Difference between revisions of "1964 AHSME Problems"

Revision as of 14:35, 9 October 2014

Problem 1

What is the value of $[\log_{10}(5\log_{10}100)]^2$ ?

$\textbf{(A)}\ \log_{10}50 \qquad \textbf{(B)}\ 25\qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 2}\qquad \textbf{(E)}\ 1 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 2

The graph of $x^2-4y^2=0$ is:

$\textbf{(A)}\ \text{a parabola} \qquad \textbf{(B)}\ \text{an ellipse} \qquad \textbf{(C)}\ \text{a pair of straight lines}\qquad \\ \textbf{(D)}\ \text{a point}}\qquad \textbf{(E)}\ \text{None of these}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 3

When a positive integer $x$ is divided by a positive integer $y$ , the quotient is $u$ and the remainder is $v$ , where $u$ and $v$ are integers. What is the remainder when $x+2uy$ is divided by $y$ ?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2u \qquad \textbf{(C)}\ 3u \qquad \textbf{(D)}\ v }\qquad \textbf{(E)}\ 2v }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 4

The expression

$\frac{P+Q}{P-Q}-\frac{P-Q}{P+Q}$

where $P=x+y$ and $Q=x-y$ , is equivalent to:

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

Solution

Problem 5

If $y$ varies directly as $x$ , and if $y=8$ when $x=4$ , the value of $y$ when $x=-8$ is:

$\textbf{(A)}\ -16} \qquad \textbf{(B)}\ -4 \qquad \textbf{(C)}\ -2 \qquad \textbf{(D)}\ 4k, k= \pm1, \pm2, \dots} \qquad \\ \textbf{(E)}\ 16k, k=\pm1,\pm2,\dots }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 6

If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is:

$\textbf{(A)}\ -27 \qquad \textbf{(B)}\ -13\frac{1}{2} \qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\frac{1}{2}}\qquad \textbf{(E)}\ 27 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 7

Let n be the number of real values of $p$ for which the roots of $x^2-px+p=0$ are equal. Then n equals:

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

Solution

Problem 8

The smaller root of the equation $\left(x-\frac{3}{4}\right)\left(x-\frac{3}{4}\right)+\left(x-\frac{3}{4}\right)\left(x-\frac{1}{2}\right) =0$ is:

$\textbf{(A)}\ -\frac{3}{4}\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ \frac{5}{8}\qquad \textbf{(D)}\ \frac{3}{4}}\qquad \textbf{(E)}\ 1 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 9

A jobber buys an article at $24 less $12\frac{1}{2}$ %. He then wishes to sell the article at a gain of $33\frac{1}{3}$ % of his cost after allowing a 20% discount on his marked price. At what price, in dollars, should the article be marked?

$\textbf{(A)}\ 25.20 \qquad \textbf{(B)}\ 30.00 \qquad \textbf{(C)}\ 33.60 \qquad \textbf{(D)}\ 40.00 }\qquad \textbf{(E)}\ \text{none of these}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 10

Given a square side of length $s$ . On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is:

$\textbf{(A)}\ s\sqrt{2} \qquad \textbf{(B)}\ s/\sqrt{2} \qquad \textbf{(C)}\ 2s \qquad \textbf{(D)}\ 2\sqrt{s} }\qquad \textbf{(E)}\ 2/\sqrt{s}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 11

Given $2^x=8^{y+1}$ and $9^y=3^{x-9}$ , find the value of $x+y$

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 27 }\qquad \textbf{(E)}\ 30 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 12

Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$ ?

$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0 }\qquad \textbf{(E)}\ \text{For some x}, x^2 \le 0}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 13

A circle is inscribed in a triangle with side lengths $8, 13$ , and $17$ . Let the segments of the side of length $8$ , made by a point of tangency, be $r$ and $s$ , with $r<s$ . What is the ratio $r:s$ ?

$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 2:5 \qquad \textbf{(C)}\ 1:2 \qquad \textbf{(D)}\ 2:3 }\qquad \textbf{(E)}\ 3:4 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 14

A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$ . Based on the cost, the percent gain on the entire transaction is:

$\textbf{(A)}\ 6.5 \qquad \textbf{(B)}\ 6.75 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 7.5 }\qquad \textbf{(E)}\ 8 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 15

A line through the point $(-a,0)$ cuts from the second quadrant a triangular region with area $T$ . The equation of the line is:

$\textbf{(A)}\ 2Tx+a^2y+2aT=0 \qquad \textbf{(B)}\ 2Tx-a^2y+2aT=0 \qquad \textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad \\ \textbf{(D)}\ 2Tx-a^2y-2aT=0 }\qquad \textbf{(E)}\ \text{none of these} }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 16

Let $f(x)=x^2+3x+2$ and let $S$ be the set of integers $\{0, 1, 2, \dots , 25 \}$ . The number of members $s$ of $S$ such that $f(s)$ has remainder zero when divided by $6$ is:

$\textbf{(A)}\ 25\qquad \textbf{(B)}\ 22\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 18 }\qquad \textbf{(E)}\ 17 }$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 17

Given the distinct points $P(x_1, y_1), Q(x_2, y_2)$ and $R(x_1+x_2, y_1+y_2)$ . Line segments are drawn connecting these points to each other and to the origin $0$ . Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure $OPRQ$ , depending upon the location of the points $P, Q$ , and $R$ , can be:

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

Solution

Problem 18

Let $n$ be the number of pairs of values of $b$ and $c$ such that $3x+by+c=0$ and $cx-2y+12=0$ have the same graph. Then $n$ is:

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

Solution

Problem 19

If $2x-3y-z=0$ and $x+3y-14z=0, z \neq 0$ , the numerical value of $\frac{x^2+3xy}{y^2+z^2}$ is:

$\textbf{(A)}\ 7\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ -20/17\qquad \textbf{(E)}\ -2$

Solution

Problem 20

The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is:

$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ -1\qquad \textbf{(E)}\ -19$

Solution

Problem 21

If $\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1$ , then $x$ equals:

$\textbf{(A)}\ 1/b^2 \qquad \textbf{(B)}\ 1/b \qquad \textbf{(C)}\ b^2 \qquad \textbf{(D)}\ b \qquad \textbf{(E)}\ \sqrt{b}$

Solution

Problem 22

Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$ . Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$ . What is the ratio of the area of $\triangle DFE$ to the area of quadrilateral $ABEF$ ?

$\textbf{(A)}\ 1:2 \qquad \textbf{(B)}\ 1:3 \qquad \textbf{(C)}\ 1:5 \qquad \textbf{(D)}\ 1:6 \qquad \textbf{(E)}\ 1:7$

Solution

Problem 23

Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$ . The product of the two numbers is:

$\textbf{(A)}\ 6\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 48\qquad \textbf{(E)}\ 96$

Solution

Problem 24

Let $y=(x-a)^2+(x-b)^2, a, b$ constants. For what value of $x$ is $y$ a minimum?

$\textbf{(A)}\ \frac{a+b}{2} \qquad \textbf{(B)}\ a+b \qquad \textbf{(C)}\ \sqrt{ab} \qquad \textbf{(D)}\ \sqrt{\frac{a^2+b^2}{2}}\qquad \textbf{(E)}\ \frac{a+b}{2ab}$

Solution

Problem 25

The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$ , is precisely:

$\textbf{(A)}\ 0, 12, -12\qquad \textbf{(B)}\ 0, 12\qquad \textbf{(C)}\ 12, -12\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 0$

Solution

Problem 26

In a ten-mile race $\textit{First}$ beats $\textit{Second}$ by $2$ miles and $\textit{First}$ beats $\textit{Third}$ by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does $\textit{Second}$ beat $\textit{Third}$ ?

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

Solution

Problem 27

If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$ , then:

$\textbf{(A)}\ 0<a<.01\qquad \textbf{(B)}\ .01<a<1 \qquad \textbf{(C)}\ 0<a<1\qquad \\ \textbf{(D)}\ 0<a \le 1\qquad \textbf{(E)}\ a>1$

Solution

Problem 28

The sum of $n$ terms of an arithmetic progression is $153$ , and the common difference is $2$ . If the first term is an integer, and $n>1$ , then the number of possible values for $n$ is:

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

Solution

Problem 29

In this figure $\angle RFS = \angle FDR, FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\tfrac{1}{2}$ inches. The length of $RS$ , in inches, is:

$[asy] import olympiad; pair F,R,S,D; F=origin; R=5*dir(aCos(9/16)); S=(7.5,0); D=4*dir(aCos(9/16)+aCos(1/8)); label("$F$",F,SW);label("$R$",R,N); label("$S$",S,SE); label("$D$",D,W); label("$7\frac{1}{2}$",(F+S)/2.5,SE); label("$4$",midpoint(F--D),SW); label("$5$",midpoint(F--R),W); label("$6$",midpoint(D--R),N); draw(F--D--R--F--S--R); markscalefactor=0.1; draw(anglemark(S,F,R)); draw(anglemark(F,D,R)); //Credit to throwaway1489 for the diagram[/asy]$

$\textbf{(A)}\ \text{undetermined} \qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\tfrac{1}{2} \qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 6\tfrac{1}{4}$

Solution

Problem 30

If $(7+4\sqrt{3})x^2+(2+\sqrt{3})x-2=0$ , the larger root minus the smaller root is:

$\textbf{(A)}\ -2+3\sqrt{3}\qquad \textbf{(B)}\ 2-\sqrt{3}\qquad \textbf{(C)}\ 6+3\sqrt{3}\qquad \textbf{(D)}\ 6-3\sqrt{3}\qquad \textbf{(E)}\ 3\sqrt{3}+2$

Solution

Problem 31

Let $f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n$ . Then $f(n+1)-f(n-1)$ , expressed in terms of $f(n)$ , equals:

$\textbf{(A)}\ \dfrac{1}{2}f(n) \qquad \textbf{(B)}\ f(n)\qquad \textbf{(C)}\ 2f(n)+1 \qquad \textbf{(D)}\ f^2(n)\qquad \textbf{(E)}\ \frac{1}{2}(f^2(n)-1)$

Solution

Problem 32

If $\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}$ , then:

$\textbf{(A)}\ a\text{ must equal }c \qquad \textbf{(B)}\ a+b+c+d\text{ must equal zero }\qquad \textbf{(C)}\ \text{either }a=c\text{ or }a+b+c+d=0,\text{ or both} \qquad \textbf{(D)}\ a+b+c+d\neq 0\text{ if }a=c \qquad \\ \textbf{(E)}\ a(b+c+d)=c(a+b+d)$

Solution

Problem 33

$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$ , in inches, equals:

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

$[asy] draw((0,0)--(6.5,0)--(6.5,4.5)--(0,4.5)--cycle); draw((2.5,1.5)--(0,0)); draw((2.5,1.5)--(0,4.5)); draw((2.5,1.5)--(6.5,4.5)); draw((2.5,1.5)--(6.5,0),linetype("8 8")); label("$A$",(0,0),dir(-135)); label("$B$",(6.5,0),dir(-45)); label("$C$",(6.5,4.5),dir(45)); label("$D$",(0,4.5),dir(135)); label("$P$",(2.5,1.5),dir(-90)); label("$3$",(1.25,0.75),dir(120)); label("$4$",(1.25,3),dir(35)); label("$5$",(4.5,3),dir(120)); //Credit to bobthesmartypants for the diagram[/asy]$

Solution

Problem 34

If $n$ is a multiple of $4$ , the sum $s=1+2i+3i^2+ \ldots +(n+1)i^{n}$ , where $i=\sqrt{-1}$ , equals:

$\textbf{(A)}\ 1+i\qquad \textbf{(B)}\ \frac{1}{2}(n+2) \qquad \textbf{(C)}\ \frac{1}{2}(n+2-ni) \qquad \\ \textbf{(D)}\ \frac{1}{2}[(n+1)(1-i)+2]\qquad \textbf{(E)}\ \frac{1}{8}(n^2+8-4ni)$

Solution

Problem 35

The sides of a triangle are of lengths $13, 14, and 15$ . The altitudes of the triangle meet at point $H$ . If $AD$ is the altitude to the side length $14$ , what is the ratio $HD:HA$ ?

Solution

Problem 36

In this figure the radius of the circle is equal to the altitude of the equilateral triangle $ABC$ . The circle is made to roll along the side $AB$ , remaining tangent to it at a variable point $T$ and intersecting lines $AC$ and $BC$ in variable points $M$ and $N$ , respectively. Let $n$ be the number of degrees in arc $MTN$ . Then $n$ , for all permissible positions of the circle:

$\textbf{(A) }\text{varies from }30^{\circ}\text{ to }90^{\circ}\quad \textbf{(B) }\text{varies from }30^{\circ}\text{ to }60^{\circ} \\ \textbf{(C) }\text{varies from }60^{\circ}\text{ to }90^{\circ} \quad \textbf{(D) }\text{remains constant at }30^{\circ}\quad \textbf{(E) }\text{remains constant at }60^{\circ}$

$[asy] pair A = (0,0), B = (1,0), C = dir(60), T = (2/3,0); pair M = intersectionpoint(A--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2)), N = intersectionpoint(B--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2)); draw((0,0)--(1,0)--dir(60)--cycle); draw(Circle((2/3,sqrt(3)/2),sqrt(3)/2)); label("$A$",A,dir(210)); label("$B$",B,dir(-30)); label("$C$",C,dir(90)); label("$M$",M,dir(190)); label("$N$",N,dir(75)); label("$T$",T,dir(-90)); //Credit to bobthesmartypants for the diagram[/asy]$

Solution

Problem 37

Given two positive number $a, b$ such that $a<b$ , let $A.M.$ be their arithmetic mean and let $G.M.$ be their positive geometric mean. Then $A.M.$ minus $G.M.$ is always less than:

$\textbf{(A) }\dfrac{(b+a)^2}{ab}\qquad \textbf{(B) }\dfrac{(b+a)^2}{8b}\qquad \textbf{(C) }\dfrac{(b-a)^2}{ab} \textbf{(D) }\dfrac{(b-a)^2}{8a}\qquad \textbf{(E) }\dfrac{(b-a)^2}{8b}$

Solution

Problem 38

The sides $PQ$ and $PR$ of $\triangle PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\frac{1}{2}$ inches. Then $QR$ , in inches, is:

$\textbf{(A) }6\qquad \textbf{(B) }7\qquad \textbf{(C) }8\qquad \textbf{(D) }9\qquad \textbf{(E) }10$

Solution

Problem 39

The magnitudes of the sides of $\triangle ABC$ are $a, b$ , and $c$ , as shown, with $c\le b\le a$ . Through interior point $P$ and the vertices $A, B, C$ , lines are drawn meeting the opposite sides in $A', B', C'$ , respectively. Let $s=AA'+BB'+CC'$ . Then, for all positions of point $P$ , $s$ is less than:

$\textbf{(A) }2a+b\qquad \textbf{(B) }2a+c\qquad \textbf{(C) }2b+c\qquad \textbf{(D) }a+2b\qquad \textbf{(E) } a+b+c$

$[asy] import math; defaultpen(fontsize(11pt)); pair A = (0,0), B = (1,3), C = (5,0), P = (1.5,1); pair X = extension(B,C,A,P), Y = extension(A,C,B,P), Z = extension(A,B,C,P); draw(A--B--C--cycle); draw(A--X); draw(B--Y); draw(C--Z); dot(P); dot(A); dot(B); dot(C); label("$A$",A,dir(210)); label("$B$",B,dir(90)); label("$C$",C,dir(-30)); label("$A'$",X,dir(-100)); label("$B'$",Y,dir(65)); label("$C'$",Z,dir(20)); label("$P$",P,dir(70)); label("$a$",X,dir(80)); label("$b$",Y,dir(-90)); label("$c$",Z,dir(110)); //Credit to bobthesmartypants for the diagram[/asy]$

Solution

Problem 40

A watch loses $2\tfrac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March $15$ . Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March $21$ , $n$ equals:

$\textbf{(A) }14\frac{14}{23}\qquad \textbf{(B) }14\frac{1}{14}\qquad \textbf{(C) }13\frac{101}{115}\qquad \textbf{(D) }13\frac{83}{115}\qquad \textbf{(E) }13\frac{13}{23}$

Solution

@@ Line 17: / Line 17: @@
 <math>\textbf{(A)}\ \text{a parabola} \qquad
 \textbf{(B)}\ \text{an ellipse} \qquad
-\textbf{(C)}\ \text{a pair of straight lines}\qquad
+\textbf{(C)}\ \text{a pair of straight lines}\qquad \\
 \textbf{(D)}\ \text{a point}}\qquad
 \textbf{(E)}\ \text{None of these}} </math>
@@ Line 59: / Line 59: @@
 \textbf{(B)}\ -4 \qquad
 \textbf{(C)}\ -2 \qquad
-\textbf{(D)}\ 4k, k= \pm1, \pm2, \dots}
+\textbf{(D)}\ 4k, k= \pm1, \pm2, \dots} \qquad \\
-\qquad
 \textbf{(E)}\ 16k, k=\pm1,\pm2,\dots }     </math>
@@ Line 93: / Line 92: @@
 == Problem 8==
-The smaller root of the equation <math>\left(x-\frac{3}{4}\right)\left(x-\frac{3}{4}\right)+\left(x-\frac{3}{4}\right)\left(x-\frac{1}{2}\right) =0</math> is:
+The smaller root of the equation
+<math>\left(x-\frac{3}{4}\right)\left(x-\frac{3}{4}\right)+\left(x-\frac{3}{4}\right)\left(x-\frac{1}{2}\right) =0</math> is:
 <math>\textbf{(A)}\ -\frac{3}{4}\qquad
@@ Line 99: / Line 99: @@
 \textbf{(C)}\ \frac{5}{8}\qquad
 \textbf{(D)}\ \frac{3}{4}}\qquad
-\textbf{(E)}\ 1 }
+\textbf{(E)}\ 1 } </math>
 [[1964 AHSME Problems/Problem 8|Solution]]
@@ Line 105: / Line 105: @@
 == Problem 9==
-A jobber buys an article at &#036;24 less </math>12\frac{1}{2}<math> %. He then wishes to sell the article at a gain of </math>33\frac{1}{3}<math> % of his cost
+A jobber buys an article at &#036;24 less <math>12\frac{1}{2}</math> %. He then wishes to sell the article at a gain of <math>33\frac{1}{3}</math> % of his cost
 after allowing a 20% discount on his marked price. At what price, in dollars, should the article be marked?
-</math>\textbf{(A)}\ 25.20 \qquad
+<math>\textbf{(A)}\ 25.20 \qquad
 \textbf{(B)}\ 30.00 \qquad
 \textbf{(C)}\ 33.60 \qquad
 \textbf{(D)}\ 40.00 }\qquad
-\textbf{(E)}\ \text{none of these}} <math>
+\textbf{(E)}\ \text{none of these}} </math>
 [[1964 AHSME Problems/Problem 9|Solution]]
@@ Line 118: / Line 118: @@
 == Problem 10==
-Given a square side of length </math>s<math>. On a diagonal as base a triangle with three unequal sides is constructed so that its area
+Given a square side of length <math>s</math>. On a diagonal as base a triangle with three unequal sides is constructed so that its area
 equals that of the square. The length of the altitude drawn to the base is:
-</math>\textbf{(A)}\ s\sqrt{2} \qquad
+<math>\textbf{(A)}\ s\sqrt{2} \qquad
 \textbf{(B)}\ s/\sqrt{2} \qquad
 \textbf{(C)}\ 2s \qquad
 \textbf{(D)}\ 2\sqrt{s} }\qquad
-\textbf{(E)}\ 2/\sqrt{s}} <math>
+\textbf{(E)}\ 2/\sqrt{s}} </math>
 [[1964 AHSME Problems/Problem 10|Solution]]
@@ Line 131: / Line 131: @@
 == Problem 11==
-Given </math>2^x=8^{y+1}<math> and </math>9^y=3^{x-9}<math>, find the value of </math>x+y<math>
+Given <math>2^x=8^{y+1}</math> and <math>9^y=3^{x-9}</math>, find the value of <math>x+y</math>
-</math>\textbf{(A)}\ 18 \qquad
+<math>\textbf{(A)}\ 18 \qquad
 \textbf{(B)}\ 21 \qquad
 \textbf{(C)}\ 24 \qquad
 \textbf{(D)}\ 27 }\qquad
-\textbf{(E)}\ 30 } <math>
+\textbf{(E)}\ 30 } </math>
 [[1964 AHSME Problems/Problem 11|Solution]]
@@ Line 143: / Line 143: @@
 == Problem 12==
-Which of the following is the negation of the statement: For all </math>x<math> of a certain set, </math>x^2>0<math>?
+Which of the following is the negation of the statement: For all <math>x</math> of a certain set, <math>x^2>0</math>?
-</math>\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad
+<math>\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad
 \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad
-\textbf{(C)}\ \text{For no x}, x^2>0\qquad
+\textbf{(C)}\ \text{For no x}, x^2>0\qquad \\
 \textbf{(D)}\ \text{For some x}, x^2>0 }\qquad
-\textbf{(E)}\ \text{For some x}, x^2 \le 0}}    <math>
+\textbf{(E)}\ \text{For some x}, x^2 \le 0}}    </math>
 [[1964 AHSME Problems/Problem 12|Solution]]
@@ Line 155: / Line 155: @@
 == Problem 13==
-A circle is inscribed in a triangle with side lengths </math>8, 13<math>, and </math>17<math>. Let the segments of the side of length </math>8<math>,
+A circle is inscribed in a triangle with side lengths <math>8, 13</math>, and <math>17</math>. Let the segments of the side of length <math>8</math>,
-made by a point of tangency, be </math>r<math> and </math>s<math>, with </math>r<s<math>. What is the ratio </math>r:s<math>?
+made by a point of tangency, be <math>r</math> and <math>s</math>, with <math>r<s</math>. What is the ratio <math>r:s</math>?
-</math>\textbf{(A)}\ 1:3 \qquad
+<math>\textbf{(A)}\ 1:3 \qquad
 \textbf{(B)}\ 2:5 \qquad
 \textbf{(C)}\ 1:2 \qquad
 \textbf{(D)}\ 2:3 }\qquad
-\textbf{(E)}\ 3:4 }   <math>
+\textbf{(E)}\ 3:4 }   </math>
 [[1964 AHSME Problems/Problem 13|Solution]]
@@ Line 168: / Line 168: @@
 == Problem 14==
-A farmer bought </math>749<math> sheep. He sold </math>700<math> of them for the price paid for the </math>749<math> sheep.
+A farmer bought <math>749</math> sheep. He sold <math>700</math> of them for the price paid for the <math>749</math> sheep.
-The remaining </math>49<math> sheep were sold at the same price per head as the other </math>700<math>.
+The remaining <math>49</math> sheep were sold at the same price per head as the other <math>700</math>.
 Based on the cost, the percent gain on the entire transaction is:
-</math>\textbf{(A)}\ 6.5 \qquad
+<math>\textbf{(A)}\ 6.5 \qquad
 \textbf{(B)}\ 6.75 \qquad
 \textbf{(C)}\ 7 \qquad
 \textbf{(D)}\ 7.5 }\qquad
-\textbf{(E)}\ 8 }  <math>
+\textbf{(E)}\ 8 }  </math>
 [[1964 AHSME Problems/Problem 14|Solution]]
@@ Line 182: / Line 182: @@
 == Problem 15==
-A line through the point </math>(-a,0)<math> cuts from the second quadrant a triangular region with area </math>T<math>. The equation of the line is:
+A line through the point <math>(-a,0)</math> cuts from the second quadrant a triangular region with area <math>T</math>. The equation of the line is:
-</math>\textbf{(A)}\ 2Tx+a^2y+2aT=0 \qquad
+<math>\textbf{(A)}\ 2Tx+a^2y+2aT=0 \qquad
 \textbf{(B)}\ 2Tx-a^2y+2aT=0 \qquad
-\textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad
+\textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad \\
 \textbf{(D)}\ 2Tx-a^2y-2aT=0 }\qquad
-\textbf{(E)}\ \text{none of these} }   <math>
+\textbf{(E)}\ \text{none of these} }   </math>
 [[1964 AHSME Problems/Problem 15|Solution]]
@@ Line 194: / Line 194: @@
 == Problem 16==
-Let </math>f(x)=x^2+3x+2<math> and let </math>S<math> be the set of integers </math>\{0, 1, 2, \dots , 25 \}<math>.
+Let <math>f(x)=x^2+3x+2</math> and let <math>S</math> be the set of integers <math>\{0, 1, 2, \dots , 25 \}</math>.
-The number of members </math>s<math> of </math>S<math> such that </math>f(s)<math> has remainder zero when divided by </math>6<math> is:
+The number of members <math>s</math> of <math>S</math> such that <math>f(s)</math> has remainder zero when divided by <math>6</math> is:
-</math>\textbf{(A)}\ 25\qquad
+<math>\textbf{(A)}\ 25\qquad
 \textbf{(B)}\ 22\qquad
 \textbf{(C)}\ 21\qquad
 \textbf{(D)}\ 18 }\qquad
-\textbf{(E)}\ 17 }     <math>
+\textbf{(E)}\ 17 }     </math>
 [[1964 AHSME Problems/Problem 16|Solution]]
@@ Line 207: / Line 207: @@
 == Problem 17==
-Given the distinct points </math>P(x_1, y_1), Q(x_2, y_2)<math> and </math>R(x_1+x_2, y_1+y_2)<math>.
+Given the distinct points <math>P(x_1, y_1), Q(x_2, y_2)</math> and <math>R(x_1+x_2, y_1+y_2)</math>.
-Line segments are drawn connecting these points to each other and to the origin </math>0<math>.
+Line segments are drawn connecting these points to each other and to the origin <math>0</math>.
-Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure </math>OPRQ<math>,
+Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure <math>OPRQ</math>,
-depending upon the location of the points </math>P, Q<math>, and </math>R<math>, can be:
+depending upon the location of the points <math>P, Q</math>, and <math>R</math>, can be:
-</math>\textbf{(A)}\ \text{(1) only}\qquad
+<math>\textbf{(A)}\ \text{(1) only}\qquad
 \textbf{(B)}\ \text{(2) only}\qquad
 \textbf{(C)}\ \text{(3) only}\qquad
 \textbf{(D)}\ \text{(1) or (2) only}\qquad
-\textbf{(E)}\ \text{all three} <math>
+\textbf{(E)}\ \text{all three} </math>
 [[1964 AHSME Problems/Problem 17|Solution]]
@@ Line 222: / Line 222: @@
 == Problem 18==
-Let </math>n<math> be the number of pairs of values of </math>b<math> and </math>c<math> such that </math>3x+by+c=0<math> and </math>cx-2y+12=0<math> have the same graph. Then </math>n<math> is:
+Let <math>n</math> be the number of pairs of values of <math>b</math> and <math>c</math> such that <math>3x+by+c=0</math> and <math>cx-2y+12=0</math> have the same graph. Then <math>n</math> is:
-</math>\textbf{(A)}\ 0\qquad
+<math>\textbf{(A)}\ 0\qquad
 \textbf{(B)}\ 1\qquad
 \textbf{(C)}\ 2\qquad
 \textbf{(D)}\ \text{finite but more than 2}\qquad
-\textbf{(E)}\ \infty <math>
+\textbf{(E)}\ \infty </math>
 [[1964 AHSME Problems/Problem 18|Solution]]
@@ Line 234: / Line 234: @@
 == Problem 19==
-If </math>2x-3y-z=0<math> and </math>x+3y-14z=0, z \neq 0<math>, the numerical value of </math>\frac{x^2+3xy}{y^2+z^2}<math> is:
+If <math>2x-3y-z=0</math> and <math>x+3y-14z=0, z \neq 0</math>, the numerical value of <math>\frac{x^2+3xy}{y^2+z^2}</math> is:
-</math>\textbf{(A)}\ 7\qquad
+<math>\textbf{(A)}\ 7\qquad
 \textbf{(B)}\ 2\qquad
 \textbf{(C)}\ 0\qquad
 \textbf{(D)}\ -20/17\qquad
-\textbf{(E)}\ -2   <math>
+\textbf{(E)}\ -2   </math>
 [[1964 AHSME Problems/Problem 19|Solution]]
@@ Line 246: / Line 246: @@
 == Problem 20==
-The sum of the numerical coefficients of all the terms in the expansion of </math>(x-2y)^{18}<math> is:
+The sum of the numerical coefficients of all the terms in the expansion of <math>(x-2y)^{18}</math> is:
-</math>\textbf{(A)}\ 0\qquad
+<math>\textbf{(A)}\ 0\qquad
 \textbf{(B)}\ 1\qquad
 \textbf{(C)}\ 19\qquad
 \textbf{(D)}\ -1\qquad
-\textbf{(E)}\ -19  <math>
+\textbf{(E)}\ -19  </math>
 [[1964 AHSME Problems/Problem 20|Solution]]
@@ Line 258: / Line 258: @@
 == Problem 21==
-If </math>\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1<math>, then </math>x<math> equals:
+If <math>\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1</math>, then <math>x</math> equals:
-</math>\textbf{(A)}\ 1/b^2 \qquad
+<math>\textbf{(A)}\ 1/b^2 \qquad
 \textbf{(B)}\ 1/b \qquad
 \textbf{(C)}\ b^2 \qquad
 \textbf{(D)}\ b \qquad
-\textbf{(E)}\ \sqrt{b} <math>
+\textbf{(E)}\ \sqrt{b} </math>
 [[1964 AHSME Problems/Problem 21|Solution]]
@@ Line 270: / Line 270: @@
 == Problem 22==
-Given parallelogram </math>ABCD<math> with </math>E<math> the midpoint of diagonal </math>BD<math>. Point </math>E<math> is connected to a point </math>F<math> in </math>DA<math> so that
+Given parallelogram <math>ABCD</math> with <math>E</math> the midpoint of diagonal <math>BD</math>. Point <math>E</math> is connected to a point <math>F</math> in <math>DA</math> so that
-</math>DF=\frac{1}{3}DA<math>. What is the ratio of the area of </math>\triangle DFE<math> to the area of quadrilateral </math>ABEF<math>?
+<math>DF=\frac{1}{3}DA</math>. What is the ratio of the area of <math>\triangle DFE</math> to the area of quadrilateral <math>ABEF</math>?
-</math>\textbf{(A)}\ 1:2 \qquad
+<math>\textbf{(A)}\ 1:2 \qquad
 \textbf{(B)}\ 1:3 \qquad
 \textbf{(C)}\ 1:5 \qquad
 \textbf{(D)}\ 1:6 \qquad
-\textbf{(E)}\ 1:7  <math>
+\textbf{(E)}\ 1:7  </math>
 [[1964 AHSME Problems/Problem 22|Solution]]
@@ Line 283: / Line 283: @@
 == Problem 23==
-Two numbers are such that their difference, their sum, and their product are to one another as </math>1:7:24<math>. The product of the two numbers is:
+Two numbers are such that their difference, their sum, and their product are to one another as <math>1:7:24</math>. The product of the two numbers is:
-</math>\textbf{(A)}\ 6\qquad
+<math>\textbf{(A)}\ 6\qquad
 \textbf{(B)}\ 12\qquad
 \textbf{(C)}\ 24\qquad
 \textbf{(D)}\ 48\qquad
-\textbf{(E)}\ 96    <math>
+\textbf{(E)}\ 96    </math>
 [[1964 AHSME Problems/Problem 23|Solution]]
@@ Line 295: / Line 295: @@
 == Problem 24==
-Let </math>y=(x-a)^2+(x-b)^2, a, b<math> constants. For what value of </math>x<math> is </math>y<math> a minimum?
+Let <math>y=(x-a)^2+(x-b)^2, a, b</math> constants. For what value of <math>x</math> is <math>y</math> a minimum?
-</math>\textbf{(A)}\ \frac{a+b}{2} \qquad
+<math>\textbf{(A)}\ \frac{a+b}{2} \qquad
 \textbf{(B)}\ a+b \qquad
 \textbf{(C)}\ \sqrt{ab} \qquad
 \textbf{(D)}\ \sqrt{\frac{a^2+b^2}{2}}\qquad
-\textbf{(E)}\ \frac{a+b}{2ab} <math>
+\textbf{(E)}\ \frac{a+b}{2ab} </math>
 [[1964 AHSME Problems/Problem 24|Solution]]
@@ Line 307: / Line 307: @@
 == Problem 25==
-The set of values of </math>m<math> for which </math>x^2+3xy+x+my-m<math> has two factors, with integer coefficients, which are linear in </math>x<math> and </math>y<math>, is precisely:
+The set of values of <math>m</math> for which <math>x^2+3xy+x+my-m</math> has two factors, with integer coefficients, which are linear in <math>x</math> and <math>y</math>, is precisely:
-</math>\textbf{(A)}\ 0, 12, -12\qquad
+<math>\textbf{(A)}\ 0, 12, -12\qquad
 \textbf{(B)}\ 0, 12\qquad
 \textbf{(C)}\ 12, -12\qquad
 \textbf{(D)}\ 12\qquad
-\textbf{(E)}\ 0   <math>
+\textbf{(E)}\ 0   </math>
 [[1964 AHSME Problems/Problem 25|Solution]]
@@ Line 319: / Line 319: @@
 == Problem 26==
-In a ten-mile race </math>First<math> beats </math>Second<math> by </math>2<math> miles and </math>First<math> beats </math>Third<math> by </math>4<math> miles.
+In a ten-mile race <math>\textit{First}</math> beats <math>\textit{Second}</math> by <math>2</math> miles and
-If the runners maintain constant speeds throughout the race, by how many miles does </math>Second<math> beat </math>Third<math>?
+<math>\textit{First}</math> beats <math>\textit{Third}</math> by <math>4</math> miles.
+If the runners maintain constant speeds throughout the race,
+by how many miles does <math>\textit{Second}</math> beat <math>\textit{Third}</math>?
-</math>\textbf{(A)}\ 2\qquad
+<math>\textbf{(A)}\ 2\qquad
 \textbf{(B)}\ 2\frac{1}{4}\qquad
 \textbf{(C)}\ 2\frac{1}{2}\qquad
 \textbf{(D)}\ 2\frac{3}{4}\qquad
-\textbf{(E)}\ 3 <math>
+\textbf{(E)}\ 3 </math>
 [[1964 AHSME Problems/Problem 26|Solution]]
@@ Line 332: / Line 334: @@
 == Problem 27==
-If </math>x<math> is a real number and </math>|x-4|+|x-3|<a<math> where </math>a>0<math>, then:
+If <math>x</math> is a real number and <math>|x-4|+|x-3|<a</math> where <math>a>0</math>, then:
-</math>\textbf{(A)}\ 0<a<.01\qquad
+<math>\textbf{(A)}\ 0<a<.01\qquad
 \textbf{(B)}\ .01<a<1 \qquad
-\textbf{(C)}\ 0<a<1\qquad
+\textbf{(C)}\ 0<a<1\qquad \\
 \textbf{(D)}\ 0<a \le 1\qquad
-\textbf{(E)}\ a>1    <math>
+\textbf{(E)}\ a>1    </math>
 [[1964 AHSME Problems/Problem 27|Solution]]
@@ Line 344: / Line 346: @@
 == Problem 28==
-The sum of </math>n<math> terms of an arithmetic progression is </math>153<math>, and the common difference is </math>2<math>.
+The sum of <math>n</math> terms of an arithmetic progression is <math>153</math>, and the common difference is <math>2</math>.
-If the first term is an integer, and </math>n>1<math>, then the number of possible values for </math>n<math> is:
+If the first term is an integer, and <math>n>1</math>, then the number of possible values for <math>n</math> is:
-</math>\textbf{(A)}\ 2\qquad
+<math>\textbf{(A)}\ 2\qquad
 \textbf{(B)}\ 3\qquad
 \textbf{(C)}\ 4\qquad
 \textbf{(D)}\ 5\qquad
-\textbf{(E)}\ 6   <math>
+\textbf{(E)}\ 6   </math>
 [[1964 AHSME Problems/Problem 28|Solution]]
@@ Line 357: / Line 359: @@
 == Problem 29==
-In this figure </math>\angle RFS = \angle FDR, FD = 4<math> inches, </math>DR = 6<math> inches, </math>FR = 5<math> inches, </math>FS = 7\dfrac{1}{2}<math> inches.
+In this figure <math>\angle RFS = \angle FDR, FD = 4</math> inches, <math>DR = 6</math> inches, <math>FR = 5</math> inches, <math>FS = 7\tfrac{1}{2}</math> inches.
-The length of </math>RS$, in inches, is:
+The length of <math>RS</math>, in inches, is:
 <asy>
@@ Line 379: / Line 381: @@
 <math>\textbf{(A)}\ \text{undetermined} \qquad
 \textbf{(B)}\ 4\qquad
-\textbf{(C)}\ 5\dfrac{1}{2} \qquad
+\textbf{(C)}\ 5\tfrac{1}{2} \qquad
 \textbf{(D)}\ 6\qquad
-\textbf{(E)}\ 6\frac{1}{4} </math>
+\textbf{(E)}\ 6\tfrac{1}{4} </math>
 [[1964 AHSME Problems/Problem 29|Solution]]
@@ Line 417: / Line 419: @@
 \textbf{(B)}\ a+b+c+d\text{ must equal zero }\qquad
 \textbf{(C)}\ \text{either }a=c\text{ or }a+b+c+d=0,\text{ or both} \qquad
-\textbf{(D)}\ a+b+c+d\neq 0\text{ if }a=c \qquad
+\textbf{(D)}\ a+b+c+d\neq 0\text{ if }a=c \qquad \\
 \textbf{(E)}\ a(b+c+d)=c(a+b+d)    </math>
@@ Line 456: / Line 458: @@
 <math>\textbf{(A)}\ 1+i\qquad
 \textbf{(B)}\ \frac{1}{2}(n+2) \qquad
-\textbf{(C)}\ \frac{1}{2}(n+2-ni) \qquad
+\textbf{(C)}\ \frac{1}{2}(n+2-ni) \qquad \\
 \textbf{(D)}\ \frac{1}{2}[(n+1)(1-i)+2]\qquad
 \textbf{(E)}\ \frac{1}{8}(n^2+8-4ni) </math>
@@ Line 565: / Line 567: @@
 == Problem 40==
-A watch loses <math>2\frac{1}{2}</math> minutes per day. It is set right at <math>1</math> P.M. on March <math>15</math>.
+A watch loses <math>2\tfrac{1}{2}</math> minutes per day. It is set right at <math>1</math> P.M. on March <math>15</math>.
 Let <math>n</math> be the positive correction, in minutes, to be added to the time shown by the watch at a given time.
 When the watch shows <math>9</math> A.M. on March <math>21</math>, <math>n</math> equals:

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Difference between revisions of "1964 AHSME Problems"

Revision as of 14:35, 9 October 2014

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40