Difference between revisions of "1956 AHSME Problems"
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+ | {{AHSC 50 Problems | ||
+ | |year=1956 | ||
+ | }} | ||
== Problem 1== | == Problem 1== | ||
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== Problem 9== | == Problem 9== | ||
− | + | When you simplify <math>\left[ \sqrt [3]{\sqrt [6]{a^9}} \right]^4\left[ \sqrt [6]{\sqrt [3]{a^9}} \right]^4</math>, the result is: | |
<math>\textbf{(A)}\ a^{16} \qquad\textbf{(B)}\ a^{12} \qquad\textbf{(C)}\ a^8 \qquad\textbf{(D)}\ a^4 \qquad\textbf{(E)}\ a^2 </math> | <math>\textbf{(A)}\ a^{16} \qquad\textbf{(B)}\ a^{12} \qquad\textbf{(C)}\ a^8 \qquad\textbf{(D)}\ a^4 \qquad\textbf{(E)}\ a^2 </math> | ||
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label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); | label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); | ||
label("$D$",D,S); label("$E$",E,NE);</asy> | label("$D$",D,S); label("$E$",E,NE);</asy> | ||
+ | The angle CDE equals: | ||
<math>\textbf{(A)}\ 7\frac{1}{2}^{\circ}\qquad | <math>\textbf{(A)}\ 7\frac{1}{2}^{\circ}\qquad | ||
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<math>\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{more than }6 </math> | <math>\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{more than }6 </math> | ||
− | |||
[[1956 AHSME Problems/Problem 27|Solution]] | [[1956 AHSME Problems/Problem 27|Solution]] | ||
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== Problem 39== | == Problem 39== | ||
− | The hypotenuse <math>c</math> and one arm a of a right triangle are consecutive integers. The square of the second arm is: | + | The hypotenuse <math>c</math> and one arm <math>a</math> of a right triangle are consecutive integers. The square of the second arm is: |
<math> \textbf{(A)}\ ca\qquad | <math> \textbf{(A)}\ ca\qquad | ||
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* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
− | {{AHSME 50p box|year=1956|before=[[1955 AHSME]]|after=[[1957 AHSME]]}} | + | {{AHSME 50p box|year=1956|before=[[1955 AHSME|1955 AHSC]]|after=[[1957 AHSME|1957 AHSC]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 22:44, 12 February 2021
1956 AHSC (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 |
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 value of when is:
Problem 2
Mr. Jones sold two pipes at each. Based on the cost, his profit one was
% and his loss on the other was %. On the sale of the pipes, he:
Problem 3
The distance light travels in one year is approximately miles. The distance light travels in years is:
Problem 4
A man has to invest. He invests at 5% and at 4%. In order to have a yearly income of , he must invest the remainder at:
Problem 5
A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is:
Problem 6
In a group of cows and chickens, the number of legs was 14 more than twice the number of heads. The number of cows was:
Problem 7
The roots of the equation will be reciprocal if:
Problem 8
If , then when
Problem 9
When you simplify , the result is:
Problem 10
A circle of radius inches has its center at the vertex of an equilateral and passes through the other two vertices. The side extended through intersects the circle at . The number of degrees of is:
Problem 11
The expression equals:
Problem 12
If is divided by the quotient is:
Problem 13
Given two positive integers and with . The percent that is less than is:
Problem 14
The points are on a circle . The tangent line at and the secant intersect at lying between and . If and , then equals:
Problem 15
The root(s) of is (are):
Problem 16
The sum of three numbers is . The ratio of the first to the second is , and the ratio of the second to the third is . The second number is:
Problem 17
The fraction was obtained by adding the two fractions and . The values of and must be, respectively:
Problem 18
If , then equals:
Problem 19
Two candles of the same height are lighted at the same time. The first is consumed in hours and the second in hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second?
Problem 20
If and , then the value of to the nearest tenth is:
Problem 21
If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is:
Problem 22
Jones covered a distance of miles on his first trip. On a later trip he traveled miles while going three times as fast. His new time compared with the old time was:
Problem 23
About the equation , with and real constants, we are told that the discriminant is zero. The roots are necessarily:
Problem 24
In the figure , , and .
The angle CDE equals:
Problem 25
The sum of all numbers of the form , where takes on integral values from to is:
Problem 26
Which one of the following combinations of given parts does not determine the indicated triangle?
Problem 27
If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by:
Problem 28
Mr. J left his entire estate to his wife, his daughter, his son, and the cook. His daughter and son got half the estate, sharing in the ratio of to . His wife got twice as much as the son. If the cook received a bequest of , then the entire estate was:
Problem 29
The points of intersection of and are joined in succession. The resulting figure is:
Problem 30
If the altitude of an equilateral triangle is , then the area is:
Problem 31
In our number system the base is ten. If the base were changed to four you would count as follows: The twentieth number would be:
Problem 32
George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time?
Problem 33
The number is equal to:
Problem 34
If is any whole number, is always divisible by
Problem 35
A rhombus is formed by two radii and two chords of a circle whose radius is feet. The area of the rhombus in square feet is:
Problem 36
If the sum is a perfect square and if is less than , then the possible values for are:
Problem 37
On a map whose scale is miles to an inch and a half, a certain estate is represented by a rhombus having a angle. The diagonal opposite is in. The area of the estate in square miles is:
Problem 38
In a right triangle with sides and , and hypotenuse , the altitude drawn on the hypotenuse is . Then:
Problem 39
The hypotenuse and one arm of a right triangle are consecutive integers. The square of the second arm is:
Problem 40
If and , then equals:
Problem 41
The equation where is satisfied by:
Problem 42
The equation has:
Problem 43
The number of scalene triangles having all sides of integral lengths, and perimeter less than is:
Problem 44
If means that and are numbers such that is less than and is less than zero, then:
Problem 45
A wheel with a rubber tire has an outside diameter of in. When the radius has been decreased a quarter of an inch, the number of revolutions in one mile will:
Problem 46
For the equation to be true where is positive, can have:
Problem 47
An engineer said he could finish a highway section in days with his present supply of a certain type of machine. However, with more of these machines the job could be done in days. If the machines all work at the same rate, how many days would it take to do the job with one machine?
Problem 48
If is a positive integer, then can be a positive integer, if and only if is:
Problem 49
Triangle is formed by three tangents to circle and ; then equals:
Problem 50
In . On square is constructed away from the triangle. If is the number of degrees in , then
See also
1956 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1955 AHSC |
Followed by 1957 AHSC | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.