Difference between revisions of "1961 AHSME Problems"
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== See also == | == See also == | ||
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* [[AMC 12 Problems and Solutions]] | * [[AMC 12 Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
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+ | {{AHSME box|year=1961|before=[[1960 AHSME]]|after=[[1962 AHSME]]}} | ||
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{{MAA Notice}} | {{MAA Notice}} |
Revision as of 01:28, 23 October 2014
Contents
[hide]- 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
When simplified, becomes:
$\textbf{(A)}\ \frac{1}{25} \qquad \textbf{(B)}\ -\frac{1}{25} \qquad \textbf{(C)}\ 25\qquad \textbf{(D)}\ -25}\qquad \textbf{(E)}\ 25\sqrt{-1}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 2
An automobile travels feet in
seconds. If this rate is maintained for
minutes, how many yards does it travel in
minutes?
$\textbf{(A)}\ \frac{a}{1080r}\qquad \textbf{(B)}\ \frac{30r}{a}\qquad \textbf{(C)}\ \frac{30a}{r}\qquad \textbf{(D)}\ \frac{10r}{a}}\qquad \textbf{(E)}\ \frac{10a}{r}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 3
If the graphs of and
are to meet at right angles, the value of
is:
$\textbf{(A)}\ \pm \frac{2}{3} \qquad \textbf{(B)}\ -\frac{2}{3}\qquad \textbf{(C)}\ -\frac{3}{2} \qquad \textbf{(D)}\ 6}\qquad \textbf{(E)}\ -6}$ (Error compiling LaTeX. Unknown error_msg)
Problem 4
Let the set consisting of the squares of the positive integers be called ; thus
is the set
.
If a certain operation on one or more members of the set always yields a member of the set,
we say that the set is closed under that operation. Then
is closed under:
$\text{(A)}\ \text{Addition}\qquad \text{(B)}\ \text{Multiplication} \qquad \text{(C)}\ \text{Division} \qquad\ \text{(D)}\ \text{Extraction of a positive integral root}}\qquad \text{(E)}\text{None of these}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 5
Let . Then
equals:
$\textbf{(A)}\ (x-2)^4 \qquad \textbf{(B)}\ (x-1)^4 \qquad \textbf{(C)}\ x^4 \qquad \textbf{(D)}\ (x+1)^4 }\qquad \textbf{(E)}\ x^4+1}$ (Error compiling LaTeX. Unknown error_msg)
Problem 6
When simplified, becomes:
$\textbf{(A)}\ 6\log{2} \qquad \textbf{(B)}\ \log{2} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0}\qquad \textbf{(E)}\ -1}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 7
When simplified, the third term in the expansion of is:
$\textbf{(A)}\ \frac{15}{x}\qquad \textbf{(B)}\ -\frac{15}{x}\qquad \textbf{(C)}\ -\frac{6x^2}{a^9} \qquad \textbf{(D)}\ \frac{20}{a^3}}\qquad \textbf{(E)}\ -\frac{20}{a^3}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 8
Let the two base angles of a triangle be and
, with
larger than
.
The altitude to the base divides the vertex angle
into two parts,
and
, with
adjacent to side
. Then:
$\textbf{(A)}\ C_1+C_2=A+B \qquad \textbf{(B)}\ C_1-C_2=B-A \qquad\ \textbf{(C)}\ C_1-C_2=A-B} \qquad \textbf{(D)}\ C_1+C_2=B-A}\qquad \textbf{(E)}\ C_1-C_2=A+B}$ (Error compiling LaTeX. Unknown error_msg)
Problem 9
Let be the result of doubling both the base and exponent of
, and
does not equal to
.
If
equals the product of
by
, then
equals:
$\textbf{(A)} a\qquad \textbf{(B)}\ 2a \qquad \textbf{(C)}\ 4a \qquad \textbf{(D)}\ 2}\qquad \textbf{(E)}\ 4}$ (Error compiling LaTeX. Unknown error_msg)
Problem 10
Each side of is
units.
is the foot of the perpendicular dropped from
on
,
and
is the midpoint of
. The length of
, in the same unit, is:
$\textbf{(A)}\ \sqrt{18} \qquad \textbf{(B)}\ \sqrt{28} \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ \sqrt{63} }\qquad \textbf{(E)}\ \sqrt{98}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 11
Two tangents are drawn to a circle from an exterior point ; they touch the circle at points
and
respectively.
A third tangent intersects segment
in
and
in
, and touches the circle at
. If
, then the perimeter of
is
$\textbf{(A)}\ 42\qquad \textbf{(B)}\ 40.5 \qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 39\frac{7}{8} }\qquad \textbf{(E)}\ \text{not determined by the given information}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 12
The first three terms of a geometric progression are . Find the fourth term.
$\textbf{(A)}\ 1\qquad \textbf{(B)}\ \sqrt[7]{2}\qquad \textbf{(C)}\ \sqrt[8]{2}\qquad \textbf{(D)}\ \sqrt[9]{2}}\qquad \textbf{(E)}\ \sqrt[10]{2}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 13
The symbol means
is a positive number or zero, and
if
is a negative number.
For all real values of
the expression
is equal to?
$\textbf{(A)}\ t^3\qquad \textbf{(B)}\ t^2+t\qquad \textbf{(C)}\ |t^2+t|\qquad \textbf{(D)}\ t\sqrt{t^2+1}}\qquad \textbf{(E)}\ |t|\sqrt{1+t^2}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 14
A rhombus is given with one diagonal twice the length of the other diagonal.
Express the side of the rhombus is terms of , where
is the area of the rhombus in square inches.
$\textbf{(A)}\ \sqrt{K}\qquad \textbf{(B)}\ \frac{1}{2}\sqrt{2K}\qquad \textbf{(C)}\ \frac{1}{3}\sqrt{3K}\qquad \textbf{(D)}\ \frac{1}{4}\sqrt{4K}}\qquad \textbf{(E)}\ \text{None of these are correct}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 15
If men working
hours a day for
days produce
articles, then the number of articles
(not necessarily an integer) produced by
men working
hours a day for
days is:
$\textbf{(A)}\ \frac{x^3}{y^2}\qquad \textbf{(B)}\ \frac{y^3}{x^2}\qquad \textbf{(C)}\ \frac{x^2}{y^3}\qquad \textbf{(D)}\ \frac{y^2}{x^3}}\qquad \textbf{(E)}\ y}$ (Error compiling LaTeX. Unknown error_msg)
Problem 16
An altitude of a triangle is increased by a length
. How much must be taken from the corresponding base
so that the area of the new triangle is one-half that of the original triangle?
$\textbf{(A)}\ \frac{bm}{h+m}\qquad \textbf{(B)}\ \frac{bh}{2h+2m}\qquad \textbf{(C)}\ \frac{b(2m+h)}{m+h}\qquad \textbf{(D)}\ \frac{b(m+h)}{2m+h}}\qquad \textbf{(E)}\ \frac{b(2m+h)}{2(h+m)}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 17
In the base ten number system the number means
.
In the Land of Mathesis, however, numbers are written in the base
.
King Rusczyk purchases an automobile there for
monetary units (abbreviated m.u).
He gives the salesman a
m.u bill, and receives, in change,
m.u. The base
is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 7\qquad \textbf{(D)}\ 8}\qquad \textbf{(E)}\ 12}$ (Error compiling LaTeX. Unknown error_msg)
Problem 18
The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is:
$\textbf{(A)}\ -12 \qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1}\qquad \textbf{(E)}\ 12}$ (Error compiling LaTeX. Unknown error_msg)
Problem 19
Consider the graphs of and
. We may say that:
$\text{(A)}\ \text{They do not intersect}\qquad \ \text{(B)}\ \text{They intersect at 1 point only}\qquad \ \text{(C)}\ \text{They intersect at 2 points only} \qquad \ \text{(D)}\ \text{They intersect at a finite number of points but greater than 2 }\qquad \ \text{(E)}\ \text{They coincide} }$ (Error compiling LaTeX. Unknown error_msg)
Problem 20
The set of points satisfying the pair of inequalities and
is contained entirely in quadrants:
$\textbf{(A)}\ \text{I and II}\qquad \textbf{(B)}\ \text{II and III}\qquad \textbf{(C)}\ \text{I and III}\qquad \textbf{(D)}\ \text{III and IV}}\qquad \textbf{(E)}\ \text{I and IV}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 21
Medians and
of
intersect in
. The midpoint of
is
.
Let the area of
be
times the area of
. Then
equals:
$\textbf{(A)}\ \frac{1}{6}\qquad \textbf{(B)}\ \frac{1}{8}\qquad \textbf{(C)}\ \frac{1}{9}\qquad \textbf{(D)}\ \frac{1}{12}}\qquad \textbf{(E)}\ \frac{1}{16}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 22
If is divisible by
, then it is also divisible by:
$\textbf{(A)}\ 3x^2-x+4\qquad \textbf{(B)}\ 3x^2-4\qquad \textbf{(C)}\ 3x^2+4\qquad \textbf{(D)}\ 3x-4 }\qquad \textbf{(E)}\ 3x+4 }$ (Error compiling LaTeX. Unknown error_msg)
Problem 23
Points and
are both in the line segment
and on the same side of its midpoint.
divides
in the ratio
,
and
divides
in the ratio
. If
, then the length of
is:
$\textbf{(A)}\ 60\qquad \textbf{(B)}\ 70\qquad \textbf{(C)}\ 75\qquad \textbf{(D)}\ 80}\qquad \textbf{(E)}\ 85}$ (Error compiling LaTeX. Unknown error_msg)
Problem 24
Thirty-one books are arranged from left to right in order of increasing prices.
The price of each book differs by from that of each adjacent book.
For the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then:
Problem 25
is isosceles with base
. Points
and
are respectively in
and
and such that
.
The number of degrees in
is:
$\textbf{(A)}\ 25\frac{5}{7}\qquad \textbf{(B)}\ 26\frac{1}{3}\qquad \textbf{(C)}\ 30\qquad \textbf{(D)}\ 40}\qquad \textbf{(E)}\ \text{Not determined by the information given}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 26
For a given arithmetic series the sum of the first terms is
, and the sum of the next
terms is
.
The first term in the series is:
$\textbf{(A)}\ -1221 \qquad \textbf{(B)}\ -21.5 \qquad \textbf{(C)}\ -20.5 \qquad \textbf{(D)}\ 3 }\qquad \textbf{(E)}\ 3.5 }$ (Error compiling LaTeX. Unknown error_msg)
Problem 27
Given two equiangular polygons and
with different numbers of sides;
each angle of
is
degrees and each angle of
is
degrees,
where
is an integer greater than
.
The number of possibilities for the pair
is:
Problem 28
If is multiplied out, the units' digit in the final product in the final product is:
$\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7}\qquad \textbf{(E)}\ 9}$ (Error compiling LaTeX. Unknown error_msg)
Problem 29
Let the roots of be
and
. The equation with roots
and
is:
$\textbf{(A)}\ x^2-bx-ac=0\qquad \textbf{(B)}\ x^2-bx+ac=0 \qquad\ \textbf{(C)}\ x^2+3bx+ca+2b^2=0 \qquad \textbf{(D)}\ x^2+3bx-ca+2b^2=0 }\qquad\ \textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0}$ (Error compiling LaTeX. Unknown error_msg)
Problem 30
If and
, then
?
$\textbf{(A)}\ \frac{a+b}{a+1}\qquad \textbf{(B)}\ \frac{2a+b}{a+1}\qquad \textbf{(C)}\ \frac{a+2b}{1+a}\qquad \textbf{(D)}\ \frac{2a+b}{1-a}}\qquad \textbf{(E)}\ \frac{a+2b}{1-a}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 31
In the ratio
is
. The bisector of the exterior angle at
intersects
extended at
(
is between
and
). The ratio
is:
$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 3:4 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:1 }\qquad \textbf{(E)}\ 7:1 }$ (Error compiling LaTeX. Unknown error_msg)
Problem 32
A regular polygon of sides is inscribed in a circle of radius
. The area of the polygon is
. Then
equals:
$\textbf{(A)}\ 8\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 15}\qquad \textbf{(E)}\ 18}$ (Error compiling LaTeX. Unknown error_msg)
Problem 33
The number of solutions of , in which
and
are integers, is:
$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3}\qquad \textbf{(E)}\ \text{More than three, but finite}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 34
Let S be the set of values assumed by the fraction .
When
is any member of the interval
. If there exists a number
such that no number of the set
is greater than
,
then
is an upper bound of
. If there exists a number
such that such that no number of the set
is less than
,
then
is a lower bound of
. We may then say:
Problem 35
The number is to be written with a factorial base of numeration, that is,
where
are integers such that
and
means
. Find
$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3}\qquad \textbf{(E)}\ 4}$ (Error compiling LaTeX. Unknown error_msg)
Problem 36
In the median from
is given perpendicular to the median from
. If
and
, find the length of
.
$\textbf{(A)}\ 4\qquad \textbf{(B)}\ \sqrt{17} \qquad \textbf{(C)}\ 4.25\qquad \textbf{(D)}\ 2\sqrt{5} }\qquad \textbf{(E)}\ 4.5}$ (Error compiling LaTeX. Unknown error_msg)
Problem 37
In racing over a distance at uniform speed,
can beat
by
yards,
can beat
by
yards,
and
can beat
by
yards. Then
, in yards, equals:
$\textbf{(A)}\ \text{Not determined by the given information}\qquad \textbf{(B)}\ 58\qquad \textbf{(C)}\ 100\qquad \textbf{(D)}\ 116}\qquad \textbf{(E)}\ 120}$ (Error compiling LaTeX. Unknown error_msg)
Problem 38
is inscribed in a semicircle of radius
so that its base
coincides with diameter
.
Point
does not coincide with either
or
. Let
. Then, for all permissible positions of
:
$\textbf{(A)}\ s^2\le8r^2\qquad \textbf{(B)}\ s^2=8r^2 \qquad \textbf{(C)}\ s^2 \ge 8r^2 \qquad\ \textbf{(D)}\ s^2\le4r^2 }\qquad \textbf{(E)}\ x^2=4r^2 }$ (Error compiling LaTeX. Unknown error_msg)
Problem 39
Any five points are taken inside or on a square with side length . Let a be the smallest possible number with the
property that it is always possible to select one pair of points from these five such that the distance between them
is equal to or less than
. Then
is:
$\textbf{(A)}\ \sqrt{3}/3\qquad \textbf{(B)}\ \sqrt{2}/2\qquad \textbf{(C)}\ 2\sqrt{2}/3\qquad \textbf{(D)}\ 1 }\qquad \textbf{(E)}\ \sqrt{2}}$ (Error compiling LaTeX. Unknown error_msg)
Problem 40
Find the minimum value of if
.
$\textbf{(A)}\ \frac{60}{13}\qquad \textbf{(B)}\ \frac{13}{5}\qquad \textbf{(C)}\ \frac{13}{12}\qquad \textbf{(D)}\ 1}\qquad \textbf{(E)}\ 0 }$ (Error compiling LaTeX. Unknown error_msg)
See also
1961 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1960 AHSME |
Followed by 1962 AHSME | |
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 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.