AHSME 1960

If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals: $\textbf{(A) }10 \textbf{(B) }9 \textbf{(C) }2 \textbf{(D) }-2 \textbf{(E) }-9$

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

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

The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is: $\textbf{(A) }\text{infinitely many} \textbf{(B) }\text{four} \textbf{(C) }\text{two} \textbf{(D) }\text{one} \textbf{(E) }\text{none}$

The circumference of a circle is $100$ inches. The side of a square inscribed in this circle, expressed in inches, is: $\textbf{(A) }\frac{25\sqrt{2}}{\pi} \textbf{(B) }\frac{50\sqrt{2}}{\pi} \textbf{(C) }\frac{100}{\pi} \textbf{(D) }\frac{100\sqrt{2}}{\pi} \textbf{(E) }50\sqrt{2}$

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

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

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

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 $(6)$ is:

$\textbf{(A) }(1) \textbf{(B) }(2) \textbf{(C) }(3) \textbf{(D) }(4) \textbf{(E) }(5)$

For a given value of $k$ the product of the roots of $x^2-3kx+2k^2-1=0$ is $7$. The roots may be characterized as: $\textbf{(A) }\text{integral and positive} \textbf{(B) }\text{integral and negative} \\ \textbf{(C) }\text{rational, but not integral} \textbf{(D) }\text{irrational} \textbf{(E) } \text{imaginary}$

The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is: $\textbf{(A) }\text{a point} \textbf{(B) }\text{a straight line} \textbf{(C) }\text{two straight lines} \textbf{(D) }\text{a circle} \textbf{(E) }\text{two circles}$

The polygon(s) formed by $y=3x+2, y=-3x+2$, and $y=-2$, is (are): $\textbf{(A) }\text{An equilateral triangle} \textbf{(B) }\text{an isosceles triangle} \textbf{(C) }\text{a right triangle} \\ \textbf{(D) }\text{a triangle and a trapezoid} \textbf{(E) }\text{a quadrilateral}$

If $a$ 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]: $\text{(A) for all a and b} \text{(B) if a }\neq\text{2b} \text{(C) if a }\neq 6 \\ \text{(D) if b }\neq 0 \text{(E) if b }\neq 3$

Triangle $I$ 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: $\textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \\ \textbf{(B)}\ P:p = R:r \text{ } \text{always} \\ \textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \\ \textbf{(D)}\ R:r = K:k \text{ } \text{always} \\ \textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes}$

In the numeration system with base $5$, 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: $\textbf{(A)}\ \text{two consecutive digits} \textbf{(B)}\ \text{two non-consecutive digits} \\ \textbf{(C)}\ \text{three consecutive digits} \textbf{(D)}\ \text{three non-consecutive digits} \\ \textbf{(E)}\ \text{four digits}$

The formula $N=8 \times 10^{8} \times x^{-3/2}$ 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: $\textbf{(A)}\ 10^4 \textbf{(B)}\ 10^6 \textbf{(C)}\ 10^8 \textbf{(D)}\ 10^{12} \textbf{(E)}\ 10^{16}$

The pair of equations $3^{x+y}=81$ and $81^{x-y}=3$ has: $\textbf{(A)}\ \text{no common solution} \\ \textbf{(B)}\ \text{the solution} \text{ } x=2, y=2 \\ \textbf{(C)}\ \text{the solution} \text{ } x=2\frac{1}{2}, y=1\frac{1}{2} \\ \textbf{(D)}\text{ a common solution in positive and negative integers} \\ \textbf{(E)}\ \text{none of these}$

Consider equation $I: x+y+z=46$ 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 $\textbf{(A)}\ \text{I can be solved in consecutive integers} \\ \textbf{(B)}\ \text{I can be solved in consecutive even integers} \\ \textbf{(C)}\ \text{II can be solved in consecutive integers} \\ \textbf{(D)}\ \text{II can be solved in consecutive even integers} \\ \textbf{(E)}\ \text{II can be solved in consecutive odd integers}$

The coefficient of $x^7$ in the expansion of $\left(\frac{x^2}{2}-\frac{2}{x}\right)^8$ is: $\textbf{(A)}\ 56 \textbf{(B)}\ -56 \textbf{(C)}\ 14 \textbf{(D)}\ -14 \textbf{(E)}\ 0$

The diagonal of square $I$ is $a+b$. The area of square $II$ with twice the area of $I$ is: $\textbf{(A)}\ (a+b)^2 \textbf{(B)}\ \sqrt{2}(a+b)^2 \textbf{(C)}\ 2(a+b) \textbf{(D)}\ \sqrt{8}(a+b) \textbf{(E)}\ 4(a+b)$

The equality $(x+m)^2-(x+n)^2=(m-n)^2$, where $m$ and $n$ are unequal non-zero constants, is satisfied by $x=am+bn$, where: $\textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value} \\ \textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values} \\ \textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value} \\ \textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values} \\ \textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value}$

The radius $R$ 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: $\textbf{(A)}\ \text{no real value of x} \\ \textbf{(B)}\ \text{one integral value of x} \\ \textbf{(C)}\ \text{one rational, but not integral, value of x} \\ \textbf{(D)}\ \text{one irrational value of x} \\ \textbf{(E)}\ \text{two real values of x}$

If $\log_{2x}216 = x$, where $x$ is real, then $x$ is: $\textbf{(A)}\ \text{A non-square, non-cube integer}$ $\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number}$ $\textbf{(C)}\ \text{An irrational number}$ $\textbf{(D)}\ \text{A perfect square}$ $\textbf{(E)}\ \text{A perfect cube}$

Let $m$ 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: $\textbf{(A)}\ 2 \textbf{(B)}\ 4 \textbf{(C)}\ 6 \textbf{(D)}\ 8 \textbf{(E)}\ 16$

Find the set of $x$-values satisfying the inequality $|\frac{5-x}{3}|11 \textbf{(E)}\ |x| < 6$

Let $S$ 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 $\textbf{(A)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may be regular} \\ \textbf{(B)}\ S=2660^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular} \\ \textbf{(C)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is regular} \\ \textbf{(D)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{is not regular} \\ \textbf{(E)}\ S=2700^{\circ} \text{ } \text{and} \text{ } P \text{ } \text{may or may not be regular}$

The equation $x-\frac{7}{x-3}=3-\frac{7}{x-3}$ has: $\textbf{(A)}\ \text{infinitely many integral roots} \textbf{(B)}\ \text{no root} \textbf{(C)}\ \text{one integral root}$ $\textbf{(D)}\ \text{two equal integral roots} \textbf{(E)}\ \text{two equal non-integral roots}$

Given the line $3x+5y=15$ and a point on this line equidistant from the coordinate axes. Such a point exists in: $\textbf{(A)}\ \text{none of the quadrants} \textbf{(B)}\ \text{quadrant I only} \textbf{(C)}\ \text{quadrants I, II only}$ $\textbf{(D)}\ \text{quadrants I, II, III only} \textbf{(E)}\ \text{each of the quadrants}$

For $x^2+2x+5$ to be a factor of $x^4+px^2+q$, the values of $p$ and $q$ must be, respectively: $\textbf{(A)}\ -2, 5 \textbf{(B)}\ 5, 25 \textbf{(C)}\ 10, 20 \textbf{(D)}\ 6, 25 \textbf{(E)}\ 14, 25$

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

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

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

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

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

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

The base of a triangle is of length $b$, and the altitude 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: $\textbf{(A)}\ \frac{bx}{h}(h-x) \textbf{(B)}\ \frac{hx}{b}(b-x) \textbf{(C)}\ \frac{bx}{h}(h-2x) \textbf{(D)}\ x(b-x) \textbf{(E)}\ x(h-x)$

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:

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

To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, $a$ and $b$ must be: $\textbf{(A)}\ \text{both rational} \textbf{(B)}\ \text{both real but not rational} \textbf{(C)}\ \text{both not real}$ $\textbf{(D)}\ \text{one real, one not real} \textbf{(E)}\ \text{one real, one not real or both not real}$

Given right $\triangle ABC$ with legs $BC=3, AC=4$. Find the length of the shorter angle trisector from $C$ to the hypotenuse: $\textbf{(A)}\ \frac{32\sqrt{3}-24}{13} \textbf{(B)}\ \frac{12\sqrt{3}-9}{13} \textbf{(C)}\ 6\sqrt{3}-8 \textbf{(D)}\ \frac{5\sqrt{10}}{6} \textbf{(E)}\ \frac{25}{12}$