Difference between revisions of "1978 AHSME Problems"

Latest revision as of 00:06, 22 February 2024

1978 AHSME (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 30-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 5 points for each correct answer, 2 points for each problem left unanswered, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. Figures are not necessarily drawn to scale. You will have 90 minutes working time to complete the test.
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Problem 1

If $1-\frac{4}{x}+\frac{4}{x^2}=0$ , then $\frac{2}{x}$ equals

$\textbf{(A) }-1\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }-1\text{ or }2\qquad \textbf{(E) }-1\text{ or }-2$

Solution

Problem 2

If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is

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

Solution

Problem 3

For all non-zero numbers $x$ and $y$ such that $x = 1/y$ , $\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$ equals

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

Solution

Problem 4

If $a = 1,~ b = 10, ~c = 100$ , and $d = 1000$ , then $(a+ b+ c-d) + (a + b- c+ d) +(a-b+ c+d)+ (-a+ b+c+d)$ is equal to

$\textbf{(A) }1111\qquad \textbf{(B) }2222\qquad \textbf{(C) }3333\qquad \textbf{(D) }1212\qquad \textbf{(E) }4242$

Solution

Problem 5

Four boys bought a boat for $\textdollar 60$ . The first boy paid one half of the sum of the amounts paid by the other boys; the second boy paid one third of the sum of the amounts paid by the other boys; and the third boy paid one fourth of the sum of the amounts paid by the other boys. How much did the fourth boy pay?

$\textbf{(A) }\textdollar 10\qquad \textbf{(B) }\textdollar 12\qquad \textbf{(C) }\textdollar 13\qquad \textbf{(D) }\textdollar 14\qquad \textbf{(E) }\textdollar 15$

Solution

Problem 6

The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations:

$x=x^2+y^2 \ \ y=2xy$ is

$\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }4$

Solution

Problem 7

Opposite sides of a regular hexagon are $12$ inches apart. The length of each side, in inches, is

$\textbf{(A) }7.5\qquad \textbf{(B) }6\sqrt{2}\qquad \textbf{(C) }5\sqrt{2}\qquad \textbf{(D) }\frac{9}{2}\sqrt{3}\qquad \textbf{(E) }4\sqrt{3}$

Solution

Problem 8

If $x\neq y$ and the sequences $x,a_1,a_2,y$ and $x,b_1,b_2,b_3,y$ each are in arithmetic progression, then $(a_2-a_1)/(b_2-b_1)$ equals

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

Solution

Problem 9

If $x<0$ , then $\left|x-\sqrt{(x-1)^2}\right|$ equals

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

Solution

Problem 10

If $\mathit{B}$ is a point on circle $\mathit{C}$ with center $\mathit{P}$ , then the set of all points $\mathit{A}$ in the plane of circle $\mathit{C}$ such that the distance between $\mathit{A}$ and $\mathit{B}$ is less than or equal to the distance between $\mathit{A}$ and any other point on circle $\mathit{C}$ is

$\textbf{(A) }\text{the line segment from }P \text{ to }B\qquad\\ \textbf{(B) }\text{the ray beginning at }P \text{ and passing through }B\qquad\\ \textbf{(C) }\text{a ray beginning at }B\qquad\\ \textbf{(D) }\text{a circle whose center is }P\qquad\\ \textbf{(E) }\text{a circle whose center is }B$

Solution

Problem 11

If $r$ is positive and the line whose equation is $x + y = r$ is tangent to the circle whose equation is $x^2 + y ^2 = r$ , then $r$ equals

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

Solution

Problem 12

In $\triangle ADE$ , $\measuredangle ADE=140^\circ$ , points $B$ and $C$ lie on sides $AD$ and $AE$ , respectively, and points $A,~B,~C,~D,~E$ are distinct.* If lengths $AB,~BC,~CD$ , and $DE$ are all equal, then the measure of $\measuredangle EAD$ is

The specification that points $A,B,C,D,E$ be distinct was not included in the original statement of the problem.

If $B=D$ , then $C=E$ and $\measuredangle EAD=20^\circ$ .

$\textbf{(A) }5^\circ\qquad \textbf{(B) }6^\circ\qquad \textbf{(C) }7.5^\circ\qquad \textbf{(D) }8^\circ\qquad \textbf{(E) }10^\circ$

Solution

Problem 13

If $a,b,c$ , and $d$ are non-zero numbers such that $c$ and $d$ are the solutions of $x^2+ax+b=0$ and $a$ and $b$ are the solutions of $x^2+cx+d=0$ , then $a+b+c+d$ equals

$\textbf{(A) }0\qquad \textbf{(B) }-2\qquad \textbf{(C) }2\qquad \textbf{(D) }4\qquad \textbf{(E) }(-1+\sqrt{5})/2$

Solution

Problem 14

If an integer $n > 8$ is a solution of the equation $x^2 - ax+b=0$ and the representation of $a$ in the base- $n$ number system is $18$ , then the base-n representation of $b$ is

$\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 80 \qquad \textbf{(D)}\ 81 \qquad \textbf{(E)}\ 280$

Solution

Problem 15

If $\sin x+\cos x=1/5$ and $0\le x<\pi$ , then $\tan x$ is

$\textbf{(A) }-\frac{4}{3}\qquad \textbf{(B) }-\frac{3}{4}\qquad \textbf{(C) }\frac{3}{4}\qquad \textbf{(D) }\frac{4}{3}\qquad\\ \textbf{(E) }\text{not completely determined by the given information}$

Solution

Problem 16

In a room containing $N$ people, $N > 3$ , at least one person has not shaken hands with everyone else in the room. What is the maximum number of people in the room that could have shaken hands with everyone else?

$\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }N-1\qquad \textbf{(D) }N\qquad \textbf{(E) }\text{none of these}$

Solution

Problem 17

If $k$ is a positive number and $f$ is a function such that, for every positive number $x$ , $\left[f(x^2+1)\right]^{\sqrt{x}}=k$ ; then, for every positive number $y$ , $\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}$ is equal to

$\textbf{(A) }\sqrt{k}\qquad \textbf{(B) }2k\qquad \textbf{(C) }k\sqrt{k}\qquad \textbf{(D) }k^2\qquad \textbf{(E) }y\sqrt{k}$

Solution

Problem 18

What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<.01$ ?

$\textbf{(A) }2499\qquad \textbf{(B) }2500\qquad \textbf{(C) }2501\qquad \textbf{(D) }10,000\qquad \textbf{(E) }\text{There is no such integer}$

Solution

Problem 19

A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le 50$ , then the probability of choosing $n$ is $p$ , and if $n > 50$ , then the probability of choosing $n$ is $3p$ . The probability that a perfect square is chosen is

$\textbf{(A) }.05\qquad \textbf{(B) }.065\qquad \textbf{(C) }.08\qquad \textbf{(D) }.09\qquad \textbf{(E) }.1$

Solution

Problem 20

If $a,b,c$ are non-zero real numbers such that $\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},$ and $x=\frac{(a+b)(b+c)(c+a)}{abc},$ and $x<0,$ then $x$ equals

$\textbf{(A) }{-}1\qquad \textbf{(B) }{-}2\qquad \textbf{(C) }{-}4\qquad \textbf{(D) }{-}6\qquad \textbf{(E) }{-}8$

Solution

Problem 21

For all positive numbers $x$ distinct from $1$ ,

$\frac{1}{\log_3(x)}+\frac{1}{\log_4(x)}+\frac{1}{\log_5(x)}$

equals

$\text{(A) }\frac{1}{\log_{60}(x)}\qquad\\ \text{(B) }\frac{1}{\log_{x}(60)}\qquad\\ \text{(C) }\frac{1}{(\log_{3}(x))(\log_{4}(x))(\log_{5}(x))}\qquad\\ \text{(D) }\frac{12}{\log_{3}(x)+\log_{4}(x)+\log_{5}(x)}\qquad\\ \text{(E) }\frac{\log_{2}(x)}{\log_{3}(x)\log_{5}(x)}+\frac{\log_{3}(x)}{\log_{2}(x)\log_{5}(x)}+\frac{\log_{5}(x)}{\log_{2}(x)\log_{3}(x)}$

Solution

Problem 22

The following four statements, and only these are found on a card: $[asy] pair A,B,C,D,E,F,G; A=(0,1); B=(0,5); C=(11,5); D=(11,1); E=(0,4); F=(0,3); G=(0,2); draw(A--B--C--D--cycle); label("On this card exactly one statement is false.", B, SE); label("On this card exactly two statements are false.", E, SE); label("On this card exactly three statements are false.", F, SE); label("On this card exactly four statements are false.", G, SE); [/asy]$

(Assume each statement is either true or false.) Among them the number of false statements is exactly

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution

Problem 23

$[asy] size(100); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,1)--(1,0)); draw((0,0)--(.5,sqrt(3)/2)--(1,0)); label("$A$",(0,0),SW); label("$B$",(1,0),SE); label("$C$",(1,1),NE); label("$D$",(0,1),NW); label("$E$",(.5,sqrt(3)/2),E); label("$F$",intersectionpoint((0,0)--(.5,sqrt(3)/2),(0,1)--(1,0)),2W); //Credit to chezbgone2 for the diagram [/asy]$

Vertex $E$ of equilateral $\triangle ABE$ is in the interior of square $ABCD$ , and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$ . If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is

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

Solution

Problem 24

If the distinct non-zero numbers $x ( y - z),~ y(z - x),~ z(x - y )$ form a geometric progression with common ratio $r$ , then $r$ satisfies the equation

$\textbf{(A) }r^2+r+1=0\qquad \textbf{(B) }r^2-r+1=0\qquad \textbf{(C) }r^4+r^2-1=0\qquad\\ \textbf{(D) }(r+1)^4+r=0\qquad \textbf{(E) }(r-1)^4+r=0$

Solution

Problem 25

Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y )$ satisfy all of the following conditions:

$\text{(i) }\frac{a}{2}\le x\le 2a\qquad \text{(ii) }\frac{a}{2}\le y\le 2a\qquad \text{(iii) }x+y\ge a\\ \\ \qquad \text{(iv) }x+a\ge y\qquad \text{(v) }y+a\ge x$

The boundary of set S is a polygon with

$\textbf{(A) }3\text{ sides}\qquad \textbf{(B) }4\text{ sides}\qquad \textbf{(C) }5\text{ sides}\qquad \textbf{(D) }6\text{ sides}\qquad \textbf{(E) }7\text{ sides}$

Solution

Problem 26

$[asy] size(100); real a=4, b=3; // import cse5; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP("A",A)--MP("B",B,N)--MP("C",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP("R",IP(CP((X+C)/2,C),B--C),NW)--MP("Q",IP(CP((X+C)/2,C),A--C+(0.1,0)))); //Credit to chezbgone2 for the diagram [/asy]$

In $\triangle ABC, AB = 10~ AC = 8$ and $BC = 6$ . Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$ . Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$ , respectively. The length of segment $QR$ is

$\textbf{(A) }4.75\qquad \textbf{(B) }4.8\qquad \textbf{(C) }5\qquad \textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }3\sqrt{3}$

Solution

Problem 27

There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 11$ , has a remainder of $1$ . What is the difference between the two smallest such integers?

$\textbf{(A) }2310\qquad \textbf{(B) }2311\qquad \textbf{(C) }27,720\qquad \textbf{(D) }27,721\qquad \textbf{(E) }\text{none of these}$

Solution

Problem 28

$[asy] size(100); import cse5; pathpen=black; pair A1=(0,0), A2=(1,0), A3=(0.5,sqrt(3)/2); D(MP("A_1",A1)--MP("A_2",A2)--MP("A_3",A3,N)--cycle); pair A4=(A1+A2)/2, A5 = (A3+A2)/2, A6 = (A4+A3)/2; D(MP("A_4",A4,S)--MP("A_6",A6,W)--A3); D(A6--MP("A_5",A5,NE)--A4); //Credit to chezbgone2 for the diagram [/asy]$

If $\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$ , then the measure of $\measuredangle A_{44}A_{45}A_{43}$ equals

$\textbf{(A) }30^\circ\qquad \textbf{(B) }45^\circ\qquad \textbf{(C) }60^\circ\qquad \textbf{(D) }90^\circ\qquad \textbf{(E) }120^\circ$

Solution

Problem 29

Sides $AB,~ BC, ~CD$ and $DA$ , respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$ . Also, $AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8$ and $DA = AA' = 9$ ; and the area of $ABCD$ is $10$ . The area of $A 'B 'C'D'$ is

$\textbf{(A) }20\qquad \textbf{(B) }40\qquad \textbf{(C) }45\qquad \textbf{(D) }50\qquad \textbf{(E) }60$

Solution

Problem 30

In a tennis tournament, $n$ women and $2n$ men play, and each player plays exactly one match with every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is $7/5$ , then $n$ equals

$\textbf{(A) }2\qquad \textbf{(B) }4\qquad \textbf{(C) }6\qquad \textbf{(D) }7\qquad \textbf{(E) }\text{none of these}$

Solution

@@ Line 1: / Line 1: @@
+{{AHSME Problems
+|year = 1978
+}}
 == Problem 1 ==
@@ Line 65: / Line 68: @@
 The number of distinct pairs <math>(x,y)</math> of real numbers satisfying both of the following equations:
-<cmath>x=x^2+y^2 \\ y=2xy</cmath>
+<cmath>x=x^2+y^2 \    \ y=2xy</cmath>
 is
@@ Line 83: / Line 86: @@
 \textbf{(B) }6\sqrt{2}\qquad
 \textbf{(C) }5\sqrt{2}\qquad
-\textbf{(D) }\frac{9}{2}\sqrt{3}\qquad
 \textbf{(D) }\frac{9}{2}\sqrt{3}\qquad
 \textbf{(E) }4\sqrt{3}    </math>
@@ Line 91: / Line 93: @@
 == Problem 8 ==
-If <math>x\neq y</math> and the sequences <math>x,a_1,a_2,y and x,b_1,b_2,b_3,y</math> each are in arithmetic progression, then <math>(a_2-a_1)/(b_2-b_1)</math> equals
+If <math>x\neq y</math> and the sequences <math>x,a_1,a_2,y</math> and <math>x,b_1,b_2,b_3,y</math> each are in arithmetic progression, then <math>(a_2-a_1)/(b_2-b_1)</math> equals
 <math>\textbf{(A) }\frac{2}{3}\qquad
@@ Line 228: / Line 230: @@
 \textbf{(C) }2501\qquad
 \textbf{(D) }10,000\qquad
-\textbf{(E) }\text{There is no such integer</math>
+\textbf{(E) }\text{There is no such integer}</math>
 [[1978 AHSME Problems/Problem 18|Solution]]
@@ Line 247: / Line 249: @@
 == Problem 20 ==
-If <math>a,b,c</math> are non-zero real numbers such that <math>\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a}</math>,
+If <math>a,b,c</math> are non-zero real numbers such that <cmath>\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},</cmath> and <cmath>x=\frac{(a+b)(b+c)(c+a)}{abc},</cmath> and <math>x<0,</math> then <math>x</math> equals
-and <math>x=\frac{(a+b)(b+c)(c+a)}{abc}</math>, and <math>x<0</math>, then <math>x</math> equals
-<math>\textbf{(A) }-1\qquad
+<math>\textbf{(A) }{-}1\qquad
-\textbf{(B) }-2\qquad
+\textbf{(B) }{-}2\qquad
-\textbf{(C) }-4\qquad
+\textbf{(C) }{-}4\qquad
-\textbf{(D) }-6\qquad
+\textbf{(D) }{-}6\qquad
-\textbf{(E) }-8    </math>
+\textbf{(E) }{-}8    </math>
 [[1978 AHSME Problems/Problem 20|Solution]]
@@ Line 260: / Line 261: @@
 == Problem 21 ==
+For all positive numbers <math>x</math> distinct from <math>1</math>,
+<cmath>\frac{1}{\log_3(x)}+\frac{1}{\log_4(x)}+\frac{1}{\log_5(x)}</cmath>
+equals
+<math>\text{(A) }\frac{1}{\log_{60}(x)}\qquad\\
+\text{(B) }\frac{1}{\log_{x}(60)}\qquad\\
+\text{(C) }\frac{1}{(\log_{3}(x))(\log_{4}(x))(\log_{5}(x))}\qquad\\
+\text{(D) }\frac{12}{\log_{3}(x)+\log_{4}(x)+\log_{5}(x)}\qquad\\
+\text{(E) }\frac{\log_{2}(x)}{\log_{3}(x)\log_{5}(x)}+\frac{\log_{3}(x)}{\log_{2}(x)\log_{5}(x)}+\frac{\log_{5}(x)}{\log_{2}(x)\log_{3}(x)}</math>
 [[1978 AHSME Problems/Problem 21|Solution]]
@@ Line 327: / Line 338: @@
 <math>\textbf{(A) }r^2+r+1=0\qquad
 \textbf{(B) }r^2-r+1=0\qquad
-\textbf{(C) }r^4+r^2-1=0\qquad
+\textbf{(C) }r^4+r^2-1=0\qquad\\
 \textbf{(D) }(r+1)^4+r=0\qquad
 \textbf{(E) }(r-1)^4+r=0     </math>
@@ Line 335: / Line 346: @@
 == Problem 25 ==
-Let <math>u</math> be a positive number. Consider the set <math>S</math> of all points whose rectangular coordinates <math>(x, y )</math> satisfy all of the following conditions:
+Let <math>a</math> be a positive number. Consider the set <math>S</math> of all points whose rectangular coordinates <math>(x, y )</math> satisfy all of the following conditions:
 <math> \text{(i) }\frac{a}{2}\le x\le 2a\qquad
@@ Line 387: / Line 398: @@
 What is the difference between the two smallest such integers?
-\textbf{(A) }2310\qquad
+<math>\textbf{(A) }2310\qquad
 \textbf{(B) }2311\qquad
 \textbf{(C) }27,720\qquad
 \textbf{(D) }27,721\qquad
-\textbf{(E) }\text{none of these}   $
+\textbf{(E) }\text{none of these}   </math>
 [[1978 AHSME Problems/Problem 27|Solution]]
@@ Line 423: / Line 434: @@
 == Problem 29 ==
-Sides <math>AB,~ BC, ~CD and DA</math>, respectively, of convex quadrilateral <math>ABCD</math> are extended past <math>B,~ C ,~ D</math> and <math>A</math> to points <math>B',~C',~ D'</math> and <math>A'</math>.
+Sides <math>AB,~ BC, ~CD</math> and <math>DA</math>, respectively, of convex quadrilateral <math>ABCD</math> are extended past <math>B,~ C ,~ D</math> and <math>A</math> to points <math>B',~C',~ D'</math> and <math>A'</math>.
 Also, <math>AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8</math> and <math>DA = AA' = 9</math>; and the area of <math>ABCD</math> is <math>10</math>. The area of <math>A 'B 'C'D'</math> is
@@ Line 449: / Line 460: @@
 == See also ==
-* [[AHSME]]
-* [[AHSME Problems and Solutions]]
 * [[AMC 12 Problems and Solutions]]
 * [[Mathematics competition resources]]
+{{AHSME box|year=1978|before=[[1977 AHSME]]|after=[[1979 AHSME]]}}
 {{MAA Notice}}

1978 AHSME (Problems • Answer Key • Resources)
Preceded by 1977 AHSME	Followed by 1979 AHSME
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Difference between revisions of "1978 AHSME Problems"

Latest revision as of 00:06, 22 February 2024

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

See also