Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1964 AHSME Problems"

(→‎Problem 35)
(→‎Problem 17)
Line 208: Line 208:
  
 
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>O</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:

Revision as of 20:34, 23 July 2019

Problem 1

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

$\textbf{(A)}\ \log_{10}50 \qquad \textbf{(B)}\ 25\qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 1$

Solution

Problem 2

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

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

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 $y$?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2u \qquad \textbf{(C)}\ 3u \qquad \textbf{(D)}\ v \qquad \textbf{(E)}\ 2v$

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:

$\textbf{(A)}\ \frac{x^2-y^2}{xy}\qquad \textbf{(B)}\ \frac{x^2-y^2}{2xy}\qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ \frac{x^2+y^2}{xy}\qquad \textbf{(E)}\ \frac{x^2+y^2}{2xy}$

Solution

Problem 5

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

$\textbf{(A)}\ -16 \qquad \textbf{(B)}\ -4 \qquad \textbf{(C)}\ -2 \qquad \textbf{(D)}\ 4k, k= \pm1, \pm2, \dots \qquad \\ \textbf{(E)}\ 16k, k=\pm1,\pm2,\dots$

Solution

Problem 6

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

$\textbf{(A)}\ -27 \qquad \textbf{(B)}\ -13\frac{1}{2} \qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\frac{1}{2}\qquad \textbf{(E)}\ 27$

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:

$\textbf{(A)}\ -\frac{3}{4}\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ \frac{5}{8}\qquad \textbf{(D)}\ \frac{3}{4}\qquad \textbf{(E)}\ 1$

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?

$\textbf{(A)}\ 25.20 \qquad \textbf{(B)}\ 30.00 \qquad \textbf{(C)}\ 33.60 \qquad \textbf{(D)}\ 40.00 \qquad \textbf{(E)}\ \text{none of these}$

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:

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

Solution

Problem 11

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

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 30$

Solution

Problem 12

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

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

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 $r:s$?

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

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:

$\textbf{(A)}\ 6.5 \qquad \textbf{(B)}\ 6.75 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 7.5 \qquad \textbf{(E)}\ 8$

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:

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

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:

$\textbf{(A)}\ 25\qquad \textbf{(B)}\ 22\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 17$

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 $O$. 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


See also

1964 AHSC (Problems • Answer Key • Resources)
Preceded by
1963 AHSME 		Followed by
1965 AHSME
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions


The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1964_AHSME_Problems&oldid=107845"