greater than is:

## Problem 2

A rectangular field is half as wide as it is long and is completely enclosed by yards of fencing. The area in terms of is:

## Problem 3

If the length of a diagonal of a square is , then the area of the square is:

## Problem 4

A barn with a flat roof is rectangular in shape, yd. wide, yd. long and yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:

## Problem 5

Mr. A owns a home worth dollars. He sells it to Mr. B at a profit based on the worth of the house. Mr. B sells the house back to Mr. A at a loss. Then:

## Problem 6

The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to:

## Problem 7

An error of is made in the measurement of a line long, while an error of only is made in a measurement of a line long. In comparison with the relative error of the first measurement, the relative error of the second measurement is:

## Problem 8

The price of an article is cut To restore it to its former value, the new price must be increased by:

## Problem 9

An equilateral triangle is drawn with a side length of A new equilateral triangle is formed by joining the midpoints of the sides of the first one. then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. the limit of the sum of the perimeters of all the triangles thus drawn is:

## Problem 10

Of the following statements, the one that is incorrect is:

## Problem 11

The limit of the sum of an infinite number of terms in a geometric progression is $\frac {a}{1 \minus{} r}$ (Error compiling LaTeX. ! Undefined control sequence.) where denotes the first term and $\minus{} 1 < r < 1$ (Error compiling LaTeX. ! Undefined control sequence.) denotes the common ratio. The limit of the sum of their squares is:

$\textbf{(A)}\ \frac {a^2}{(1 \minus{} r)^2} \qquad\textbf{(B)}\ \frac {a^2}{1 \plus{} r^2} \qquad\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2} \qquad\textbf{(D)}\ \frac {4a^2}{1 \plus{} r^2} \qquad\textbf{(E)}\ \text{none of these}$ (Error compiling LaTeX. ! Undefined control sequence.)

## Problem 12

At o'clock, the hour and minute hands of a clock form an angle of:

## Problem 13

can do a piece of work in days. is more efficient than . The number of days it takes to do the same piece of work is:

## Problem 14

In connection with proof in geometry, indicate which one of the following statements is *incorrect*:

## Problem 15

The largest number by which the expression is divisible for all possible integral values of , is:

## Problem 16

If in applying the quadratic formula to a quadratic equation

it happens that , then the graph of will certainly:

## Problem 17

Indicate in which one of the following equations is neither directly nor inversely proportional to :

## Problem 18

The expression is to be factored into two linear prime binomial factors with integer coefficients. This can be one if is:

## Problem 19

A six place number is formed by repeating a three place number; for example, or , etc. Any number of this form is always exactly divisible by:

## Problem 20

When simplified and expressed with negative exponents, the expression is equal to:

## Problem 21

Given: and . The inequality which is not always correct is:

## Problem 22

The values of in the equation: are:

## Problem 23

The radius of a cylindrical box is inches and the height is inches. The number of inches that may be added to either the radius or the height to give the same nonzero increase in volume is:

## Problem 24

when simplified is:

## Problem 25

The apothem of a square having its area numerically equal to its perimeter is compared with the apothem of an equilateral triangle having its area numerically equal to its perimeter. The first apothem will be:

## Problem 26

In the equation the roots are equal when:

## Problem 27

Through a point inside a triangle, three lines are drawn from the vertices to the opposite sides forming six triangular sections. Then:

## Problem 28

The pressure of wind on a sail varies jointly as the area of the sail and the square of the velocity of the wind. The pressure on a square foot is pound when the velocity is miles per hour. The velocity of the wind when the pressure on a square yard is pounds is:

## Problem 29

Of the following sets of data the only one that does not determine the shape of a triangle is:

## Problem 30

If two poles and high are apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is:

## Problem 31

A total of handshakes were exchanged at the conclusion of a party. Assuming that each participant was equally polite toward all the others, the number of people present was:

## Problem 32

If is inscribed in a semicircle whose diameter is , then must be

## Problem 33

The roots of the equation can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:

*[Note: Abscissas means x-coordinate.]*

## Problem 34

The value of is:

## Problem 35

If and , then

## Problem 36

Which of the following methods of proving a geometric figure a locus is not correct?

## Problem 37

A number which when divided by leaves a remainder of , when divided by leaves a remainder of , by leaves a remainder of , etc., down to where, when divided by , it leaves a remainder of , is:

## Problem 38

A rise of feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from to is approximately:

## Problem 39

A stone is dropped into a well and the report of the stone striking the bottom is heard seconds after it is dropped. Assume that the stone falls feet in t seconds and that the velocity of sound is feet per second. The depth of the well is:

## Problem 40

equals:

## Problem 41

The formula expressing the relationship between and in the table is:

## Problem 42

If , then:

## Problem 43

Of the following statements, the only one that is incorrect is:

ab

## Problem 44

If , where are other than zero, then equals:

## Problem 45

If you are given and , then the only logarithm that cannot be found without the use of tables is:

## Problem 46

is a fixed diameter of a circle whose center is . From , any point on the circle, a chord CD is drawn perpendicular to AB. Then, as moves over a semicircle, the bisector of angle cuts the circle in a point that always:

## Problem 47

If and are the roots of the equation , the value of is:

## Problem 48

The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as:

## Problem 49

The medians of a right triangle which are drawn from the vertices of the acute angles are and . The value of the hypotenuse is:

## Problem 50

Tom, Dick and Harry started out on a -mile journey. Tom and Harry went by automobile at the rate of mph, while Dick walked at the rate of mph. After a certain distance, Harry got off and walked on at mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was: