1960 AHSME Problems

Problem 1

If $2$ is a solution (root) of $x^3+hx+10=0$ , then $h$ equals:

$\textbf{(A)}10\qquad \textbf{(B )}9 \qquad \textbf{(C )}2\qquad \textbf{(D )}-2\qquad \textbf{(E )}-9$

Problem 2

It takes $5$ seconds for a clock to strike $6$ o'clock beginning at $6:00$ o'clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike $12$ o'clock?

$\textbf{(A)}9\frac{1}{5}\qquad \textbf{(B )}10\qquad \textbf{(C )}11\qquad \textbf{(D )}14\frac{2}{5}\qquad \textbf{(E )}\text{none of these}$

Solution

Problem 3

Applied to a bill for $\textdollar{10,000}$ the difference between a discount of $40$ % and two successive discounts of $36$ % and $4$ %, expressed in dollars, is:

$\textbf{(A)}0\qquad \textbf{(B )}144\qquad \textbf{(C )}256\qquad \textbf{(D )}400\qquad \textbf{(E )}416$

Solution

Problem 4

Each of two angles of a triangle is $60^{\circ}$ and the included side is $4$ inches. The area of the triangle, in square inches, is:

$\textbf{(A)}8\sqrt{3}\qquad \textbf{(B )}8\qquad \textbf{(C )}4\sqrt{3}\qquad \textbf{(D )}4\qquad \textbf{(E )}2\sqrt{3}$

Solution

Problem 5

The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is:

$\textbf{(A)}\text{infinitely many}\qquad \textbf{(B )}\text{four}\qquad \textbf{(C )}\text{two}\qquad \textbf{(D )}\text{one}\qquad \textbf{(E )}\text{none}$

Solution

Problem 6

The circumference of a circle is $100$ inches. The side of a square inscribed in this circle, expressed in inches, is:

$\textbf{(A)}\frac{25\sqrt{2}}{\pi}\qquad \textbf{(B )}\frac{50\sqrt{2}}{\pi}\qquad \textbf{(C )}\frac{100}{\pi}\qquad \textbf{(D )}\frac{100\sqrt{2}}{\pi}\qquad \textbf{(E )}50\sqrt{2}$

Solution

Problem 7

Circle $I$ passes through the center of, and is tangent to, circle $II$ . The area of circle $I$ is $4$ square inches. Then the area of circle $II$ , in square inches, is:

$\textbf{(A)}8\qquad \textbf{(B )}8\sqrt{2}\qquad \textbf{(C )}8\sqrt{\pi}\qquad \textbf{(D )}16\qquad \textbf{(E )}16\sqrt{2}$

Solution

Problem 8

The number $2.5252525\ldots$ can be written as a fraction. When reduced to lowest terms the sum of the numerator and denominator of this fraction is:

$\textbf{(A)}7\qquad \textbf{(B)} 29\qquad \textbf{(C )}141\qquad \textbf{(D )}349\qquad \textbf{(E )}\text{none of these}$

Solution

Problem 9

The fraction $\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}$ is (with suitable restrictions of the values of $a, b$ , and $c$ ):

$\text{(A) irreducible}\qquad \text{(B) reducible to negative 1}\qquad \text{(C) reducible to a polynomial of three terms}\qquad \text{(D) reducible to}\frac{a-b+c}{a+b-c}\qquad \text{(E) reducible to}\frac{a+b-c}{a-b+c}$

Solution

Problem 10

Given the following six statements: $\text{(1) All women are good drivers}\\ \text{(2) Some women are good drivers}\\ \text{(3) No men are good drivers}\\ \text{(4) All men are bad drivers}\\ \text{(5) At least one man is a bad driver}\\ \text{(6) All men are good drivers.}$ $The statement that negates statement$ \text{(6)} $is:$ \textbf{(A)}(1)\qquad \textbf{(B )}(2)\qquad \textbf{(C )}(3)\qquad \textbf{(D )}(4)\qquad \textbf{(E )}(5) $[[1960 AHSME Problems/Problem 10|Solution]]

== Problem 11==

For a given value of$ (Error compiling LaTeX. Unknown error_msg)k $the product of the roots of$ x^2-3kx+2k^2-1=0 $is$ 7 $. The roots may be characterized as:$ \textbf{(A)}\text{integral and positive} \qquad \textbf{(B )}\text{integral and negative} \qquad \textbf{(C )}\text{rational, but not integral} \qquad \textbf{(D )}\text{irrational} \qquad \textbf{(E )} \text{imaginary} $[[1960 AHSME Problems/Problem 11|Solution]]

== Problem 12==

The locus of the centers of all circles of given radius$ (Error compiling LaTeX. Unknown error_msg)a $, in the same plane, passing through a fixed point, is:$ \textbf{(A)}\text{a point}\qquad \textbf{(B )}\text{ a straight line}\qquad \textbf{(C )}\text{two straight lines}\qquad \textbf{(D )}\text{a circle}\qquad \textbf{(E )}\text{two circles} $[[1960 AHSME Problems/Problem 12|Solution]]

== Problem 13==

The polygon(s) formed by$ (Error compiling LaTeX. Unknown error_msg)y=3x+2, y=-3x+2 $, and$ y=-2 $, is (are):$ \textbf{(A)}\text{An equilateral triangle}\qquad \textbf{(B )}\text{an isosceles triangle}\qquad \textbf{(C )}\text{a right triangle}\qquad \textbf{(D )}\text{a triangle and a trapezoid}\qquad \textbf{(E )}\text{a quadrilateral} $[[1960 AHSME Problems/Problem 13|Solution]]

== Problem 14==

If$ (Error compiling LaTeX. Unknown error_msg)a $and$ b $are real numbers, the equation$ 3x-5+a=bx+1 $has a unique solution$ x $[The symbol$ a \neq 0 $means that$ a $is different from zero]:$ \textbf{(A)}\text{for all a and b} \qquad \textbf{(B )}\text{if a }\neq\text{2b}\qquad \textbf{(C )}\text{if a }\neq 6\qquad \textbf{(D )}\text{if b }\neq 0\qquad \textbf{(E )}\text{if b }\neq 3 $[[1960 AHSME Problems/Problem 14|Solution]]

== Problem 15==

Triangle$ (Error compiling LaTeX. Unknown error_msg)I $is equilateral with side$ A $, perimeter$ P $, area$ K $, and circumradius$ R $(radius of the circumscribed circle). Triangle$ II $is equilateral with side$ a $, perimeter$ p $, area$ k $, and circumradius$ r $. If$ A $is different from$ a $, then:$ \textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad \textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad \textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad \textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad \textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes} $[[1960 AHSME Problems/Problem 15|Solution]]

== Problem 16==

In the numeration system with base$ (Error compiling LaTeX. Unknown error_msg)5 $, counting is as follows:$ 1, 2, 3, 4, 10, 11, 12, 13, 14, 20,\ldots $. The number whose description in the decimal system is$ 69 $, when described in the base$ 5 $system, is a number with:$ \textbf{(A)}\ \text{two consecutive digits} \qquad \textbf{(B)}\ \text{two non-consecutive digits} \qquad \textbf{(C)}\ \text{three consecutive digits} \qquad \textbf{(D)}\ \text{three non-consecutive digits} \qquad \textbf{(E)}\ \text{four digits} $[[1960 AHSME Problems/Problem 16|Solution]]

== Problem 17==

The formula$ (Error compiling LaTeX. Unknown error_msg)N=8 \times 10^{8} \times x^{-3/2} $gives, for a certain group, the number of individuals whose income exceeds$ x $dollars. The lowest income, in dollars, of the wealthiest$ 800 $individuals is at least:$ \textbf{(A)}\ 10^4\qquad \textbf{(B)}\ 10^6\qquad \textbf{(C)}\ 10^8\qquad \textbf{(D)}\ 10^{12} \qquad \textbf{(E)}\ 10^{16} $[[1960 AHSME Problems/Problem 17|Solution]]

== Problem 18==

The pair of equations$ (Error compiling LaTeX. Unknown error_msg)3^{x+y}=81 $and$ 81^{x-y}=3 $has:$ \textbf{(A)}\ \text{no common solution} \qquad \textbf{(B)}\ \text{the solution} \text{ } x=2, y=2\qquad \textbf{(C)}\ \text{the solution} \text{ } x=2\frac{1}{2}, y=1\frac{1}{2} \qquad \textbf{(D)}\text{ a common solution in positive and negative integers} \qquad \textbf{(E)}\ \text{none of these} $[[1960 AHSME Problems/Problem 18|Solution]]

== Problem 19==

Consider equation$ (Error compiling LaTeX. Unknown error_msg)I: x+y+z=46 $where$ x, y $, and$ z $are positive integers, and equation$ II: x+y+z+w=46 $, where$ x, y, z $, and$ w $are positive integers. Then$ \text{(A) I can be solved in consecutive integers} \qquad \text{(B) I can be solved in consecutive even integers} \qquad \text{(C) II can be solved in consecutive integers} \qquad \text{(D) II can be solved in consecutive even integers} \qquad \text{(E) II can be solved in consecutive odd integers} $[[1960 AHSME Problems/Problem 19|Solution]]

== Problem 20==

The coefficient of$ (Error compiling LaTeX. Unknown error_msg)x^7 $in the expansion of$ (\frac{x^2}{2}-\frac{2}{x})^8 $is:$ \textbf{(A)}\ 56\qquad \textbf{(B)}\ -56\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ -14\qquad \textbf{(E)}\ 0 $[[1960 AHSME Problems/Problem 20|Solution]]

== Problem 21==

The diagonal of square$ (Error compiling LaTeX. Unknown error_msg)I $is$ a+b $. The perimeter of square$ II $with twice the area of$ I $is:$ \textbf{(A)}\ (a+b)^2\qquad \textbf{(B)}\ \sqrt{2}(a+b)^2\qquad \textbf{(C)}\ 2(a+b)\qquad \textbf{(D)}\ \sqrt{8}(a+b) \qquad \textbf{(E)}\ 4(a+b) $[[1960 AHSME Problems/Problem 21|Solution]]

== Problem 22==

The equality$ (Error compiling LaTeX. Unknown error_msg)(x+m)^2-(x+n)^2=(m-n)^2 $, where$ m $and$ n $are unequal non-zero constants, is satisfied by$ x=am+bn $, where:$ \textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad \textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad \textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad \textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad \textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value} $[[1960 AHSME Problems/Problem 22|Solution]]

== Problem 23==

The radius$ (Error compiling LaTeX. Unknown error_msg)R $of a cylindrical box is$ 8 $inches, the height$ H $is$ 3 $inches. The volume$ V = \pi R^2H $is to be increased by the same fixed positive amount when$ R $is increased by$ x $inches as when$ H $is increased by$ x $inches. This condition is satisfied by:$ \textbf{(A)}\ \text{no real value of} \text{ } x\qquad \textbf{(B)}\ \text{one integral value of} \text{ } x\qquad \textbf{(C)}\ \text{one rational, but not integral, value of} \text{ } x\qquad \textbf{(D)}\ \text{one irrational value of} \text{ } x\qquad \textbf{(E)}\ \text{two real values of} \text{ } x $[[1960 AHSME Problems/Problem 23|Solution]]

== Problem 24==

If$ (Error compiling LaTeX. Unknown error_msg)\log_{2x}216 = x $, where$ x $is real, then$ x $is:$ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad \textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad \textbf{(C)}\ \text{An irrational number} \qquad \textbf{(D)}\ \text{A perfect square}\qquad \textbf{(E)}\ \text{A perfect cube} $[[1960 AHSME Problems/Problem 24|Solution]]

== Problem 25==

Let$ (Error compiling LaTeX. Unknown error_msg)m $and$ n $be any two odd numbers, with$ n $less than$ m $. The largest integer which divides all possible numbers of the form$ m^2-n^2 $is:$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16 $[[1960 AHSME Problems/Problem 25|Solution]]

== Problem 26==

Find the set of$ (Error compiling LaTeX. Unknown error_msg)x $-values satisfying the inequality$ |\frac{5-x}{3}|<2 $. [The symbol$ |a| $means$ +a $if$ a $is positive,$ -a $if$ a $is negative,$ 0 $if$ a $is zero. The notation$ 1<a<2 $means that a can have any value between$ 1 $and$ 2 $, excluding$ 1 $and$ 2 $. ]$ \textbf{(A)}\ 1 < x < 11\qquad \textbf{(B)}\ -1 < x < 11\qquad \textbf{(C)}\ x< 11\qquad \textbf{(D)}\ x>11\qquad \textbf{(E)}\ |x| < 6 $[[1960 AHSME Problems/Problem 26|Solution]]

== Problem 27==

Let$ (Error compiling LaTeX. Unknown error_msg)S $be the sum of the interior angles of a polygon$ P $for which each interior angle is$ 7\frac{1}{2} $times the exterior angle at the same vertex. Then$ \textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular}\qquad \textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad \textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular}\qquad \textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular}\qquad \textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular} $[[1960 AHSME Problems/Problem 27|Solution]]

== Problem 28==

The equation$ (Error compiling LaTeX. Unknown error_msg)x-\frac{7}{x-3}=3-\frac{7}{x-3} $has:$ \textbf{(A)}\ \text{infinitely many integral roots}\qquad \textbf{(B)}\ \text{no root}\qquad \textbf{(C)}\ \text{one integral root}\qquad \textbf{(D)}\ \text{two equal integral roots} \qquad \textbf{(E)}\ \text{two equal non-integral roots} $[[1960 AHSME Problems/Problem 28|Solution]]

== Problem 29==

Five times$ (Error compiling LaTeX. Unknown error_msg)A $'s money added to$ B $'s money is more than$ \texdollar{51.00} $. Three times$ A $'s money minus$ B $'s money is$ \textdollar{21.00} $. If$ a $represents$ A $'s money in dollars and$ b $represents$ B $'s money in dollars, then:$ \textbf{(A)}\ a>9, b>6 \qquad \textbf{(B)}\ a>9, b<6 \qquad \textbf{(C)}\ a>9, b=6\qquad \textbf{(D)}\ a>9, \text{but we can put no bounds on} \text{ } b\qquad \textbf{(E)}\ 2a=3b $[[1960 AHSME Problems/Problem 29|Solution]]

== Problem 30==

Given the line$ (Error compiling LaTeX. Unknown error_msg)3x+5y=15 $and a point on this line equidistant from the coordinate axes. Such a point exists in:$ \textbf{(A)}\ \text{none of the quadrants}\qquad \textbf{(B)}\ \text{quadrant I only}\qquad \textbf{(C)}\ \text{quadrants I, II only}\qquad \textbf{(D)}\ \text{quadrants I, II, III only} \qquad \textbf{(E)}\ \text{each of the quadrants} $[[1960 AHSME Problems/Problem 30|Solution]]

== Problem 31==

For$ (Error compiling LaTeX. Unknown error_msg)x^2+2x+5 $to be a factor of$ x^4+px^2+q $, the values of$ p $and$ q $must be, respectively:$ \textbf{(A)}\ -2, 5\qquad \textbf{(B)}\ 5, 25\qquad \textbf{(C)}\ 10, 20\qquad \textbf{(D)}\ 6, 25\qquad \textbf{(E)}\ 14, 25 $[[1960 AHSME Problems/Problem 31|Solution]]

== Problem 32==

In this figure the center of the circle is$ (Error compiling LaTeX. Unknown error_msg)O $.$ AB \perp BC $,$ ADOE $is a straight line,$ AP = AD $, and$ AB$ has a length twice the radius. Then:

$[asy] size(150); defaultpen(linewidth(0.8)+fontsize(10)); real e=350,c=55; pair O=origin,E=dir(e),C=dir(c),B=dir(180+c),D=dir(180+e), rot=rotate(90,B)*O,A=extension(E,D,B,rot); path tangent=A--B; pair P=waypoint(tangent,abs(A-D)/abs(A-B)); draw(unitcircle^^C--B--A--E); dot(A^^B^^C^^D^^E^^P,linewidth(2)); label("$O$",O,dir(290)); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,NE); label("$D$",D,dir(120)); label("$E$",E,SE); label("$P$",P,SW);[/asy]$

$\textbf{(A)} AP^2 = PB \times AB\qquad \textbf{(B)}\ AP \times DO = PB \times AD\qquad \textbf{(C)}\ AB^2 = AD \times DE\qquad \textbf{(D)}\ AB \times AD = OB \times AO\qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 33

You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times\ldots \times 61$ of all prime numbers less than or equal to $61$ , and $n$ takes, successively, the values $2, 3, 4,\ldots, 59$ . Let $N$ be the number of primes appearing in this sequence. Then $N$ is:

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 57\qquad \textbf{(E)}\ 58$

Solution

Problem 34

Two swimmers, at opposite ends of a $90$ -foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.

$\textbf{(A)}\ 24\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 18$

Solution

Problem 35

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$ . It is found that $t_1$ , the length of the tangent, is the mean proportional between $m$ and $n$ . If $m$ and $t$ are integers, then $t$ may have the following number of values:

$\textbf{(A)}\ \text{zero}\qquad \textbf{(B)}\ \text{one}\qquad \textbf{(C)}\ \text{two}\qquad \textbf{(D)}\ \text{three}\qquad \textbf{(E)}\ \text{infinitely many}$

Solution

Problem 36

Let $s_1, s_2, s_3$ be the respective sums of $n, 2n, 3n$ terms of the same arithmetic progression with $a$ as the first term and $d$ as the common difference. Let $R=s_3-s_2-s_1$ . Then $R$ is dependent on:

$\textbf{(A)}\ a\text{ }\text{and}\text{ }d\qquad \textbf{(B)}\ d\text{ }\text{and}\text{ }n\qquad \textbf{(C)}\ a\text{ }\text{and}\text{ }n\qquad \textbf{(D)}\ a, d,\text{ }\text{and}\text{ }n\qquad \textbf{(E)}\ \text{neither} \text{ } a \text{ } \text{nor} \text{ } d \text{ } \text{nor} \text{ } n$

Solution

Problem 37

The base of a triangle is of length $b$ , and the latitude is of length $h$ . A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:

$\textbf{(A)}\ \frac{bx}{h}(h-x)\qquad \textbf{(B)}\ \frac{hx}{b}(b-x)\qquad \textbf{(C)}\ \frac{bx}{h}(h-2x)\qquad \textbf{(D)}\ x(b-x)\qquad \textbf{(E)}\ x(h-x)$

Solution

Problem 38

In this diagram $AB$ and $AC$ are the equal sides of an isosceles $\triangle ABC$ , in which is inscribed equilateral $\triangle DEF$ . Designate $\angle BFD$ by $a$ , $\angle ADE$ by $b$ , and $\angle FEC$ by $c$ . Then:

$[asy] size(150); defaultpen(linewidth(0.8)+fontsize(10)); pair A=(5,12),B=origin,C=(10,0),D=(5/3,4),E=(10-5*.45,12*.45),F=(6,0); draw(A--B--C--cycle^^D--E--F--cycle); draw(anglemark(E,D,A,1,45)^^anglemark(F,E,C,1,45)^^anglemark(D,F,B,1,45)); label("$b$",(D.x+.2,D.y+.25),dir(30)); label("$c$",(E.x,E.y-.4),S); label("$a$",(F.x-.4,F.y+.1),dir(150)); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); label("$D$",D,dir(150)); label("$E$",E,dir(60)); label("$F$",F,S);[/asy]$

$\textbf{(A)}\ b=\frac{a+c}{2}\qquad \textbf{(B)}\ b=\frac{a-c}{2}\qquad \textbf{(C)}\ a=\frac{b-c}{2} \qquad \textbf{(D)}\ a=\frac{b+c}{2}\qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 39

To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$ , $a$ and $b$ must be:

$\textbf{(A)}\ \text{both rational}\qquad \textbf{(B)}\ \text{both real but not rational}\qquad \textbf{(C)}\ \text{both not real}\qquad \textbf{(D)}\ \text{one real, one not real}\qquad \textbf{(E)}\ \text{one real, one not real or both not real}$

Solution

Problem 40

Given right $\triangle ABC$ with legs $BC=3, AC=4$ . Find the length of the shorter angle trisector from $C$ to the hypotenuse:

$\textbf{(A)}\ \frac{32\sqrt{3}-24}{13}\qquad \textbf{(B)}\ \frac{12\sqrt{3}-9}{13}\qquad \textbf{(C)}\ 6\sqrt{3}-8\qquad \textbf{(D)}\ \frac{5\sqrt{10}}{6}\qquad \textbf{(E)}\ \frac{25}{12}$

Solution

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1960 AHSME Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40