AHSME 1952

If the radius of a circle is a rational number, its area is given by a number which is: $\textbf{(A)\ } \text{rational} \textbf{(B)\ } \text{irrational} \textbf{(C)\ } \text{integral} \textbf{(D)\ } \text{a perfect square } \textbf{(E)\ } \text{none of these}$

Two high school classes took the same test. One class of $20$ students made an average grade of $80\%$; the other class of $30$ students made an average grade of $70\%$. The average grade for all students in both classes is: $\textbf{(A)}\ 75\% \textbf{(B)}\ 74\% \textbf{(C)}\ 72\% \textbf{(D)}\ 77\% \textbf{(E)\ }\text{none of these}$

The expression $a^3-a^{-3}$ equals: $\textbf{(A) \ }\left(a-\frac{1}{a}\right)\left(a^2+1+\frac{1}{a^2}\right) \textbf{(B) \ }\left(\frac{1}{a}-a\right)\left(a^2-1+\frac{1}{a^2}\right) \textbf{(C) \ }\left(a-\frac{1}{a}\right)\left(a^2-2+\frac{1}{a^2}\right)$ $\textbf{(D) \ }\left(\frac{1}{a}-a\right)\left(\frac{1}{a^2}+1+a^2\right) \textbf{(E) \ }\text{none of these}$

The cost $C$ of sending a parcel post package weighing $P$ pounds, $P$ an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. The formula for the cost is: $\textbf{(A) \ }C=10+3P \textbf{(B) \ }C=10P+3 \textbf{(C) \ }C=10+3(P-1)$ $\textbf{(D) \ }C=9+3P \textbf{(E) \ }C=10P-7$

The points $(6,12)$ and $(0,-6)$ are connected by a straight line. Another point on this line is: $\textbf{(A) \ }(3,3) \textbf{(B) \ }(2,1) \textbf{(C) \ }(7,16) \textbf{(D) \ }(-1,-4) \textbf{(E) \ }(-3,-8)$

The difference of the roots of $x^2-7x-9=0$ is: $\textbf{(A) \ }+7 \textbf{(B) \ }+\frac{7}{2} \textbf{(C) \ }+9 \textbf{(D) \ }2\sqrt{85} \textbf{(E) \ }\sqrt{85}$

When simplified, $(x^{-1}+y^{-1})^{-1}$ is equal to: $\textbf{(A) \ }x+y \textbf{(B) \ }\frac{xy}{x+y} \textbf{(C) \ }xy \textbf{(D) \ }\frac{1}{xy} \textbf{(E) \ }\frac{x+y}{xy}$

Two equal circles in the same plane cannot have the following number of common tangents. $\textbf{(A) \ }1 \textbf{(B) \ }2 \textbf{(C) \ }3 \textbf{(D) \ }4 \textbf{(E) \ }\text{none of these}$

If $m=\frac{cab}{a-b}$, then $b$ equals: $\textbf{(A) \ }\frac{m(a-b)}{ca} \textbf{(B) \ }\frac{cab-ma}{-m} \textbf{(C) \ }\frac{1}{1+c} \textbf{(D) \ }\frac{ma}{m+ca} \textbf{(E) \ }\frac{m+ca}{ma}$

An automobile went up a hill at a speed of $10$ miles an hour and down the same distance at a speed of $20$ miles an hour. The average speed for the round trip was: $\textbf{(A) \ }12\frac{1}{2}\text{mph} \textbf{(B) \ }13\frac{1}{3}\text{mph} \textbf{(C) \ }14\frac{1}{2}\text{mph} \textbf{(D) \ }15\text{mph} \textbf{(E) \ }\text{none of these}$

If $y=f(x)=\frac{x+2}{x-1}$, then it is incorrect to say: $\textbf{(A)\ }x=\frac{y+2}{y-1} \textbf{(B)\ }f(0)=-2 \textbf{(C)\ }f(1)=0$ $\textbf{(D)\ }f(-2)=0 \textbf{(E)\ }f(y)=x$

The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\frac{1}{2}$. The first term of the progression is: $\textbf{(A) \ }3 \text{ or } 1\frac{1}{2} \textbf{(B) \ }1 \textbf{(C) \ }2\frac{1}{2} \textbf{(D) \ }6 \textbf{(E) \ }9\text{ or }3$

The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when: $\textbf{(A) \ }x=-p \textbf{(B) \ }x=\frac{p}{2} \textbf{(C) \ }x=-2p \textbf{(D) \ }x=\frac{p^2}{4q}$ $\textbf{(E) \ }x=\frac{-p}{2}$

The sides of a triangle are in the ratio $6:8:9$. Then: $\textbf{(A) \ }\text{the triangle is obtuse}$ $\textbf{(B) \ }\text{the angles are in the ratio }6:8:9$ $\textbf{(C) \ }\text{the triangle is acute}$ $\textbf{(D) \ }\text{the angle opposite the largest side is double the angle opposite the smallest side}$ $\textbf{(E) \ }\text{none of these}$

If the base of a rectangle is increased by $10\%$ and the area is unchanged, then the altitude is decreased by: $\textbf{(A) \ }9\% \textbf{(B) \ }10\% \textbf{(C) \ }11\% \textbf{(D) \ }11\frac{1}{9}\% \textbf{(E) \ }9\frac{1}{11}\%$

A merchant bought some goods at a discount of $20\%$ of the list price. He wants to mark them at such a price that he can give a discount of $20\%$ of the marked price and still make a profit of $20\%$ of the selling price. The per cent of the list price at which he should mark them is: $\textbf{(A) \ }20 \textbf{(B) \ }100 \textbf{(C) \ }125 \textbf{(D) \ }80 \textbf{(E) \ }120$

$\log p+\log q=\log(p+q)$ only if: $\textbf{(A) \ }p=q=\text{zero} \textbf{(B) \ }p=\frac{q^2}{1-q} \textbf{(C) \ }p=q=1$ $\textbf{(D) \ }p=\frac{q}{q-1} \textbf{(E) \ }p=\frac{q}{q+1}$

Angle $B$ of triangle $ABC$ is trisected by $BD$ and $BE$ which meet $AC$ at $D$ and $E$ respectively. Then: $\textbf{(A) \ }\frac{AD}{EC}=\frac{AE}{DC} \textbf{(B) \ }\frac{AD}{EC}=\frac{AB}{BC} \textbf{(C) \ }\frac{AD}{EC}=\frac{BD}{BE}$ $\textbf{(D) \ }\frac{AD}{EC}=\frac{(AB)(BD)}{(BE)(BC)} \textbf{(E) \ }\frac{AD}{EC}=\frac{(AE)(BD)}{(DC)(BE)}$

If $\frac{x}{y}=\frac{3}{4}$, then the incorrect expression in the following is: $\textbf{(A) \ }\frac{x+y}{y}=\frac{7}{4} \textbf{(B) \ }\frac{y}{y-x}=\frac{4}{1} \textbf{(C) \ }\frac{x+2y}{x}=\frac{11}{3}$ $\textbf{(D) \ }\frac{x}{2y}=\frac{3}{8} \textbf{(E) \ }\frac{x-y}{y}=\frac{1}{4}$

The sides of a regular polygon of $n$ sides, $n>4$, are extended to form a star. The number of degrees at each point of the star is: $\textbf{(A) \ }\frac{360}{n} \textbf{(B) \ }\frac{(n-4)180}{n} \textbf{(C) \ }\frac{(n-2)180}{n}$ $\textbf{(D) \ }180-\frac{90}{n} \textbf{(E) \ }\frac{180}{n}$

On hypotenuse $AB$ of a right triangle $ABC$ a second right triangle $ABD$ is constructed with hypotenuse $AB$. If $\overline{BC}=1$, $\overline{AC}=b$, and $\overline{AD}=2$, then $\overline{BD}$ equals: $\textbf{(A) \ }\sqrt{b^2+1} \textbf{(B) \ }\sqrt{b^2-3} \textbf{(C) \ }\sqrt{b^2+1}+2$ $\textbf{(D) \ }b^2+5 \textbf{(E) \ }\sqrt{b^2+3}$

If $\frac{x^2-bx}{ax-c}=\frac{m-1}{m+1}$ has roots which are numerically equal but of opposite signs, the value of $m$ must be: $\textbf{(A) \ }\frac{a-b}{a+b} \textbf{(B) \ }\frac{a+b}{a-b} \textbf{(C) \ }c \textbf{(D) \ }\frac{1}{c} \textbf{(E) \ }1$

In the figure, it is given that angle $C = 90^{\circ}$, $\overline{AD} = \overline{DB}$, $DE \perp AB$, $\overline{AB} = 20$, and $\overline{AC} = 12$. The area of quadrilateral $ADEC$ is:

$\textbf{(A)}\ 75 \textbf{(B)}\ 58\frac{1}{2} \textbf{(C)}\ 48 \textbf{(D)}\ 37\frac{1}{2} \textbf{(E)}\ \text{none of these}$

A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately: $\textbf{(A)}\ \text{200 yd.} \textbf{(B)}\ \text{352 yd.} \textbf{(C)}\ \text{300 yd.} \textbf{(D)}\ \text{245 yd.} \textbf{(E)}\ \text{512 yd.}$

If $\left(r+\frac1r\right)^2=3$, then $r^3+\frac1{r^3}$ equals $\textbf{(A)}\ 1 \textbf{(B)}\ 2 \textbf{(C)}\ 0 \textbf{(D)}\ 3 \textbf{(E)}\ 6$

The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is: $\textbf{(A)}\ 1:2 \textbf{(B)}\ 1:3 \textbf{(C)}\ 1:\sqrt3 \textbf{(D)}\ \sqrt3:2 \textbf{(E)}\ 2:3$

In the table shown, the formula relating x and y is: $$\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5\\ \hline y & 3 & 7 & 13 & 21 & 31\\ \hline\end{array}$$ $\text{(A) } y = 4x - 1 \text{(B) } y = x^3 - x^2 + x + 2 \\ \text{(C) } y = x^2 + x + 1 \text{(D) } y = (x^2 + x + 1)(x - 1) \\ \text{(E) } \text{none of these}$

In a circle of radius $5$ units, $CD$ and $AB$ are perpendicular diameters. A chord $CH$ cutting $AB$ at $K$ is $8$ units long. The diameter $AB$ is divided into two segments whose dimensions are: $\textbf{(A)}\ 1.25, 8.75 \textbf{(B)}\ 2.75,7.25 \textbf{(C)}\ 2,8 \textbf{(D)}\ 4,6 \textbf{(E)}\ \text{none of these}$

When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is: $\textbf{(A)}\ 1: 2 \textbf{(B)}\ 2: 1 \textbf{(C)}\ 1: 4 \textbf{(D)}\ 4: 1 \textbf{(E)}\ 1: 1$

Given $12$ points in a plane no three of which are collinear, the number of lines they determine is: $\textbf{(A)}\ 24 \textbf{(B)}\ 54 \textbf{(C)}\ 120 \textbf{(D)}\ 66 \textbf{(E)}\ \text{none of these}$

$K$ takes $30$ minutes less time than $M$ to travel a distance of $30$ miles. $K$ travels $\frac {1}{3}$ mile per hour faster than $M$. If $x$ is $K$'s rate of speed in miles per hours, then $K$'s time for the distance is: $\textbf{(A)}\ \dfrac{x + \frac {1}{3}}{30} \textbf{(B)}\ \dfrac{x - \frac {1}{3}}{30} \textbf{(C)}\ \frac{30}{x+\frac{1}{3}} \textbf{(D)}\ \frac{30}{x} \textbf{(E)}\ \frac{x}{30}$

A circle and a square have the same perimeter. Then: $\text{(A) their areas are equal} \\ \text{(B) the area of the circle is the greater} \\ \text{(C) the area of the square is the greater} \\ \text{(D) the area of the circle is } \pi \text{ times the area of the square} \\ \text{(E) none of these}$

The price of an article was increased $p\%$. Later the new price was decreased $p\%$. If the last price was one dollar, the original price was: $\textbf{(A)}\ \frac{1-p^2}{200} \textbf{(B)}\ \frac{\sqrt{1-p^2}}{100} \textbf{(C)}\ \text{one dollar} \\ \textbf{(D)}\ 1-\frac{p^2}{10000-p^2} \textbf{(E)}\ \frac{10000}{10000-p^2}$

With a rational denominator, the expression $\frac {\sqrt {2}}{\sqrt {2} + \sqrt {3} - \sqrt {5}}$ is equivalent to: $\textbf{(A)}\ \frac {3 + \sqrt {6} + \sqrt {15}}{6} \textbf{(B)}\ \frac {\sqrt {6} - 2 + \sqrt {10}}{6} \textbf{(C)}\ \frac{2+\sqrt{6}+\sqrt{10}}{10} \\ \textbf{(D)}\ \frac {2 + \sqrt {6} - \sqrt {10}}{6} \textbf{(E)}\ \text{none of these}$

To be continuous at $x = - 1$, the value of $\frac {x^3 + 1}{x^2 - 1}$ is taken to be: $\textbf{(A)}\ - 2 \textbf{(B)}\ 0 \textbf{(C)}\ \frac {3}{2} \textbf{(D)}\ \infty \textbf{(E)}\ -\frac{3}{2}$

Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is: $\textbf{(A)}\ 21\frac{1}{3}\pi-32\sqrt{3} \textbf{(B)}\ 32\sqrt{3}+21\frac{1}{3}\pi \textbf{(C)}\ 32\sqrt{3}+42\frac{2}{3}\pi \\ \textbf{(D)}\ 16\sqrt {3} + 42\frac {2}{3}\pi \textbf{(E)}\ 42\frac {2}{3}\pi$

The area of a trapezoidal field is $1400$ square yards. Its altitude is $50$ yards. Find the two bases, if the number of yards in each base is an integer divisible by $8$. The number of solutions to this problem is: $\textbf{(A)}\ \text{none} \textbf{(B)}\ \text{one} \textbf{(C)}\ \text{two} \textbf{(D)}\ \text{three} \textbf{(E)}\ \text{more than three}$

If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is: $\textbf{(A)}\ \frac {\sqrt {8d^2 - p^2}}{2} \textbf{(B)}\ \frac {\sqrt {8d^2 + p^2}}{2} \textbf{(C)}\ \frac{\sqrt{6d^2-p^2}}{2} \\ \textbf{(D)}\ \frac {\sqrt {6d^2 + p^2}}{2} \textbf{(E)}\ \frac {8d^2 - p^2}{4}$

In order to draw a graph of $ax^2+bx+c$, a table of values was constructed. These values of the function for a set of equally spaced increasing values of $x$ were $3844, 3969, 4096, 4227, 4356, 4489, 4624$, and $4761$. The one which is incorrect is: $\text{(A) } 4096 \text{(B) } 4356 \text{(C) } 4489 \text{(D) } 4761 \text{(E) } \text{none of these}$

Increasing the radius of a cylinder by $6$ units increased the volume by $y$ cubic units. Increasing the height of the cylinder by $6$ units also increases the volume by $y$ cubic units. If the original height is $2$, then the original radius is: $\text{(A) } 2 \text{(B) } 4 \text{(C) } 6 \text{(D) } 6\pi \text{(E) } 8$

Let $D$ represent a repeating decimal. If $P$ denotes the $r$ figures of $D$ which do not repeat themselves, and $Q$ denotes the $s$ figures of $D$ which do repeat themselves, then the incorrect expression is: $\text{(A) } D = .PQQQ\ldots \\ \text{(B) } 10^rD = P.QQQ\ldots \\ \text{(C) } 10^{r + s}D = PQ.QQQ\ldots \\ \text{(D) } 10^r(10^s - 1)D = Q(P - 1) \\ \text{(E) } 10^r\cdot10^{2s}D = PQQ.QQQ\ldots$

The diameter of a circle is divided into $n$ equal parts. On each part a semicircle is constructed. As $n$ becomes very large, the sum of the lengths of the arcs of the semicircles approaches a length: $\textbf{(A) }$ equal to the semi-circumference of the original circle $\textbf{(B) }$ equal to the diameter of the original circle $\textbf{(C) }$ greater than the diameter, but less than the semi-circumference of the original circle $\textbf{(D) }$ that is infinite $\textbf{(E) }$ greater than the semi-circumference

If an integer of two digits is $k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by $\textbf{(A) \ } 9-k \textbf{(B) \ } 10-k \textbf{(C) \ } 11-k \textbf{(D) \ } k-1 \textbf{(E) \ } k+1$

If $a$ and $b$ are two unequal positive numbers, then: $\text{(A) } \frac{2ab}{a+b}>\sqrt{ab}>\frac{a+b}{2} \text{(B) } \sqrt{ab}>\frac{2ab}{a+b}>\frac{a+b}{2} \\ \text{(C) } \frac{2ab}{a+b}>\frac{a+b}{2}>\sqrt{ab} \text{(D) } \frac{a+b}{2}>\frac{2ab}{a+b}>\sqrt{ab} \\ \text{(E) } \frac {a + b}{2} > \sqrt {ab} > \frac {2ab}{a + b}$

The base of a new rectangle equals the sum of the diagonal and the greater side of a given rectangle, while the altitude of the new rectangle equals the difference of the diagonal and the greater side of the given rectangle. The area of the new rectangle is: $\text{(A) greater than the area of the given rectangle} \\ \text{(B) equal to the area of the given rectangle} \\ \text{(C) equal to the area of a square with its side equal to the smaller side of the given rectangle} \\ \text{(D) equal to the area of a square with its side equal to the greater side of the given rectangle} \\ \text{(E) equal to the area of a rectangle whose dimensions are the diagonal and the shorter side of the given rectangle}$

In the set of equations $z^x = y^{2x}, 2^z = 2\cdot4^x, x + y + z = 16$, the integral roots in the order $x,y,z$ are: $\textbf{(A) } 3,4,9 \textbf{(B) } 9,-5,-12 \textbf{(C) } 12,-5,9 \textbf{(D) } 4,3,9 \textbf{(E) } 4,9,3$

Two cyclists, $k$ miles apart, and starting at the same time, would be together in $r$ hours if they traveled in the same direction, but would pass each other in $t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is: $\text{(A) } \frac {r + t}{r - t} \text{(B) } \frac {r}{r - t} \text{(C) } \frac {r + t}{r} \text{(D) } \frac{r}{t} \text{(E) } \frac{r+k}{t-k}$

In the figure, $\overline{CD}$, $\overline{AE}$ and $\overline{BF}$ are one-third of their respective sides. It follows that $\overline{AN_2}: \overline{N_2N_1}: \overline{N_1D} = 3: 3: 1$, and similarly for lines BE and CF. Then the area of triangle $N_1N_2N_3$ is: $\text{(A) } \frac {1}{10} \triangle ABC \text{(B) } \frac {1}{9} \triangle ABC \text{(C) } \frac{1}{7}\triangle ABC \text{(D) } \frac{1}{6}\triangle ABC \text{(E) } \text{none of these}$

A line initially 1 inch long grows according to the following law, where the first term is the initial length. $$1+\frac{1}{4}\sqrt{2}+\frac{1}{4}+\frac{1}{16}\sqrt{2}+\frac{1}{16}+\frac{1}{64}\sqrt{2}+\frac{1}{64}+\cdots$$ If the growth process continues forever, the limit of the length of the line is: $\textbf{(A) } \infty \textbf{(B) } \frac{4}{3} \textbf{(C) } \frac{8}{3} \textbf{(D) } \frac{1}{3}(4+\sqrt{2}) \textbf{(E) } \frac{2}{3}(4+\sqrt{2})$