Difference between revisions of "1950 AHSME Problems"
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+ | {{AHSC 50 Problems | ||
+ | |year=1950 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
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== Problem 5 == | == Problem 5 == | ||
− | If five geometric means are inserted between <math>8</math> and <math>5832</math>, the fifth term in the | + | If five geometric means are inserted between <math>8</math> and <math>5832</math>, the fifth term in the geometric series is: |
<math> \textbf{(A)}\ 648\qquad\textbf{(B)}\ 832\qquad\textbf{(C)}\ 1168\qquad\textbf{(D)}\ 1944\qquad\textbf{(E)}\ \text{None of these} </math> | <math> \textbf{(A)}\ 648\qquad\textbf{(B)}\ 832\qquad\textbf{(C)}\ 1168\qquad\textbf{(D)}\ 1944\qquad\textbf{(E)}\ \text{None of these} </math> | ||
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== Problem 6 == | == Problem 6 == | ||
− | The values of <math>y</math> which will satisfy the equations < | + | The values of <math>y</math> which will satisfy the equations <cmath>\begin{array}{rcl} 2x^{2}+6x+5y+1&=&0\\ 2x+y+3&=&0 \end{array}</cmath> may be found by solving: |
<math> \textbf{(A)}\ y^{2}+14y-7=0\qquad\textbf{(B)}\ y^{2}+8y+1=0\qquad\textbf{(C)}\ y^{2}+10y-7=0\qquad\\ \textbf{(D)}\ y^{2}+y-12=0\qquad \textbf{(E)}\ \text{None of these equations} </math> | <math> \textbf{(A)}\ y^{2}+14y-7=0\qquad\textbf{(B)}\ y^{2}+8y+1=0\qquad\textbf{(C)}\ y^{2}+10y-7=0\qquad\\ \textbf{(D)}\ y^{2}+y-12=0\qquad \textbf{(E)}\ \text{None of these equations} </math> | ||
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If the digit <math>1</math> is placed after a two digit number whose tens' digit is <math>t</math>, and units' digit is <math>u</math>, the new number is: | If the digit <math>1</math> is placed after a two digit number whose tens' digit is <math>t</math>, and units' digit is <math>u</math>, the new number is: | ||
− | <math> \textbf{(A)}\ 10t+u+1\qquad\textbf{(B)}\ 100t+10u+1\qquad\textbf{(C)}\ | + | <math> \textbf{(A)}\ 10t+u+1\qquad\textbf{(B)}\ 100t+10u+1\qquad\textbf{(C)}\ 1000t+10u+1\qquad\textbf{(D)}\ t+u+1\qquad\\ \textbf{(E)}\ \text{None of these answers} </math> |
[[1950 AHSME Problems/Problem 7|Solution]] | [[1950 AHSME Problems/Problem 7|Solution]] | ||
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== Problem 9 == | == Problem 9 == | ||
− | The area of | + | The largest area of a triangle that can be inscribed in a semi-circle whose radius is <math>r</math> is: |
<math> \textbf{(A)}\ r^{2}\qquad\textbf{(B)}\ r^{3}\qquad\textbf{(C)}\ 2r^{2}\qquad\textbf{(D)}\ 2r^{3}\qquad\textbf{(E)}\ \frac{1}{2}r^{2} </math> | <math> \textbf{(A)}\ r^{2}\qquad\textbf{(B)}\ r^{3}\qquad\textbf{(C)}\ 2r^{2}\qquad\textbf{(D)}\ 2r^{3}\qquad\textbf{(E)}\ \frac{1}{2}r^{2} </math> | ||
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== Problem 11 == | == Problem 11 == | ||
− | If in the formula <math> C =\frac{en}{R+nr} </math>, <math>n</math> is increased while <math>e</math>, <math>R</math> and <math>r</math> are kept constant, then <math>C</math>: | + | If in the formula <math> C =\frac{en}{R+nr} </math>, where <math>e</math>, <math>n</math>, <math>R</math> and <math>r</math> are all positive, <math>n</math> is increased while <math>e</math>, <math>R</math> and <math>r</math> are kept constant, then <math>C</math>: |
<math> \textbf{(A)}\ \text{Increases}\qquad\textbf{(B)}\ \text{Decreases}\qquad\textbf{(C)}\ \text{Remains constant}\qquad\textbf{(D)}\ \text{Increases and then decreases}\qquad\\ \textbf{(E)}\ \text{Decreases and then increases} </math> | <math> \textbf{(A)}\ \text{Increases}\qquad\textbf{(B)}\ \text{Decreases}\qquad\textbf{(C)}\ \text{Remains constant}\qquad\textbf{(D)}\ \text{Increases and then decreases}\qquad\\ \textbf{(E)}\ \text{Decreases and then increases} </math> | ||
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== Problem 14 == | == Problem 14 == | ||
− | For the simultaneous equations <cmath>2x-3y=8 | + | For the simultaneous equations <cmath>2x-3y=8</cmath><cmath>6y-4x=9</cmath> |
<math> \textbf{(A)}\ x=4,y=0\qquad\textbf{(B)}\ x=0,y=\frac{3}{2}\qquad\textbf{(C)}\ x=0,y=0\qquad\\ \textbf{(D)}\ \text{There is no solution}\qquad\textbf{(E)}\ \text{There are an infinite number of solutions} </math> | <math> \textbf{(A)}\ x=4,y=0\qquad\textbf{(B)}\ x=0,y=\frac{3}{2}\qquad\textbf{(C)}\ x=0,y=0\qquad\\ \textbf{(D)}\ \text{There is no solution}\qquad\textbf{(E)}\ \text{There are an infinite number of solutions} </math> | ||
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== Problem 18 == | == Problem 18 == | ||
+ | Of the following | ||
− | |||
(1) <math> a(x-y)=ax-ay </math> | (1) <math> a(x-y)=ax-ay </math> | ||
− | (2) <math> a | + | |
+ | (2) <math> a^{x-y}=a^x-a^y </math> | ||
+ | |||
(3) <math> \log (x-y)=\log x-\log y </math> | (3) <math> \log (x-y)=\log x-\log y </math> | ||
+ | |||
(4) <math> \frac{\log x}{\log y}=\log{x}-\log{y} </math> | (4) <math> \frac{\log x}{\log y}=\log{x}-\log{y} </math> | ||
− | |||
− | <math> \textbf{(A)}\ \text{Only 1 and 4 are true}\qquad\\ \textbf{(B)}\ \text{Only 1 and 5 are true}\qquad\\ \textbf{(C)}\ \text{Only 1 and 3 are true}\qquad\\ \textbf{(D)}\ \text{Only 1 and 2 are true}\qquad\\ \textbf{(E)}\ \text{Only 1 is true} </math> | + | (5) <math> a(xy)=ax \cdot ay </math> |
+ | |||
+ | <math>\textbf{(A)}\ \text{Only 1 and 4 are true}\qquad\\\textbf{(B)}\ \text{Only 1 and 5 are true}\qquad\\\textbf{(C)}\ \text{Only 1 and 3 are true}\qquad\\\textbf{(D)}\ \text{Only 1 and 2 are true}\qquad\\\textbf{(E)}\ \text{Only 1 is true} </math> | ||
[[1950 AHSME Problems/Problem 18|Solution]] | [[1950 AHSME Problems/Problem 18|Solution]] | ||
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== Problem 20 == | == Problem 20 == | ||
− | When <math>x^{13} | + | When <math>x^{13}+1</math> is divided by <math>x-1</math>, the remainder is: |
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \text{None of these answers} </math> | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \text{None of these answers} </math> | ||
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<math> \textbf{(A)}\ \text{2 real roots}\qquad\textbf{(B)}\ \text{1 real and 1 imaginary root}\qquad\textbf{(C)}\ \text{2 imaginary roots}\qquad\textbf{(D)}\ \text{No roots}\qquad\textbf{(E)}\ \text{1 real root} </math> | <math> \textbf{(A)}\ \text{2 real roots}\qquad\textbf{(B)}\ \text{1 real and 1 imaginary root}\qquad\textbf{(C)}\ \text{2 imaginary roots}\qquad\textbf{(D)}\ \text{No roots}\qquad\textbf{(E)}\ \text{1 real root} </math> | ||
− | [[1950 AHSME Problems/Problem | + | [[1950 AHSME Problems/Problem 24|Solution]] |
== Problem 25 == | == Problem 25 == | ||
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== Problem 36 == | == Problem 36 == | ||
− | A merchant buys goods at <math>25\%</math> | + | A merchant buys goods at <math>25\%</math> off the list price. He desires to mark the goods so that he can give a discount of <math>20\%</math> on the marked price and still clear a profit of <math>25\%</math> on the selling price. What percent of the list price must he mark the goods? |
<math> \textbf{(A)}\ 125\%\qquad\textbf{(B)}\ 100\%\qquad\textbf{(C)}\ 120\%\qquad\textbf{(D)}\ 80\%\qquad\textbf{(E)}\ 75\% </math> | <math> \textbf{(A)}\ 125\%\qquad\textbf{(B)}\ 100\%\qquad\textbf{(C)}\ 120\%\qquad\textbf{(D)}\ 80\%\qquad\textbf{(E)}\ 75\% </math> | ||
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If <math> y =\log_{a}{x} </math>, <math>a>1</math>, which of the following statements is incorrect? | If <math> y =\log_{a}{x} </math>, <math>a>1</math>, which of the following statements is incorrect? | ||
− | <math> \textbf{(A)}\ \text{If }x=1,y=0\qquad\\ \textbf{(B)}\ \text{If }x=a,y=1\qquad\\ \textbf{(C)}\ \text{If }x=-1,y\text{ is imaginary (complex)}\qquad\\ \textbf{(D)}\ \text{If }0<x< | + | <math> \textbf{(A)}\ \text{If }x=1,y=0\qquad\\ \textbf{(B)}\ \text{If }x=a,y=1\qquad\\ \textbf{(C)}\ \text{If }x=-1,y\text{ is imaginary (complex)}\qquad\\ \textbf{(D)}\ \text{If }0<x<1,y\text{ is always less than 0 and decreases without limit as }x\text{ approaches zero}\qquad\\ \textbf{(E)}\ \text{Only some of the above statements are correct} </math> |
[[1950 AHSME Problems/Problem 37|Solution]] | [[1950 AHSME Problems/Problem 37|Solution]] | ||
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Given the series <math> 2+1+\frac{1}{2}+\frac{1}{4}+... </math> and the following five statements: | Given the series <math> 2+1+\frac{1}{2}+\frac{1}{4}+... </math> and the following five statements: | ||
+ | |||
(1) the sum increases without limit | (1) the sum increases without limit | ||
+ | |||
(2) the sum decreases without limit | (2) the sum decreases without limit | ||
+ | |||
(3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small | (3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small | ||
+ | |||
(4) the difference between the sum and 4 can be made less than any positive quantity no matter how small | (4) the difference between the sum and 4 can be made less than any positive quantity no matter how small | ||
+ | |||
(5) the sum approaches a limit | (5) the sum approaches a limit | ||
− | Of these | + | |
+ | Of these statements, the correct ones are: | ||
<math> \textbf{(A)}\ \text{Only }3\text{ and }4\qquad\textbf{(B)}\ \text{Only }5\qquad\textbf{(C)}\ \text{Only }2\text{ and }4\qquad\textbf{(D)}\ \text{Only }2,3\text{ and }4\qquad\textbf{(E)}\ \text{Only }4\text{ and }5 </math> | <math> \textbf{(A)}\ \text{Only }3\text{ and }4\qquad\textbf{(B)}\ \text{Only }5\qquad\textbf{(C)}\ \text{Only }2\text{ and }4\qquad\textbf{(D)}\ \text{Only }2,3\text{ and }4\qquad\textbf{(E)}\ \text{Only }4\text{ and }5 </math> | ||
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== Problem 42 == | == Problem 42 == | ||
− | The equation <math> x^{x^{x | + | The equation <math> x^{x^{x^{.^{.^.}}}}=2 </math> is satisfied when <math>x</math> is equal to: |
<math> \textbf{(A)}\ \infty\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt[4]{2}\qquad\textbf{(D)}\ \sqrt{2}\qquad\textbf{(E)}\ \text{None of these} </math> | <math> \textbf{(A)}\ \infty\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt[4]{2}\qquad\textbf{(D)}\ \sqrt{2}\qquad\textbf{(E)}\ \text{None of these} </math> | ||
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The graph of <math>y=\log x</math> | The graph of <math>y=\log x</math> | ||
− | <math> \textbf{(A)}\text{Cuts the }y\text{-axis}\qquad\\ \textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis}\qquad\\ \textbf{(C)}\ \text{Cuts the }x\text{-axis}\qquad\\ \textbf{(D)}\ \text{Cuts neither axis}\qquad\\ \textbf{(E)}\ \text{Cuts all circles whose center is at the origin} </math> | + | <math> \textbf{(A)}\ \text{Cuts the }y\text{-axis}\qquad\\ \textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis}\qquad\\ \textbf{(C)}\ \text{Cuts the }x\text{-axis}\qquad\\ \textbf{(D)}\ \text{Cuts neither axis}\qquad\\ \textbf{(E)}\ \text{Cuts all circles whose center is at the origin} </math> |
[[1950 AHSME Problems/Problem 44|Solution]] | [[1950 AHSME Problems/Problem 44|Solution]] | ||
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== Problem 50 == | == Problem 50 == | ||
− | A privateer discovers a merchantman <math>10</math> miles to leeward at <math>11\text{:}45 \text{ a.m.}</math> and with a good breeze bears down upon her at <math>11</math> mph, while the merchantman can only make <math>8</math> mph in | + | A privateer discovers a merchantman <math>10</math> miles to leeward at <math>11\text{:}45 \text{ a.m.}</math> and with a good breeze bears down upon her at <math>11</math> mph, while the merchantman can only make <math>8</math> mph in his attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only <math>17</math> miles while the merchantman makes <math>15</math>. The privateer will overtake the merchantman at: |
<math> \textbf{(A)}\ 3\text{:}45\text{ p.m.}\qquad\textbf{(B)}\ 3\text{:}30\text{ p.m.}\qquad\textbf{(C)}\ 5\text{:}00\text{ p.m.}\qquad\textbf{(D)}\ 2\text{:}45\text{ p.m.}\qquad\textbf{(E)}\ 5\text{:}30\text{ p.m.} </math> | <math> \textbf{(A)}\ 3\text{:}45\text{ p.m.}\qquad\textbf{(B)}\ 3\text{:}30\text{ p.m.}\qquad\textbf{(C)}\ 5\text{:}00\text{ p.m.}\qquad\textbf{(D)}\ 2\text{:}45\text{ p.m.}\qquad\textbf{(E)}\ 5\text{:}30\text{ p.m.} </math> | ||
[[1950 AHSME Problems/Problem 50|Solution]] | [[1950 AHSME Problems/Problem 50|Solution]] | ||
+ | |||
+ | == See also == | ||
+ | |||
+ | * [[AMC 12 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{AHSME 50p box|year=1950|before=First AHSC|after=[[1951 AHSME|1951 AHSC]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 19:50, 3 May 2023
1950 AHSC (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
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
If is divided into three parts proportional to
,
, and
, the smallest part is:
Problem 2
Let . When
,
. When
,
is equal to:
Problem 3
The sum of the roots of the equation is equal to:
Problem 4
Reduced to lowest terms, is equal to:
Problem 5
If five geometric means are inserted between and
, the fifth term in the geometric series is:
Problem 6
The values of which will satisfy the equations
may be found by solving:
Problem 7
If the digit is placed after a two digit number whose tens' digit is
, and units' digit is
, the new number is:
Problem 8
If the radius of a circle is increased , the area is increased:
Problem 9
The largest area of a triangle that can be inscribed in a semi-circle whose radius is is:
Problem 10
After rationalizing the numerator of , the denominator in simplest form is:
Problem 11
If in the formula , where
,
,
and
are all positive,
is increased while
,
and
are kept constant, then
:
Problem 12
As the number of sides of a polygon increases from to
, the sum of the exterior angles formed by extending each side in succession:
Problem 13
The roots of are:
Problem 14
For the simultaneous equations
Problem 15
The real factors of are:
Problem 16
The number of terms in the expansion of when simplified is:
Problem 17
The formula which expresses the relationship between and
as shown in the accompanying table is:
Problem 18
Of the following
(1)
(2)
(3)
(4)
(5)
Problem 19
If men can do a job in
days, then
men can do the job in:
Problem 20
When is divided by
, the remainder is:
Problem 21
The volume of a rectangular solid each of whose side, front, and bottom faces are ,
, and
respectively is:
Problem 22
Successive discounts of and
are equivalent to a single discount of:
Problem 23
A man buys a house for dollars and rents it. He puts
of each month's rent aside for repairs and upkeep; pays
dollars a year taxes and realizes
on his investment. The monthly rent is:
Problem 24
The equation has:
Problem 25
The value of is equal to:
Problem 26
if , then
Problem 27
A car travels miles from
to
at
miles per hour but returns the same distance at
miles per hour. The average speed for the round trip is closest to:
Problem 28
Two boys and
start at the same time to ride from Port Jervis to Poughkeepsie,
miles away.
travels
miles an hour slower than
.
reaches Poughkeepsie and at once turns back meeting
miles from Poughkeepsie. The rate of
was:
Problem 29
A manufacturer built a machine which will address envelopes in
minutes. He wishes to build another machine so that when both are operating together they will address
envelopes in
minutes. The equation used to find how many minutes
it would require the second machine to address
envelopes alone is:
Problem 30
From a group of boys and girls, girls leave. There are then left two boys for each girl. After this
boys leave. There are then
girls for each boy. The number of girls in the beginning was:
Problem 31
John ordered pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by
. The ratio of the number of pairs of black socks to the number of pairs of blue socks in the original order was:
Problem 32
A foot ladder is placed against a vertical wall of a building. The foot of the ladder is
feet from the base of the building. If the top of the ladder slips
feet, then the foot of the ladder will slide:
Problem 33
The number of circular pipes with an inside diameter of inch which will carry the same amount of water as a pipe with an inside diameter of
inches is:
Problem 34
When the circumference of a toy balloon is increased from inches to
inches, the radius is increased by:
Problem 35
In triangle ,
inches,
inches,
inches. The radius of the inscribed circle is:
Problem 36
A merchant buys goods at off the list price. He desires to mark the goods so that he can give a discount of
on the marked price and still clear a profit of
on the selling price. What percent of the list price must he mark the goods?
Problem 37
If ,
, which of the following statements is incorrect?
Problem 38
If the expression has the value
for all values of
and
, then the equation
:
Problem 39
Given the series and the following five statements:
(1) the sum increases without limit
(2) the sum decreases without limit
(3) the difference between any term of the sequence and zero can be made less than any positive quantity no matter how small
(4) the difference between the sum and 4 can be made less than any positive quantity no matter how small
(5) the sum approaches a limit
Of these statements, the correct ones are:
Problem 40
The limit of as
approaches
as a limit is:
Problem 41
The least value of the function with
is:
Problem 42
The equation is satisfied when
is equal to:
Problem 43
The sum to infinity of is:
Problem 44
The graph of
Problem 45
The number of diagonals that can be drawn in a polygon of sides is:
Problem 46
In triangle ,
,
, and
. If sides
and
are doubled while
remains the same, then:
Problem 47
A rectangle inscribed in a triangle has its base coinciding with the base of the triangle. If the altitude of the triangle is
, and the altitude
of the rectangle is half the base of the rectangle, then:
Problem 48
A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is:
Problem 49
A triangle has a fixed base that is
inches long. The median from
to side
is
inches long and can have any position emanating from
. The locus of the vertex
of the triangle is:
Problem 50
A privateer discovers a merchantman miles to leeward at
and with a good breeze bears down upon her at
mph, while the merchantman can only make
mph in his attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only
miles while the merchantman makes
. The privateer will overtake the merchantman at:
See also
1950 AHSC (Problems • Answer Key • Resources) | ||
Preceded by First AHSC |
Followed by 1951 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.