1961 AHSME Problems

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Problem 1

When simplified, $(-\frac{1}{125})^{-2/3}$ 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)

Solution

Problem 2

An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ 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)

Solution

Problem 3

If the graphs of $2y+x+3=0$ and $3y+ax+2=0$ are to meet at right angles, the value of $a$ 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)

Solution

Problem 4

Let the set consisting of the squares of the positive integers be called $u$; thus $u$ is the set $1, 4, 9, 16 \ldots$. 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 $u$ 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)

Solution

Problem 5

Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ 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)

Solution

Problem 6

When simplified, $\log{8} \div \log{\frac{1}{8}}$ 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)

Solution

Problem 7

When simplified, the third term in the expansion of $(\frac{a}{\sqrt{x}}-\frac{\sqrt{x}}{a^2})^6$ 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)

Solution

Problem 8

Let the two base angles of a triangle be $A$ and $B$, with $B$ larger than $A$. The altitude to the base divides the vertex angle $C$ into two parts, $C_1$ and $C_2$, with $C_2$ adjacent to side $a$. 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)

Solution

Problem 9

Let $r$ be the result of doubling both the base and exponent of $a^b$, and $b$ does not equal to $0$. If $r$ equals the product of $a^b$ by $x^b$, then $x$ 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)

Solution

Problem 10

Each side of $\triangle ABC$ is $12$ units. $D$ is the foot of the perpendicular dropped from $A$ on $BC$, and $E$ is the midpoint of $AD$. The length of $BE$, 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)

Solution

Problem 11

Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\triangle APR$ 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)

Solution

Problem 12

The first three terms of a geometric progression are $\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$. 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)

Solution

Problem 13

The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. For all real values of $t$ the expression $\sqrt{t^4+t^2}$ 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)

Solution

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 $K$, where $K$ 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)

Solution

Problem 15

If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles (not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ 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)

Solution

Problem 16

An altitude $h$ of a triangle is increased by a length $m$. How much must be taken from the corresponding base $b$ 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)

Solution

Problem 17

In the base ten number system the number $526$ means $5 \times 10^2+2 \times 10 + 6$. In the Land of Mathesis, however, numbers are written in the base $r$. King Rusczyk purchases an automobile there for $440$ monetary units (abbreviated m.u). He gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The base $r$ 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)

Solution

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)

Solution

Problem 19

Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. 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)

Solution

Problem 20

The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ 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)

Solution

Problem 21

Medians $AD$ and $CE$ of $\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$. Let the area of $\triangle MNE$ be $k$ times the area of $\triangle ABC$. Then $k$ 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)

Solution

Problem 22

If $3x^3-9x^2+kx-12$ is divisible by $x-3$, 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)

Solution

Problem 23

Points $P$ and $Q$ are both in the line segment $AB$ and on the same side of its midpoint. $P$ divides $AB$ in the ratio $2:3$, and $Q$ divides $AB$ in the ratio $3:4$. If $PQ=2$, then the length of $AB$ 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)

Solution

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:

$\textbf{(A)}\ \text{The adjacent book referred to is at the left of the middle book}\qquad \textbf{(B)}\ \text{The middle book sells for &#036;36 }\qquad \textbf{(C)}\ \text{The cheapest book sells for &#036;4 }\qquad \textbf{(D)}\ \text{The most expensive book sells for &#036;64 }\qquad \textbf{(E)}\ \text{None of these is correct }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 25

$\triangle ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively in $CB$ and $AB$ and such that $AC=AP=PQ=QB$. The number of degrees in $\angle B$ 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)

Solution

Problem 26

For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. 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)

Solution

Problem 27

Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is:

$\textbf{(A)}\ \infty\qquad \textbf{(B)}\ \text{finite, but greater than 2}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 0$

Solution

Problem 28

If $2137^{753}$ 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)

Solution

Problem 29

Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ 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)

Solution

Problem 30

If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=$?

$\textbf{(A)}\ \frac{a+b}{a+1}\qquad \textbf{(B)}\ \frac{2a+b}{a+1}\qquad \textbf{(C)}\ \frac{a+2b}{1+a}\qquad \textbf{(D)}\ \frac{2a+b}{1-a}}\qquad \textbf{(E)}\ \frac{a+2b}{1-a}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 31

In $\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ 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)n$sides is inscribed in a circle of radius$R$. The area of the polygon is$3R^2$. Then$n$equals:$\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}=55$, in which$x$and$y$are integers, is:$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3}\qquad \textbf{(E)}\ \text{More than three, but finite}} $[[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}$. When$x$is any member of the interval$x \ge 0$. If there exists a number$M$such that no number of the set$S$is greater than$M$,  then$M$is an upper bound of$S$. If there exists a number$m$such that such that no number of the set$S$is less than$m$,  then$m$is a lower bound of$S$. We may then say:$\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)695$is to be written with a factorial base of numeration, that is,$695=a_1+a_2\times2!+a_3\times3!+ \ldots a_n \times n!$where$a_1, a_2, a_3 ... a_n$are integers such that$0 \le a_k \le k,$and$n!$means$n(n-1)(n-2)...2 \times 1$. Find$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 ABC$the median from$A$is given perpendicular to the median from$B$. If$BC=7$and$AC=6$, find the length of$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)d$at uniform speed,$A$can beat$B$by$20$yards,$B$can beat$C$by$10$yards,  and$A$can beat$C$by$28$yards. Then$d$, 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} $[[1961 AHSME Problems/Problem 37|Solution]]

== Problem 38==$ (Error compiling LaTeX. Unknown error_msg)\triangle ABC$is inscribed in a semicircle of radius$r$so that its base$AB$coincides with diameter$AB$.  Point$C$does not coincide with either$A$or$B$. Let$s=AC+BC$. Then, for all permissible positions of$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)1$. Let a be the smallest possible number with the  property that it is always possible to select one pair of points from these five such that the distance between them  is equal to or less than$a$. Then$a$is:$\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}} $[[1961 AHSME Problems/Problem 39|Solution]]

== Problem 40==

Find the minimum value of$ (Error compiling LaTeX. Unknown error_msg)\sqrt{x^2+y^2}$if$5x+12y=60$.$\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 } $

Solution

See also

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