Difference between revisions of "1964 AHSME Problems"
m (→See also) |
Made in 2016 (talk | contribs) (→See also) |
||
(4 intermediate revisions by 3 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{AHSC 40 Problems | ||
+ | |year = 1964 | ||
+ | }} | ||
== Problem 1== | == Problem 1== | ||
Line 6: | Line 9: | ||
\textbf{(B)}\ 25\qquad | \textbf{(B)}\ 25\qquad | ||
\textbf{(C)}\ 10 \qquad | \textbf{(C)}\ 10 \qquad | ||
− | \textbf{(D)}\ 2 | + | \textbf{(D)}\ 2\qquad |
− | \textbf{(E)}\ 1 | + | \textbf{(E)}\ 1 </math> |
[[1964 AHSME Problems/Problem 1|Solution]] | [[1964 AHSME Problems/Problem 1|Solution]] | ||
Line 18: | Line 21: | ||
\textbf{(B)}\ \text{an ellipse} \qquad | \textbf{(B)}\ \text{an ellipse} \qquad | ||
\textbf{(C)}\ \text{a pair of straight lines}\qquad \\ | \textbf{(C)}\ \text{a pair of straight lines}\qquad \\ | ||
− | \textbf{(D)}\ \text{a point | + | \textbf{(D)}\ \text{a point}\qquad |
− | \textbf{(E)}\ \text{None of these | + | \textbf{(E)}\ \text{None of these}</math> |
[[1964 AHSME Problems/Problem 2|Solution]] | [[1964 AHSME Problems/Problem 2|Solution]] | ||
Line 31: | Line 34: | ||
\textbf{(B)}\ 2u \qquad | \textbf{(B)}\ 2u \qquad | ||
\textbf{(C)}\ 3u \qquad | \textbf{(C)}\ 3u \qquad | ||
− | \textbf{(D)}\ v | + | \textbf{(D)}\ v \qquad |
− | \textbf{(E)}\ 2v | + | \textbf{(E)}\ 2v </math> |
[[1964 AHSME Problems/Problem 3|Solution]] | [[1964 AHSME Problems/Problem 3|Solution]] | ||
Line 48: | Line 51: | ||
\textbf{(C)}\ 1 \qquad | \textbf{(C)}\ 1 \qquad | ||
\textbf{(D)}\ \frac{x^2+y^2}{xy}\qquad | \textbf{(D)}\ \frac{x^2+y^2}{xy}\qquad | ||
− | \textbf{(E)}\ \frac{x^2+y^2}{2xy | + | \textbf{(E)}\ \frac{x^2+y^2}{2xy} </math> |
[[1964 AHSME Problems/Problem 4|Solution]] | [[1964 AHSME Problems/Problem 4|Solution]] | ||
Line 56: | Line 59: | ||
If <math>y</math> varies directly as <math>x</math>, and if <math>y=8</math> when <math>x=4</math>, the value of <math>y</math> when <math>x=-8</math> is: | If <math>y</math> varies directly as <math>x</math>, and if <math>y=8</math> when <math>x=4</math>, the value of <math>y</math> when <math>x=-8</math> is: | ||
− | <math>\textbf{(A)}\ -16 | + | <math>\textbf{(A)}\ -16 \qquad |
\textbf{(B)}\ -4 \qquad | \textbf{(B)}\ -4 \qquad | ||
\textbf{(C)}\ -2 \qquad | \textbf{(C)}\ -2 \qquad | ||
− | \textbf{(D)}\ 4k, k= \pm1, \pm2, \dots | + | \textbf{(D)}\ 4k, k= \pm1, \pm2, \dots \qquad \\ |
− | \textbf{(E)}\ 16k, k=\pm1,\pm2,\dots | + | \textbf{(E)}\ 16k, k=\pm1,\pm2,\dots </math> |
[[1964 AHSME Problems/Problem 5|Solution]] | [[1964 AHSME Problems/Problem 5|Solution]] | ||
Line 71: | Line 74: | ||
\textbf{(B)}\ -13\frac{1}{2} \qquad | \textbf{(B)}\ -13\frac{1}{2} \qquad | ||
\textbf{(C)}\ 12\qquad | \textbf{(C)}\ 12\qquad | ||
− | \textbf{(D)}\ 13\frac{1}{2 | + | \textbf{(D)}\ 13\frac{1}{2}\qquad |
− | \textbf{(E)}\ 27 | + | \textbf{(E)}\ 27 </math> |
[[1964 AHSME Problems/Problem 6|Solution]] | [[1964 AHSME Problems/Problem 6|Solution]] | ||
Line 98: | Line 101: | ||
\textbf{(B)}\ \frac{1}{2}\qquad | \textbf{(B)}\ \frac{1}{2}\qquad | ||
\textbf{(C)}\ \frac{5}{8}\qquad | \textbf{(C)}\ \frac{5}{8}\qquad | ||
− | \textbf{(D)}\ \frac{3}{4 | + | \textbf{(D)}\ \frac{3}{4}\qquad |
− | \textbf{(E)}\ 1 | + | \textbf{(E)}\ 1 </math> |
[[1964 AHSME Problems/Problem 8|Solution]] | [[1964 AHSME Problems/Problem 8|Solution]] | ||
Line 105: | Line 108: | ||
== Problem 9== | == Problem 9== | ||
− | A jobber buys an article at | + | A jobber buys an article at <math>\$24</math> less <math>12\frac{1}{2}\%</math>. He then wishes to sell the article at a gain of <math>33\frac{1}{3}\%</math> of his cost |
− | after allowing a 20% discount on his marked price. At what price, in dollars, should the article be marked? | + | after allowing a <math>20\%</math> discount on his marked price. At what price, in dollars, should the article be marked? |
<math>\textbf{(A)}\ 25.20 \qquad | <math>\textbf{(A)}\ 25.20 \qquad | ||
\textbf{(B)}\ 30.00 \qquad | \textbf{(B)}\ 30.00 \qquad | ||
\textbf{(C)}\ 33.60 \qquad | \textbf{(C)}\ 33.60 \qquad | ||
− | \textbf{(D)}\ 40.00 | + | \textbf{(D)}\ 40.00 \qquad |
− | \textbf{(E)}\ \text{none of these | + | \textbf{(E)}\ \text{none of these} </math> |
[[1964 AHSME Problems/Problem 9|Solution]] | [[1964 AHSME Problems/Problem 9|Solution]] | ||
Line 124: | Line 127: | ||
\textbf{(B)}\ s/\sqrt{2} \qquad | \textbf{(B)}\ s/\sqrt{2} \qquad | ||
\textbf{(C)}\ 2s \qquad | \textbf{(C)}\ 2s \qquad | ||
− | \textbf{(D)}\ 2\sqrt{s | + | \textbf{(D)}\ 2\sqrt{s} \qquad |
− | \textbf{(E)}\ 2/\sqrt{s | + | \textbf{(E)}\ 2/\sqrt{s}</math> |
[[1964 AHSME Problems/Problem 10|Solution]] | [[1964 AHSME Problems/Problem 10|Solution]] | ||
Line 136: | Line 139: | ||
\textbf{(B)}\ 21 \qquad | \textbf{(B)}\ 21 \qquad | ||
\textbf{(C)}\ 24 \qquad | \textbf{(C)}\ 24 \qquad | ||
− | \textbf{(D)}\ 27 | + | \textbf{(D)}\ 27 \qquad |
− | \textbf{(E)}\ 30 | + | \textbf{(E)}\ 30 </math> |
[[1964 AHSME Problems/Problem 11|Solution]] | [[1964 AHSME Problems/Problem 11|Solution]] | ||
Line 148: | Line 151: | ||
\textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad | \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad | ||
\textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ | \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ | ||
− | \textbf{(D)}\ \text{For some x}, x^2>0 | + | \textbf{(D)}\ \text{For some x}, x^2>0\qquad |
− | \textbf{(E)}\ \text{For some x}, x^2 \le 0 | + | \textbf{(E)}\ \text{For some x}, x^2 \le 0 </math> |
[[1964 AHSME Problems/Problem 12|Solution]] | [[1964 AHSME Problems/Problem 12|Solution]] | ||
Line 161: | Line 164: | ||
\textbf{(B)}\ 2:5 \qquad | \textbf{(B)}\ 2:5 \qquad | ||
\textbf{(C)}\ 1:2 \qquad | \textbf{(C)}\ 1:2 \qquad | ||
− | \textbf{(D)}\ 2:3 | + | \textbf{(D)}\ 2:3 \qquad |
− | \textbf{(E)}\ 3:4 | + | \textbf{(E)}\ 3:4 </math> |
[[1964 AHSME Problems/Problem 13|Solution]] | [[1964 AHSME Problems/Problem 13|Solution]] | ||
Line 175: | Line 178: | ||
\textbf{(B)}\ 6.75 \qquad | \textbf{(B)}\ 6.75 \qquad | ||
\textbf{(C)}\ 7 \qquad | \textbf{(C)}\ 7 \qquad | ||
− | \textbf{(D)}\ 7.5 | + | \textbf{(D)}\ 7.5 \qquad |
− | \textbf{(E)}\ 8 | + | \textbf{(E)}\ 8 </math> |
[[1964 AHSME Problems/Problem 14|Solution]] | [[1964 AHSME Problems/Problem 14|Solution]] | ||
Line 187: | Line 190: | ||
\textbf{(B)}\ 2Tx-a^2y+2aT=0 \qquad | \textbf{(B)}\ 2Tx-a^2y+2aT=0 \qquad | ||
\textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad \\ | \textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad \\ | ||
− | \textbf{(D)}\ 2Tx-a^2y-2aT=0 | + | \textbf{(D)}\ 2Tx-a^2y-2aT=0 \qquad |
− | \textbf{(E)}\ \text{none of these | + | \textbf{(E)}\ \text{none of these} </math> |
[[1964 AHSME Problems/Problem 15|Solution]] | [[1964 AHSME Problems/Problem 15|Solution]] | ||
Line 200: | Line 203: | ||
\textbf{(B)}\ 22\qquad | \textbf{(B)}\ 22\qquad | ||
\textbf{(C)}\ 21\qquad | \textbf{(C)}\ 21\qquad | ||
− | \textbf{(D)}\ 18 | + | \textbf{(D)}\ 18 \qquad |
− | \textbf{(E)}\ 17 | + | \textbf{(E)}\ 17 </math> |
[[1964 AHSME Problems/Problem 16|Solution]] | [[1964 AHSME Problems/Problem 16|Solution]] | ||
Line 208: | Line 211: | ||
Given the distinct points <math>P(x_1, y_1), Q(x_2, y_2)</math> and <math>R(x_1+x_2, y_1+y_2)</math>. | Given the distinct points <math>P(x_1, y_1), Q(x_2, y_2)</math> and <math>R(x_1+x_2, y_1+y_2)</math>. | ||
− | Line segments are drawn connecting these points to each other and to the origin <math> | + | Line segments are drawn connecting these points to each other and to the origin <math>O</math>. |
Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure <math>OPRQ</math>, | Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure <math>OPRQ</math>, | ||
depending upon the location of the points <math>P, Q</math>, and <math>R</math>, can be: | depending upon the location of the points <math>P, Q</math>, and <math>R</math>, can be: | ||
Line 466: | Line 469: | ||
== Problem 35== | == Problem 35== | ||
− | The sides of a triangle are of lengths <math>13, 14, and 15</math>. The altitudes of the triangle meet at point <math>H</math>. | + | The sides of a triangle are of lengths <math>13, 14,</math> and <math>15</math>. The altitudes of the triangle meet at point <math>H</math>. |
If <math>AD</math> is the altitude to the side length <math>14</math>, what is the ratio <math>HD:HA</math>? | If <math>AD</math> is the altitude to the side length <math>14</math>, what is the ratio <math>HD:HA</math>? | ||
Line 586: | Line 589: | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
− | {{AHSME 40p box|year=1964|before=[[1963 AHSME]]|after=[[1965 AHSME]]}} | + | {{AHSME 40p box|year=1964|before=[[1963 AHSME|1963 AHSC]]|after=[[1965 AHSME|1965 AHSC]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 14:16, 20 February 2020
1964 AHSC (Answer Key) Printable versions: • 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 |
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
What is the value of ?
Problem 2
The graph of is:
Problem 3
When a positive integer is divided by a positive integer
, the quotient is
and the remainder is
, where
and
are integers.
What is the remainder when
is divided by
?
Problem 4
The expression
where and
, is equivalent to:
Problem 5
If varies directly as
, and if
when
, the value of
when
is:
Problem 6
If are in geometric progression, the fourth term is:
Problem 7
Let n be the number of real values of for which the roots of
are equal. Then n equals:
Problem 8
The smaller root of the equation
is:
Problem 9
A jobber buys an article at less
. He then wishes to sell the article at a gain of
of his cost
after allowing a
discount on his marked price. At what price, in dollars, should the article be marked?
Problem 10
Given a square side of length . On a diagonal as base a triangle with three unequal sides is constructed so that its area
equals that of the square. The length of the altitude drawn to the base is:
Problem 11
Given and
, find the value of
Problem 12
Which of the following is the negation of the statement: For all of a certain set,
?
Problem 13
A circle is inscribed in a triangle with side lengths , and
. Let the segments of the side of length
,
made by a point of tangency, be
and
, with
. What is the ratio
?
Problem 14
A farmer bought sheep. He sold
of them for the price paid for the
sheep.
The remaining
sheep were sold at the same price per head as the other
.
Based on the cost, the percent gain on the entire transaction is:
Problem 15
A line through the point cuts from the second quadrant a triangular region with area
. The equation of the line is:
Problem 16
Let and let
be the set of integers
.
The number of members
of
such that
has remainder zero when divided by
is:
Problem 17
Given the distinct points and
.
Line segments are drawn connecting these points to each other and to the origin
.
Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure
,
depending upon the location of the points
, and
, can be:
Problem 18
Let be the number of pairs of values of
and
such that
and
have the same graph. Then
is:
Problem 19
If and
, the numerical value of
is:
Problem 20
The sum of the numerical coefficients of all the terms in the expansion of is:
Problem 21
If , then
equals:
Problem 22
Given parallelogram with
the midpoint of diagonal
. Point
is connected to a point
in
so that
. What is the ratio of the area of
to the area of quadrilateral
?
Problem 23
Two numbers are such that their difference, their sum, and their product are to one another as . The product of the two numbers is:
Problem 24
Let constants. For what value of
is
a minimum?
Problem 25
The set of values of for which
has two factors, with integer coefficients, which are linear in
and
, is precisely:
Problem 26
In a ten-mile race beats
by
miles and
beats
by
miles.
If the runners maintain constant speeds throughout the race,
by how many miles does
beat
?
Problem 27
If is a real number and
where
, then:
Problem 28
The sum of terms of an arithmetic progression is
, and the common difference is
.
If the first term is an integer, and
, then the number of possible values for
is:
Problem 29
In this figure inches,
inches,
inches,
inches.
The length of
, in inches, is:
Problem 30
If , the larger root minus the smaller root is:
Problem 31
Let .
Then
, expressed in terms of
, equals:
Problem 32
If , then:
Problem 33
is a point interior to rectangle
and such that
inches,
inches, and
inches. Then
, in inches, equals:
Problem 34
If is a multiple of
, the sum
, where
, equals:
Problem 35
The sides of a triangle are of lengths and
. The altitudes of the triangle meet at point
.
If
is the altitude to the side length
, what is the ratio
?
Problem 36
In this figure the radius of the circle is equal to the altitude of the equilateral triangle .
The circle is made to roll along the side
, remaining tangent to it at a variable point
and intersecting lines
and
in variable points
and
, respectively.
Let
be the number of degrees in arc
. Then
, for all permissible positions of the circle:
Problem 37
Given two positive number such that
, let
be their arithmetic mean and let
be their positive geometric mean.
Then
minus
is always less than:
Problem 38
The sides and
of
are respectively of lengths
inches, and
inches. The median
is
inches.
Then
, in inches, is:
Problem 39
The magnitudes of the sides of are
, and
, as shown, with
.
Through interior point
and the vertices
, lines are drawn meeting the opposite sides in
, respectively.
Let
. Then, for all positions of point
,
is less than:
Problem 40
A watch loses minutes per day. It is set right at
P.M. on March
.
Let
be the positive correction, in minutes, to be added to the time shown by the watch at a given time.
When the watch shows
A.M. on March
,
equals:
See also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1963 AHSC |
Followed by 1965 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 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.