1961 AHSME Problems
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 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:
$\textbf{(A)}\ \text{Addition}\qquad \textbf{(B)}\ \text{Multiplication} \qquad \textbf{(C)}\ \text{Division} \qquad\ \textbf{(D)}\ \text{Extraction of a positive integral root}}\qquad \textbf{(E)}\text{None of these}}$ (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:
$\textbf{(A)}\ \text{They do not intersect}\qquad\ \textbf{(B)}\ \text{They intersect at 1 point only}\qquad\ \textbf{(C)}\ \text{They intersect at 2 points only} \qquad\ \textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2 }\qquad\ \textbf{(E)}\ \text{They coincide} }$ (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 $2 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:
$\textbf{(A)}\ \text{The adjacent book referred to is at the left of the middle book}\qquad \textbf{(B)}\ \text{The middle book sells for $36 }\qquad \textbf{(C)}\ \text{The cheapest book sells for $4 }\qquad \textbf{(D)}\ \text{The most expensive book sells for $64 }\qquad \textbf{(E)}\ \text{None of these is correct }$ (Error compiling LaTeX. Unknown error_msg)
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:
$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 3:4 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:1 }\qquad \textbf{(E)}\ 7:1 }
[[1961 AHSME Problems/Problem 31|Solution]]
== Problem 32==
A regular polygon of$ (Error compiling LaTeX. Unknown error_msg)nR
3R^2
n
\textbf{(A)}\ 8\qquad
\textbf{(B)}\ 10\qquad
\textbf{(C)}\ 12\qquad
\textbf{(D)}\ 15}\qquad
\textbf{(E)}\ 18} $[[1961 AHSME Problems/Problem 32|Solution]]
== Problem 33==
The number of solutions of$ (Error compiling LaTeX. Unknown error_msg)2^{2x}-3^{2y}=55x
y
\textbf{(A)}\ 0\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2\qquad
\textbf{(D)}\ 3}\qquad
\textbf{(E)}\ \text{More than three, but finite}} $[[1961 AHSME Problems/Problem 33|Solution]]
== Problem 34==
Let S be the set of values assumed by the fraction$ (Error compiling LaTeX. Unknown error_msg)\frac{2x+3}{x+2}x
x \ge 0
M
S
M
M
S
m
S
m
m
S
\textbf{(A)}\ \text{m is in S, but M is not in S}\qquad\
\textbf{(B)}\ \text{M is in S, but m is not in S}\qquad\
\textbf{(C)}\ \text{Both m and M are in S}\qquad\
\textbf{(D)}\ \text{Neither m nor M are in S}\qquad\
\textbf{(E)}\ \text{M does not exist either in or outside S} $[[1961 AHSME Problems/Problem 34|Solution]]
== Problem 35==
The number$ (Error compiling LaTeX. Unknown error_msg)695695=a_1+a_2\times2!+a_3\times3!+ \ldots a_n \times n!
a_1, a_2, a_3 ... a_n
0 \le a_k \le k,
n!
n(n-1)(n-2)...2 \times 1
a_4$$ (Error compiling LaTeX. Unknown error_msg)\textbf{(A)}\ 0\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2\qquad
\textbf{(D)}\ 3}\qquad
\textbf{(E)}\ 4} $[[1961 AHSME Problems/Problem 35|Solution]]
== Problem 36==
In$ (Error compiling LaTeX. Unknown error_msg)\triangle ABCA
B
BC=7
AC=6
AB
\textbf{(A)}\ 4\qquad
\textbf{(B)}\ \sqrt{17} \qquad
\textbf{(C)}\ 4.25\qquad
\textbf{(D)}\ 2\sqrt{5} }\qquad
\textbf{(E)}\ 4.5} $[[1961 AHSME Problems/Problem 36|Solution]]
== Problem 37==
In racing over a distance$ (Error compiling LaTeX. Unknown error_msg)dA
B
20
B
C
10
A
C
28
d
\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} $[[1961 AHSME Problems/Problem 37|Solution]]
== Problem 38==$ (Error compiling LaTeX. Unknown error_msg)\triangle ABCr
AB
AB
C
A
B
s=AC+BC
C
\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 } $[[1961 AHSME Problems/Problem 38|Solution]]
== Problem 39==
Any five points are taken inside or on a square with side length$ (Error compiling LaTeX. Unknown error_msg)1a
a
\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}} $[[1961 AHSME Problems/Problem 39|Solution]]
== Problem 40==
Find the minimum value of$ (Error compiling LaTeX. Unknown error_msg)\sqrt{x^2+y^2}5x+12y=60
\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 } $
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.