AHSME 1962

The expression $\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to: $\textbf{(A)}\ \frac{4y-1}{8} \textbf{(B)}\ 8 \textbf{(C)}\ \frac{15}{2} \textbf{(D)}\ \frac{15}{8} \textbf{(E)}\ \frac{1}{8}$

The expression $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to: $\textbf{(A)}\ \frac{\sqrt{3}}{6} \textbf{(B)}\ \frac{-\sqrt{3}}{6} \textbf{(C)}\ \frac{\sqrt{-3}}{6} \textbf{(D)}\ \frac{5\sqrt{3}}{6} \textbf{(E)}\ 1$

The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: $\textbf{(A)}\ -2 \textbf{(B)}\ 0 \textbf{(C)}\ 2 \textbf{(D)}\ 4 \textbf{(E)}\ \text{undetermined}$

If $8^x = 32$, then $x$ equals: $\textbf{(A)}\ 4 \textbf{(B)}\ \frac{5}{3} \textbf{(C)}\ \frac{3}{2} \textbf{(D)}\ \frac{3}{5} \textbf{(E)}\ \frac{1}{4}$

If the radius of a circle is increased by $1$ unit, the ratio of the new circumference to the new diameter is: $\textbf{(A)}\ \pi+2 \textbf{(B)}\ \frac{2\pi+1}{2} \textbf{(C)}\ \pi \textbf{(D)}\ \frac{2\pi-1}{2} \textbf{(E)}\ \pi-2$

A square and an equilateral triangle have equal perimeters. The area of the triangle is $9 \sqrt{3}$ square inches. Expressed in inches the diagonal of the square is: $\textbf{(A)}\ \frac{9}{2} \textbf{(B)}\ 2\sqrt{5} \textbf{(C)}\ 4\sqrt{2} \textbf{(D)}\ \frac{9\sqrt{2}}{2} \textbf{(E)}\ \text{none of these}$

Let the bisectors of the exterior angles at $B$ and $C$ of triangle $ABC$ meet at D$.$ Then, if all measurements are in degrees, angle $BDC$ equals: $\textbf{(A)}\ \frac{1}{2}(90-A) \textbf{(B)}\ 90-A \textbf{(C)}\ \frac{1}{2}(180-A)$ $\textbf{(D)}\ 180-A \textbf{(E)}\ 180-2A$

Given the set of $n$ numbers; $n > 1$, of which one is $1 - \frac {1}{n}$ and all the others are $1$. The arithmetic mean of the $n$ numbers is: $\textbf{(A)}\ 1 \textbf{(B)}\ n-\frac{1}{n} \textbf{(C)}\ n-\frac{1}{n^2} \textbf{(D)}\ 1-\frac{1}{n^2} \textbf{(E)}\ 1-\frac{1}{n}-\frac{1}{n^2}$

When $x^9-x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is: $\textbf{(A)}\ \text{more than 5} \textbf{(B)}\ 5 \textbf{(C)}\ 4 \textbf{(D)}\ 3 \textbf{(E)}\ 2$

A man drives $150$ miles to the seashore in $3$ hours and $20$ minutes. He returns from the shore to the starting point in $4$ hours and $10$ minutes. Let $r$ be the average rate for the entire trip. Then the average rate for the trip going exceeds $r$ in miles per hour, by: $\textbf{(A)}\ 5 \textbf{(B)}\ 4\frac{1}{2} \textbf{(C)}\ 4 \textbf{(D)}\ 2 \textbf{(E)}\ 1$

The difference between the larger root and the smaller root of $x^2 - px + (p^2 - 1)/4 = 0$ is: $\textbf{(A)}\ 0 \textbf{(B)}\ 1 \textbf{(C)}\ 2 \textbf{(D)}\ p \textbf{(E)}\ p+1$

When $\left ( 1 - \frac{1}{a} \right ) ^6$ is expanded the sum of the last three coefficients is: $\textbf{(A)}\ 22 \textbf{(B)}\ 11 \textbf{(C)}\ 10 \textbf{(D)}\ -10 \textbf{(E)}\ -11$

$R$ varies directly as $S$ and inversely as $T$. When $R = \frac{4}{3}$ and $T = \frac {9}{14}$, $S = \frac37$. Find $S$ when $R = \sqrt {48}$ and $T = \sqrt {75}$. $\textbf{(A)}\ 28 \textbf{(B)}\ 30 \textbf{(C)}\ 40 \textbf{(D)}\ 42 \textbf{(E)}\ 60$

Let $s$ be the limiting sum of the geometric series $4- \frac83 + \frac{16}{9} - \dots$, as the number of terms increases without bound. Then $s$ equals: $\textbf{(A)}\ \text{a number between 0 and 1} \textbf{(B)}\ 2.4 \textbf{(C)}\ 2.5 \textbf{(D)}\ 3.6 \textbf{(E)}\ 12$

Given triangle $ABC$ with base $AB$ fixed in length and position. As the vertex $C$ moves on a straight line, the intersection point of the three medians moves on: $\textbf{(A)}\ \text{a circle} \textbf{(B)}\ \text{a parabola} \textbf{(C)}\ \text{an ellipse} \textbf{(D)}\ \text{a straight line} \textbf{(E)}\ \text{a curve here not listed}$

Given rectangle $R_1$ with one side $2$ inches and area $12$ square inches. Rectangle $R_2$ with diagonal $15$ inches is similar to $R_1$. Expressed in square inches the area of $R_2$ is: $\textbf{(A)}\ \frac{9}2 \textbf{(B)}\ 36 \textbf{(C)}\ \frac{135}{2} \textbf{(D)}\ 9\sqrt{10} \textbf{(E)}\ \frac{27\sqrt{10}}{4}$

If $a = \log_8 225$ and $b = \log_2 15$, then $a$, in terms of $b$, is: $\textbf{(A)}\ \frac{b}{2} \textbf{(B)}\ \frac{2b}{3} \textbf{(C)}\ b \textbf{(D)}\ \frac{3b}{2} \textbf{(E)}\ 2b$

A regular dodecagon ($12$ sides) is inscribed in a circle with radius $r$ inches. The area of the dodecagon, in square inches, is: $\textbf{(A)}\ 3r^2 \textbf{(B)}\ 2r^2 \textbf{(C)}\ \frac{3r^2\sqrt{3}}{4} \textbf{(D)}\ r^2\sqrt{3} \textbf{(E)}\ 3r^2\sqrt{3}$

If the parabola $y = ax^2 + bx + c$ passes through the points $( - 1, 12)$, $(0, 5)$, and $(2, - 3)$, the value of $a + b + c$ is: $\textbf{(A)}\ -4 \textbf{(B)}\ -2 \textbf{(C)}\ 0 \textbf{(D)}\ 1 \textbf{(E)}\ 2$

The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: $\textbf{(A)}\ 108 \textbf{(B)}\ 90 \textbf{(C)}\ 72 \textbf{(D)}\ 54 \textbf{(E)}\ 36$

It is given that one root of $2x^2 + rx + s = 0$, with $r$ and $s$ real numbers, is $3+2i (i = \sqrt{-1})$. The value of $s$ is: $\textbf{(A)}\ \text{undetermined} \textbf{(B)}\ 5 \textbf{(C)}\ 6 \textbf{(D)}\ -13 \textbf{(E)}\ 26$

The number $121_b$, written in the integral base $b$, is the square of an integer, for $\textbf{(A)}\ b = 10,\text{ only} \textbf{(B)}\ b = 10\text{ and }b = 5,\text{ only}$ $\textbf{(C)}\ 2\leq b\leq 10 \textbf{(D)}\ b > 2 \textbf{(E)}\ \text{no value of }b$

In triangle $ABC$, $CD$ is the altitude to $AB$ and $AE$ is the altitude to $BC$. If the lengths of $AB$, $CD$, and $AE$ are known, the length of $DB$ is:

$\textbf{(A)}\ \text{not determined by the information given}$ $\textbf{(B)}\ \text{determined only if A is an acute angle}$ $\textbf{(C)}\ \text{determined only if B is an acute angle}$ $\textbf{(D)}\ \text{determined only if ABC is an acute triangle}$ $\textbf{(E)}\ \text{none of these is correct}$

Three machines $\text{P, Q, and R,}$ working together, can do a job in $x$ hours. When working alone, $\text{P}$ needs an additional $6$ hours to do the job; $\text{Q}$, one additional hour; and $R$, $x$ additional hours. The value of $x$ is: $\textbf{(A)}\ \frac{2}3 \textbf{(B)}\ \frac{11}{12} \textbf{(C)}\ \frac{3}2 \textbf{(D)}\ 2 \textbf{(E)}\ 3$

Given square $ABCD$ with side $8$ feet. A circle is drawn through vertices $A$ and $D$ and tangent to side $BC$. The radius of the circle, in feet, is: $\textbf{(A)}\ 4 \textbf{(B)}\ 4\sqrt{2} \textbf{(C)}\ 5 \textbf{(D)}\ 5\sqrt{2} \textbf{(E)}\ 6$

For any real value of $x$ the maximum value of $8x - 3x^2$ is: $\textbf{(A)}\ 0 \textbf{(B)}\ \frac{8}3 \textbf{(C)}\ 4 \textbf{(D)}\ 5 \textbf{(E)}\ \frac{16}{3}$

Let $a @ b$ represent the operation on two numbers, $a$ and $b$, which selects the larger of the two numbers, with $a@a = a$. Let $a ! b$ represent the operator which selects the smaller of the two numbers, with $a ! a = a$. Which of the following three rules is (are) correct? $\textbf{(1)}\ a@b = b@a \textbf{(2)}\ a@(b@c) = (a@b)@c \textbf{(3)}\ a ! (b@c) = (a ! b) @ (a ! c)$

$\textbf{(A)}\ (1)\text{ only} \textbf{(B)}\ (2)\text{ only} \textbf{(C)}\ \text{(1) and (2) only}$ $\textbf{(D)}\ \text{(1) and (3) only} \textbf{(E)}\ \text{all three}$

The set of $x$-values satisfying the equation $x^{\log_{10} x} = \frac{x^3}{100}$ consists of: $\textbf{(A)}\ \frac{1}{10} \textbf{(B)}\ \text{10, only} \textbf{(C)}\ \text{100, only} \textbf{(D)}\ \text{10 or 100, only} \textbf{(E)}\ \text{more than two real numbers.}$

Which of the following sets of $x$-values satisfy the inequality $2x^2 + x \frac{3}2\text{ or }x <-2 \textbf{(C)}\ x <\frac{3}2$ $\textbf{(D)}\ \frac{3}2 < x < 2 \textbf{(E)}\ x <-2$

Consider the statements: $\textbf{(1)}\ \text{p and q are both true} \textbf{(2)}\ \text{p is true and q is false} \textbf{(3)}\ \text{p is false and q is true} \textbf{(4)}\ \text{p is false and q is false.}$ How many of these imply the negative of the statement "p and q are both true?" $\textbf{(A)}\ 0 \textbf{(B)}\ 1 \textbf{(C)}\ 2 \textbf{(D)}\ 3 \textbf{(E)}\ 4$

The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. How many such pairs are there? $\textbf{(A)}\ 1 \textbf{(B)}\ 2 \textbf{(C)}\ 3 \textbf{(D)}\ 4 \textbf{(E)}\ \text{infinitely many}$

If $x_{k+1} = x_k + \frac12$ for $k=1, 2, \dots, n-1$ and $x_1=1$, find $x_1 + x_2 + \dots + x_n$. $\textbf{(A)}\ \frac{n+1}{2} \textbf{(B)}\ \frac{n+3}{2} \textbf{(C)}\ \frac{n^2-1}{2} \textbf{(D)}\ \frac{n^2+n}{4} \textbf{(E)}\ \frac{n^2+3n}{4}$

The set of $x$-values satisfying the inequality $2 \leq |x-1| \leq 5$ is: $\textbf{(A)}\ -4\leq x\leq-1\text{ or }3\leq x\leq 6$ $\textbf{(B)}\ 3\leq x\leq 6\text{ or }-6\leq x\leq-3 \textbf{(C)}\ x\leq-1\text{ or }x\geq 3$ $\textbf{(D)}\ -1\leq x\leq 3 \textbf{(E)}\ -4\leq x\leq 6$

For what real values of $K$ does $x = K^2 (x-1)(x-2)$ have real roots? $\textbf{(A)}\ \text{none} \textbf{(B)}\ -21\text{ or }K<-2 \textbf{(E)}\ \text{all}$

A man on his way to dinner short after $6: 00$ p.m. observes that the hands of his watch form an angle of $110^{\circ}$. Returning before $7: 00$ p.m. he notices that again the hands of his watch form an angle of $110^{\circ}$. The number of minutes that he has been away is: $\textbf{(A)}\ 36\frac{2}3 \textbf{(B)}\ 40 \textbf{(C)}\ 42 \textbf{(D)}\ 42.4 \textbf{(E)}\ 45$

If both $x$ and $y$ are both integers, how many pairs of solutions are there of the equation $(x-8)(x-10) = 2^y$? $\textbf{(A)}\ 0 \textbf{(B)}\ 1 \textbf{(C)}\ 2 \textbf{(D)}\ 3 \textbf{(E)}\ \text{more than 3}$

$ABCD$ is a square with side of unit length. Points $E$ and $F$ are taken respectively on sides $AB$ and $AD$ so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. In square units this maximum area is: $\textbf{(A)}\ \frac{1}2 \textbf{(B)}\ \frac{9}{16} \textbf{(C)}\ \frac{19}{32} \textbf{(D)}\ \frac{5}{8} \textbf{(E)}\ \frac{2}3$

The population of Nosuch Junction at one time was a perfect square. Later, with an increase of $100$, the population was one more than a perfect square. Now, with an additional increase of $100$, the population is again a perfect square.

The original population is a multiple of: $\textbf{(A)}\ 3 \textbf{(B)}\ 7 \textbf{(C)}\ 9 \textbf{(D)}\ 11 \textbf{(E)}\ 17$

Two medians of a triangle with unequal sides are $3$ inches and $6$ inches. Its area is $3 \sqrt{15}$ square inches. The length of the third median in inches, is: $\textbf{(A)}\ 4 \textbf{(B)}\ 3\sqrt{3} \textbf{(C)}\ 3\sqrt{6} \textbf{(D)}\ 6\sqrt{3} \textbf{(E)}\ 6\sqrt{6}$

The limiting sum of the infinite series, $\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots$ whose $n$th term is $\frac{n}{10^n}$ is: $\textbf{(A)}\ \frac{1}9 \textbf{(B)}\ \frac{10}{81} \textbf{(C)}\ \frac{1}8 \textbf{(D)}\ \frac{17}{72} \textbf{(E)}\ \text{larger than any finite quantity}$