Difference between revisions of "1973 AHSME Problems"

Latest revision as of 18:25, 22 August 2024

1973 AHSME (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 35-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 ? points for each correct answer, ? points for each problem left unanswered, and ? 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 ? minutes working time to complete the test.
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

Problem 1

A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length

$\textbf{(A)}\ 3\sqrt3\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 6\sqrt3\qquad\textbf{(D)}\ 12\sqrt3\qquad\textbf{(E)}\ \text{ none of these}$

Solution

Problem 2

One thousand unit cubes are fastened together to form a large cube with edge length 10 units; this is painted and then separated into the original cubes. The number of these unit cubes which have at least one face painted is

$\textbf{(A)}\ 600\qquad\textbf{(B)}\ 520\qquad\textbf{(C)}\ 488\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 400$

Solution

Problem 3

The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, the largest possible difference between the two primes is

$\textbf{(A)}\ 112\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 88\qquad\textbf{(E)}\ 80$

Solution

Problem 4

Two congruent 30-60-90 are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is

$\textbf{(A)}\ 6\sqrt3\qquad\textbf{(B)}\ 8\sqrt3\qquad\textbf{(C)}\ 9\sqrt3\qquad\textbf{(D)}\ 12\sqrt3\qquad\textbf{(E)}\ 24$

Solution

Problem 5

Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean),

$\text{I. Averaging is associative }$

$\text{II. Averaging is commutative }$

$\text{III. Averaging distributes over addition }$

$\text{IV. Addition distributes over averaging }$

$\text{V. Averaging has an identity element }$

those which are always true are

$\textbf{(A)}\ \text{All}\qquad\textbf{(B)}\ \text{I and II only}\qquad\textbf{(C)}\ \text{II and III only}$

$\textbf{(D)}\ \text{II and IV only}\qquad\textbf{(E)}\ \text{II and V only}$

Solution

Problem 6

If 554 is the base $b$ representation of the square of the number whose base $b$ representation is 24, then $b$ , when written in base 10, equals

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 16$

Solution

Problem 7

The sum of all integers between 50 and 350 which end in 1 is

$\textbf{(A)}\ 5880\qquad\textbf{(B)}\ 5539\qquad\textbf{(C)}\ 5208\qquad\textbf{(D)}\ 4877\qquad\textbf{(E)}\ 4566$

Solution

Problem 8

If 1 pint of paint is needed to paint a statue 6 ft. high, then the number of pints it will take to paint (to the same thickness) 540 statues similar to the original but only 1 ft. high is

$\textbf{(A)}\ 90\qquad\textbf{(B)}\ 72\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 15$

Solution

Problem 9

In $\triangle ABC$ with right angle at $C$ , altitude $CH$ and median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$ , then the area of $\triangle ABC$ is

$\textbf{(A)}\ 6K\qquad\textbf{(B)}\ 4\sqrt3\ K\qquad\textbf{(C)}\ 3\sqrt3\ K\qquad\textbf{(D)}\ 3K\qquad\textbf{(E)}\ 4K$

Solution

Problem 10

If $n$ is a real number, then the simultaneous system

$nx+y = 1$

$ny+z = 1$

$x+nz = 1$

has no solution if and only if $n$ is equal to

$\textbf{(A)}\ -1\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 0\text{ or }1\qquad\textbf{(E)}\ \frac{1}2$

Solution

Problem 11

A circle with a circumscribed and an inscribed square centered at the origin of a rectangular coordinate system with positive and axes and is shown in each figure to below.

$[asy] size((400)); draw((0,0)--(22,0), EndArrow); draw((10,-10)--(10,12), EndArrow); draw((25,0)--(47,0), EndArrow); draw((35,-10)--(35,12), EndArrow); draw((-25,0)--(-3,0), EndArrow); draw((-15,-10)--(-15,12), EndArrow); draw((-50,0)--(-28,0), EndArrow); draw((-40,-10)--(-40,12), EndArrow); draw(Circle((-40,0),6)); draw(Circle((-15,0),6)); draw(Circle((10,0),6)); draw(Circle((35,0),6)); draw((-34,0)--(-40,6)--(-46,0)--(-40,-6)--(-34,0)--(-34,6)--(-46,6)--(-46,-6)--(-34,-6)--cycle); draw((-6.5,0)--(-15,8.5)--(-23.5,0)--(-15,-8.5)--cycle); draw((-10.8,4.2)--(-19.2,4.2)--(-19.2,-4.2)--(-10.8,-4.2)--cycle); draw((14.2,4.2)--(5.8,4.2)--(5.8,-4.2)--(14.2,-4.2)--cycle); draw((16,6)--(4,6)--(4,-6)--(16,-6)--cycle); draw((41,0)--(35,6)--(29,0)--(35,-6)--cycle); draw((43.5,0)--(35,8.5)--(26.5,0)--(35,-8.5)--cycle); label("I", (-49,9)); label("II", (-24,9)); label("III", (1,9)); label("IV", (26,9)); label("X", (-28,0), S); label("X", (-3,0), S); label("X", (22,0), S); label("X", (47,0), S); label("Y", (-40,12), E); label("Y", (-15,12), E); label("Y", (10,12), E); label("Y", (35,12), E);[/asy]$

The inequalities

$|x|+|y|\leq\sqrt{2(x^{2}+y^{2})}\leq 2\mbox{Max}(|x|, |y|)$

are represented geometrically* by the figure numbered

$\textbf{(A)}\ I\qquad\textbf{(B)}\ II\qquad\textbf{(C)}\ III\qquad\textbf{(D)}\ IV\qquad\textbf{(E)}\ \mbox{none of these}$

* An inequality of the form $f(x, y) \leq g(x, y)$ , for all $x$ and $y$ is represented geometrically by a figure showing the containment

$\{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\\ \{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}$

for a typical real number $a$ .

Solution

Problem 12

The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is

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

Solution

Problem 13

The fraction $\frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}}$ is equal to

$\textbf{(A)}\ \frac{2\sqrt2}{3} \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ \frac{2\sqrt3}3 \qquad \textbf{(D)}\ \frac43 \qquad \textbf{(E)}\ \frac{16}{9}$

Solution

Problem 14

Each valve $A$ , $B$ , and $C$ , when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves $A$ and $C$ open it takes 1.5 hours, and with only valves $B$ and $C$ open it takes 2 hours. The number of hours required with only valves $A$ and $B$ open is

$\textbf{(A)}\ 1.1\qquad\textbf{(B)}\ 1.15\qquad\textbf{(C)}\ 1.2\qquad\textbf{(D)}\ 1.25\qquad\textbf{(E)}\ 1.75$

Solution

Problem 15

A sector with acute central angle $\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is

$\textbf{(A)}\ 3\cos\theta \qquad \textbf{(B)}\ 3\sec\theta \qquad \textbf{(C)}\ 3 \cos \frac12 \theta \qquad \textbf{(D)}\ 3 \sec \frac12 \theta \qquad \textbf{(E)}\ 3$

Solution

Problem 16

If the sum of all the angles except one of a convex polygon is $2190^{\circ}$ , then the number of sides of the polygon must be

$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 21$

Solution

Problem 17

If $\theta$ is an acute angle and $\sin \frac12 \theta = \sqrt{\frac{x-1}{2x}}$ , then $\tan \theta$ equals

$\textbf{(A)}\ x \qquad \textbf{(B)}\ \frac1{x} \qquad \textbf{(C)}\ \frac{\sqrt{x-1}}{x+1} \qquad \textbf{(D)}\ \frac{\sqrt{x^2-1}}{x} \qquad \textbf{(E)}\ \sqrt{x^2-1}$

Solution

Problem 18

If $p \geq 5$ is a prime number, then $24$ divides $p^2 - 1$ without remainder

$\textbf{(A)}\ \text{never} \qquad \textbf{(B)}\ \text{sometimes only} \qquad \textbf{(C)}\ \text{always} \qquad$

$\textbf{(D)}\ \text{only if } p =5 \qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 19

Define $n_a!$ for $n$ and $a$ positive to be

$n_a ! = n (n-a)(n-2a)(n-3a)...(n-ka)$

where $k$ is the greatest integer for which $n>ka$ . Then the quotient $72_8!/18_2!$ is equal to

$\textbf{(A)}\ 4^5 \qquad \textbf{(B)}\ 4^6 \qquad \textbf{(C)}\ 4^8 \qquad \textbf{(D)}\ 4^9 \qquad \textbf{(E)}\ 4^{12}$

Solution

Problem 20

A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is

$\textbf{(A)}\ 4+\sqrt{185} \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ \sqrt{32}+\sqrt{137}$

Solution

Problem 21

The number of sets of two or more consecutive positive integers whose sum is 100 is

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

Solution

Problem 22

The set of all real solutions of the inequality

$|x - 1| + |x + 2| < 5$

is

$\textbf{(A)}\ x \in ( - 3,2) \qquad \textbf{(B)}\ x \in ( - 1,2) \qquad \textbf{(C)}\ x \in ( - 2,1) \qquad$

$\textbf{(D)}\ x \in \left( - \frac32,\frac72\right) \qquad \textbf{(E)}\ \O \text{ (empty})$

[Note: I updated the notation on this problem.]

Solution

Problem 23

There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, then the probability that the under-side is also red is

$\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac23 \qquad \textbf{(E)}\ \frac34$

Solution

Problem 24

The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to $$3.15$ . The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to $$4.20$ at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to

$\textbf{(A)}\ $1.70 \qquad \textbf{(B)}\ $1.65 \qquad \textbf{(C)}\ $1.20 \qquad \textbf{(D)}\ $1.05 \qquad \textbf{(E)}\ $0.95$

Solution

Problem 25

A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is

$\textbf{(A)}\ 36\pi-34 \qquad \textbf{(B)}\ 30\pi - 15 \qquad \textbf{(C)}\ 36\pi - 33 \qquad$

$\textbf{(D)}\ 35\pi - 9\sqrt3 \qquad \textbf{(E)}\ 30\pi - 9\sqrt3$

Solution

Problem 26

The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is

$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 8$

Solution

Problem 27

Cars A and B travel the same distance. Car A travels half that distance at $u$ miles per hour and half at $v$ miles per hour. Car B travels half the time at $u$ miles per hour and half at $v$ miles per hour. The average speed of Car A is $x$ miles per hour and that of Car B is $y$ miles per hour. Then we always have

$\textbf{(A)}\ x \leq y\qquad \textbf{(B)}\ x \geq y \qquad \textbf{(C)}\ x=y \qquad \textbf{(D)}\ x<y\qquad \textbf{(E)}\ x>y$

Solution

Problem 28

If $a$ , $b$ , and $c$ are in geometric progression (G.P.) with $1 < a < b < c$ and $n>1$ is an integer, then $\log_an$ , $\log_bn$ , $\log_cn$ form a sequence

$\textbf{(A)}\ \text{which is a G.P} \qquad$

$\textbf{(B)}\ \text{whichi is an arithmetic progression (A.P)} \qquad$

$\textbf{(C)}\ \text{in which the reciprocals of the terms form an A.P} \qquad$

$\textbf{(D)}\ \text{in which the second and third terms are the }n\text{th powers of the first and second respectively} \qquad$

$\textbf{(E)}\ \text{none of these}$

Solution

Problem 29

Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is

$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ \text{infinity} \qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 30

Let $[t]$ denote the greatest integer $\leq t$ where $t \geq 0$ and $S = \{(x,y): (x-T)^2 + y^2 \leq T^2 \text{ where } T = t - [t]\}$ . Then we have

$\textbf{(A)}\ \text{the point } (0,0) \text{ does not belong to } S \text{ for any } t \qquad$

$\textbf{(B)}\ 0 \leq \text{Area } S \leq \pi \text{ for all } t \qquad$

$\textbf{(C)}\ S \text{ is contained in the first quadrant for all } t \geq 5 \qquad$

$\textbf{(D)}\ \text{the center of } S \text{ for any } t \text{ is on the line } y=x \qquad$

$\textbf{(E)}\ \text{none of the other statements is true}$

Solution

Problem 31

In the following equation, each of the letters represents uniquely a different digit in base ten:

$(YE) \cdot (ME) = TTT$

The sum $E+M+T+Y$ equals

$\textbf{(A)}\ 19 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 24$

Solution

Problem 32

The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length $\sqrt{15}$ is

$\textbf{(A)}\ 9 \qquad \textbf{(B)}\ 9/2 \qquad \textbf{(C)}\ 27/2 \qquad \textbf{(D)}\ \frac{9\sqrt3}{2} \qquad \textbf{(E)}\ \text{none of these}$

Solution

Problem 33

When one ounce of water is added to a mixture of acid and water, the new mixture is $20\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\frac13\%$ acid. The percentage of acid in the original mixture is

$\textbf{(A)}\ 22\% \qquad \textbf{(B)}\ 24\% \qquad \textbf{(C)}\ 25\% \qquad \textbf{(D)}\ 30\% \qquad \textbf{(E)}\ 33\frac13 \%$

Solution

Problem 34

A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was

$\textbf{(A)}\ 54 \text{ or } 18 \qquad \textbf{(B)}\ 60 \text{ or } 15 \qquad \textbf{(C)}\ 63 \text{ or } 12 \qquad \textbf{(D)}\ 72 \text{ or } 36 \qquad \textbf{(E)}\ 75 \text{ or } 20$

Solution

Problem 35

In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$ . Chords $MP$ , $PQ$ , and $NR$ are each $s$ units long and chord $MN$ is $d$ units long.

$[asy] draw(Circle((0,0),10)); draw((0,0)--(10,0)--(8.5,5.3)--(-8.5,5.3)--(-3,9.5)--(3,9.5)); dot((0,0)); dot((10,0)); dot((8.5,5.3)); dot((-8.5,5.3)); dot((-3,9.5)); dot((3,9.5)); label("1", (5,0), S); label("s", (8,2.6)); label("d", (0,4)); label("s", (-5,7)); label("s", (0,8.5)); label("O", (0,0),W); label("R", (10,0), E); label("M", (-8.5,5.3), W); label("N", (8.5,5.3), E); label("P", (-3,9.5), NW); label("Q", (3,9.5), NE); [/asy]$

Of the three equations

$\textbf{I.}\ d-s=1, \qquad \textbf{II.}\ ds=1, \qquad \textbf{III.}\ d^2-s^2=\sqrt{5}$

those which are necessarily true are

$\textbf{(A)}\ \textbf{I}\ \text{only} \qquad\textbf{(B)}\ \textbf{II}\ \text{only} \qquad\textbf{(C)}\ \textbf{III}\ \text{only} \qquad\textbf{(D)}\ \textbf{I}\ \text{and}\ \textbf{II}\ \text{only} \qquad\textbf{(E)}\ \textbf{I, II}\ \text{and} \textbf{ III}$

Solution

1973 AHSME (Problems • Answer Key • Resources)
Preceded by First AHSME, see 1972 AHSC	Followed by 1974 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
All AHSME Problems and Solutions

@@ Line 1: / Line 1: @@
+{{AHSME 35 Problems
+|year = 1973
+}}
 ==Problem 1==
@@ Line 32: / Line 35: @@
 <math>\text{I. Averaging is associative }</math>
 <math>\text{II. Averaging is commutative }</math>
 <math>\text{III. Averaging distributes over addition }</math>
 <math>\text{IV. Addition distributes over averaging }</math>
 <math>\text{V. Averaging has an identity element }</math>
 those which are always true are
-<math> \textbf{(A)}\ \text{All}\qquad\textbf{(B)}\ \text{I and II only}\qquad\textbf{(C)}\ \text{II and III only}\qquad\textbf{(D)}\ \text{II and IV only}\qquad\textbf{(E)}\ \text{II and V only} </math>
+<math> \textbf{(A)}\ \text{All}\qquad\textbf{(B)}\ \text{I and II only}\qquad\textbf{(C)}\ \text{II and III only}</math>
+<math>\textbf{(D)}\ \text{II and IV only}\qquad\textbf{(E)}\ \text{II and V only} </math>
 [[1973 AHSME Problems/Problem 5|Solution]]
 ==Problem 6==
@@ Line 76: / Line 86: @@
 ==Problem 10==
-	 If  is a real number, then the simultaneous system
+If <math>n</math> is a real number, then the simultaneous system
- <math> nx+y = 1 </math>
+<math> nx+y = 1 </math>
+<math> ny+z = 1 </math>
+<math> x+nz = 1 </math>
-has no solution if and only if  is equal to
+has no solution if and only if <math>n</math> is equal to
 <math> \textbf{(A)}\ -1\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 0\text{ or }1\qquad\textbf{(E)}\ \frac{1}2 </math>
 [[1973 AHSME Problems/Problem 10|Solution]]
 ==Problem 11==
-	 A circle with a circumscribed and an inscribed square centered at the origin  of a rectangular coordinate system with positive  and  axes  and  is shown in each figure  to  below.
+A circle with a circumscribed and an inscribed square centered at the origin  of a rectangular coordinate system with positive  and  axes  and  is shown in each figure  to  below.
 <asy>
@@ Line 132: / Line 146: @@
 <math> \textbf{(A)}\ I\qquad\textbf{(B)}\ II\qquad\textbf{(C)}\ III\qquad\textbf{(D)}\ IV\qquad\textbf{(E)}\ \mbox{none of these} </math>
-*An inequality of the form , for all  and  is represented geometrically by a figure showing the containment
+<nowiki>*</nowiki> An inequality of the form <math> f(x, y) \leq g(x, y)</math>, for all <math>x</math> and <math>y</math> is represented geometrically by a figure showing the containment
-for a typical real number .
+<math>\{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\\ \{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}</math>
+for a typical real number <math>a</math>.
 [[1973 AHSME Problems/Problem 11|Solution]]
 ==Problem 12==
-	 The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is
+The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is
 <math> \textbf{(A)}\ 3: 2\qquad\textbf{(B)}\ 3: 1\qquad\textbf{(C)}\ 2: 3\qquad\textbf{(D)}\ 2: 1\qquad\textbf{(E)}\ 1: 2 </math>
 [[1973 AHSME Problems/Problem 12|Solution]]
 ==Problem 13==
-	 The fraction  is equal to
+The fraction <math> \frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}} </math> is equal to
-	<math> \frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}} </math>
+<math> \textbf{(A)}\ \frac{2\sqrt2}{3} \qquad
+\textbf{(B)}\ 1 \qquad
+\textbf{(C)}\ \frac{2\sqrt3}3 \qquad
+\textbf{(D)}\ \frac43 \qquad
+\textbf{(E)}\ \frac{16}{9}</math>
 [[1973 AHSME Problems/Problem 13|Solution]]
 ==Problem 14==
-	 Each valve , , and , when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves  and  open it takes 1.5 hours, and with only valves  and  open it takes 2 hours. The number of hours required with only valves  and  open is
+Each valve <math>A</math>, <math>B</math>, and <math>C</math>, when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves <math>A</math> and <math>C</math> open it takes 1.5 hours, and with only valves <math>B</math> and <math>C</math> open it takes 2 hours. The number of hours required with only valves <math>A</math> and <math>B</math> open is
 	<math> \textbf{(A)}\ 1.1\qquad\textbf{(B)}\ 1.15\qquad\textbf{(C)}\ 1.2\qquad\textbf{(D)}\ 1.25\qquad\textbf{(E)}\ 1.75 </math>
 [[1973 AHSME Problems/Problem 14|Solution]]
 ==Problem 15==
-	 A sector with acute central angle  is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is
+A sector with acute central angle <math> \theta</math> is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is
+<math> \textbf{(A)}\ 3\cos\theta \qquad
+\textbf{(B)}\ 3\sec\theta \qquad
+\textbf{(C)}\ 3 \cos \frac12 \theta \qquad
+\textbf{(D)}\ 3 \sec \frac12 \theta \qquad
+\textbf{(E)}\ 3</math>
 [[1973 AHSME Problems/Problem 15|Solution]]
 ==Problem 16==
-	 If the sum of all the angles except one of a convex polygon is , then the number of sides of the polygon must be
+If the sum of all the angles except one of a convex polygon is <math> 2190^{\circ}</math>, then the number of sides of the polygon must be
+<math> \textbf{(A)}\ 13 \qquad
+\textbf{(B)}\ 15 \qquad
+\textbf{(C)}\ 17 \qquad
+\textbf{(D)}\ 19 \qquad
+\textbf{(E)}\ 21</math>
 [[1973 AHSME Problems/Problem 16|Solution]]
 ==Problem 17==
-	 If  is an acute angle and , then  equals
+If <math>\theta</math> is an acute angle and <math> \sin \frac12 \theta = \sqrt{\frac{x-1}{2x}}</math>, then <math> \tan \theta</math> equals
+<math> \textbf{(A)}\ x \qquad
+\textbf{(B)}\ \frac1{x} \qquad
+\textbf{(C)}\ \frac{\sqrt{x-1}}{x+1} \qquad
+\textbf{(D)}\ \frac{\sqrt{x^2-1}}{x} \qquad
+\textbf{(E)}\ \sqrt{x^2-1}</math>
 [[1973 AHSME Problems/Problem 17|Solution]]
 ==Problem 18==
+If <math> p \geq 5</math> is a prime number, then <math>24</math> divides <math>p^2 - 1</math> without remainder
-	 If  is a prime number, then  divides  without remainder
+<math> \textbf{(A)}\ \text{never} \qquad
+\textbf{(B)}\ \text{sometimes only} \qquad
+\textbf{(C)}\ \text{always} \qquad</math>
+<math> \textbf{(D)}\ \text{only if } p =5 \qquad
+\textbf{(E)}\ \text{none of these}</math>
 [[1973 AHSME Problems/Problem 18|Solution]]
 ==Problem 19==
+Define <math> n_a!</math> for <math>n</math> and <math>a</math> positive to be
-	 Define  for  and  positive to be
+<cmath> n_a ! = n (n-a)(n-2a)(n-3a)...(n-ka)</cmath>
+where <math>k</math> is the greatest integer for which <math>n>ka</math>. Then the quotient <math> 72_8!/18_2!</math> is equal to
-where  is the greatest integer for which . Then the quotient  is equal to
+<math> \textbf{(A)}\ 4^5 \qquad
+\textbf{(B)}\ 4^6 \qquad
+\textbf{(C)}\ 4^8 \qquad
+\textbf{(D)}\ 4^9 \qquad
+\textbf{(E)}\ 4^{12}</math>
 [[1973 AHSME Problems/Problem 19|Solution]]
 ==Problem 20==
-	 A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is
+A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is
+<math> \textbf{(A)}\ 4+\sqrt{185} \qquad
+\textbf{(B)}\ 16 \qquad
+\textbf{(C)}\ 17 \qquad
+\textbf{(D)}\ 18 \qquad
+\textbf{(E)}\ \sqrt{32}+\sqrt{137}</math>
 [[1973 AHSME Problems/Problem 20|Solution]]
 ==Problem 21==
-	 The number of sets of two or more consecutive positive integers whose sum is 100 is
+The number of sets of two or more consecutive positive integers whose sum is 100 is
+<math> \textbf{(A)}\ 1 \qquad
+\textbf{(B)}\ 2 \qquad
+\textbf{(C)}\ 3 \qquad
+\textbf{(D)}\ 4 \qquad
+\textbf{(E)}\ 5</math>
 [[1973 AHSME Problems/Problem 21|Solution]]
 ==Problem 22==
-	 The set of all real solutions of the inequality
+The set of all real solutions of the inequality
+<cmath> |x - 1| + |x + 2| < 5</cmath>
 is
+<math> \textbf{(A)}\ x \in ( - 3,2) \qquad \textbf{(B)}\ x \in ( - 1,2) \qquad \textbf{(C)}\ x \in ( - 2,1) \qquad</math>
+<math> \textbf{(D)}\ x \in \left( - \frac32,\frac72\right) \qquad \textbf{(E)}\ \O \text{ (empty})</math>
 [Note: I updated the notation on this problem.]
 [[1973 AHSME Problems/Problem 22|Solution]]
 ==Problem 23==
-	 There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, then the probability that the under-side is also red is
+There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, then the probability that the under-side is also red is
+<math> \textbf{(A)}\ \frac14 \qquad
+\textbf{(B)}\ \frac13 \qquad
+\textbf{(C)}\ \frac12 \qquad
+\textbf{(D)}\ \frac23 \qquad
+\textbf{(E)}\ \frac34</math>
 [[1973 AHSME Problems/Problem 23|Solution]]
 ==Problem 24==
-	 The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to . The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to  at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to
+The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to <math> \$3.15</math>. The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to <math> \$4.20</math> at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to
+<math> \textbf{(A)}\ \$1.70 \qquad
+\textbf{(B)}\ \$1.65 \qquad
+\textbf{(C)}\ \$1.20 \qquad
+\textbf{(D)}\ \$1.05 \qquad
+\textbf{(E)}\ \$0.95</math>
 [[1973 AHSME Problems/Problem 24|Solution]]
 ==Problem 25==
-	 A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is
+A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is
+<math> \textbf{(A)}\ 36\pi-34 \qquad
+\textbf{(B)}\ 30\pi - 15 \qquad
+\textbf{(C)}\ 36\pi - 33 \qquad</math>
+<math> \textbf{(D)}\ 35\pi - 9\sqrt3 \qquad
+\textbf{(E)}\ 30\pi - 9\sqrt3</math>
 [[1973 AHSME Problems/Problem 25|Solution]]
 ==Problem 26==
+The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is
-	 The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is
+<math> \textbf{(A)}\ 20 \qquad
+\textbf{(B)}\ 18 \qquad
+\textbf{(C)}\ 12 \qquad
+\textbf{(D)}\ 10 \qquad
+\textbf{(E)}\ 8</math>
 [[1973 AHSME Problems/Problem 26|Solution]]
 ==Problem 27==
-	 Cars A and B travel the same distance. Care A travels half that distance at  miles per hour and half at miles per hour. Car B travels half the time at  miles per hour and half at  miles per hour. The average speed of Car A is  miles per hour and that of Car B is  miles per hour. Then we always have
+Cars A and B travel the same distance. Car A travels half that distance at <math>u</math> miles per hour and half at <math>v</math> miles per hour. Car B travels half the time at <math>u</math> miles per hour and half at <math>v</math> miles per hour. The average speed of Car A is <math>x</math> miles per hour and that of Car B is <math>y</math> miles per hour. Then we always have
+<math> \textbf{(A)}\ x \leq y\qquad
+\textbf{(B)}\ x \geq y \qquad
+\textbf{(C)}\ x=y \qquad
+\textbf{(D)}\ x<y\qquad
+\textbf{(E)}\ x>y</math>
 [[1973 AHSME Problems/Problem 27|Solution]]
 ==Problem 28==
-	 If , , and  are in geometric progression (G.P.) with  and  is an integer, then , ,  form a sequence
+If <math>a</math>, <math>b</math>, and <math>c</math> are in geometric progression (G.P.) with <math> 1 < a < b < c</math> and <math>n>1</math> is an integer, then <math> \log_an</math>, <math> \log_bn</math>, <math> \log_cn</math> form a sequence
+<math> \textbf{(A)}\ \text{which is a G.P} \qquad</math>
+<math> \textbf{(B)}\ \text{whichi is an arithmetic progression (A.P)} \qquad</math>
+<math> \textbf{(C)}\ \text{in which the reciprocals of the terms form an A.P} \qquad</math>
+<math> \textbf{(D)}\ \text{in which the second and third terms are the }n\text{th powers of the first and second respectively} \qquad</math>
+<math> \textbf{(E)}\ \text{none of these}</math>
 [[1973 AHSME Problems/Problem 28|Solution]]
 ==Problem 29==
+Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is
-	 Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first me at the point A again, then the number of times they meet, excluding the start and finish, is
+<math> \textbf{(A)}\ 13 \qquad
+\textbf{(B)}\ 25 \qquad
+\textbf{(C)}\ 44 \qquad
+\textbf{(D)}\ \text{infinity} \qquad
+\textbf{(E)}\ \text{none of these}</math>
 [[1973 AHSME Problems/Problem 29|Solution]]
 ==Problem 30==
-	 Let  denote the greatest integer  where  and . Then we have
+Let <math> [t]</math> denote the greatest integer <math> \leq t</math> where <math> t \geq 0</math> and <math> S = \{(x,y): (x-T)^2 + y^2 \leq T^2 \text{ where } T = t - [t]\}</math>. Then we have
+<math> \textbf{(A)}\ \text{the point } (0,0) \text{ does not belong to } S \text{ for any } t \qquad</math>
+<math> \textbf{(B)}\ 0 \leq \text{Area } S \leq \pi \text{ for all } t \qquad</math>
+<math> \textbf{(C)}\ S \text{ is contained in the first quadrant for all } t \geq 5 \qquad</math>
+<math> \textbf{(D)}\ \text{the center of } S \text{ for any } t \text{ is on the line } y=x \qquad</math>
+<math> \textbf{(E)}\ \text{none of the other statements is true}</math>
 [[1973 AHSME Problems/Problem 30|Solution]]
@@ Line 261: / Line 394: @@
 ==Problem 31==
+In the following equation, each of the letters represents uniquely a different digit in base ten:
-	 In the following equation, each of the letters represents uniquely a different digit in base ten:
+<cmath> (YE) \cdot (ME) = TTT</cmath>
+The sum <math> E+M+T+Y</math> equals
-The sum  equals
+<math> \textbf{(A)}\ 19 \qquad
+\textbf{(B)}\ 20 \qquad
+\textbf{(C)}\ 21 \qquad
+\textbf{(D)}\ 22 \qquad
+\textbf{(E)}\ 24</math>
 [[1973 AHSME Problems/Problem 31|Solution]]
 ==Problem 32==
-	 The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length  is
+The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length <math>\sqrt{15}</math> is
+<math> \textbf{(A)}\ 9 \qquad
+\textbf{(B)}\ 9/2 \qquad
+\textbf{(C)}\ 27/2 \qquad
+\textbf{(D)}\ \frac{9\sqrt3}{2} \qquad
+\textbf{(E)}\ \text{none of these}</math>
 [[1973 AHSME Problems/Problem 32|Solution]]
 ==Problem 33==
-	 When one ounce of water is added to a mixture of acid and water, the new mixture is  acid. When one ounce of acid is added to the new mixture, the result is  acid. The percentage of acid in the original mixture is
+When one ounce of water is added to a mixture of acid and water, the new mixture is <math> 20\%</math> acid. When one ounce of acid is added to the new mixture, the result is <math> 33\frac13\%</math> acid. The percentage of acid in the original mixture is
+<math> \textbf{(A)}\ 22\% \qquad
+\textbf{(B)}\ 24\% \qquad
+\textbf{(C)}\ 25\% \qquad
+\textbf{(D)}\ 30\% \qquad
+\textbf{(E)}\ 33\frac13 \%</math>
 [[1973 AHSME Problems/Problem 33|Solution]]
 ==Problem 34==
-	 A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was
+A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was
+<math> \textbf{(A)}\ 54 \text{ or } 18 \qquad
+\textbf{(B)}\ 60 \text{ or } 15 \qquad
+\textbf{(C)}\ 63 \text{ or } 12 \qquad
+\textbf{(D)}\ 72 \text{ or } 36 \qquad
+\textbf{(E)}\ 75 \text{ or } 20</math>
 [[1973 AHSME Problems/Problem 34|Solution]]
 ==Problem 35==
-	 In the unit circle shown in the figure, chords  and  are parallel to the unit radius  of the circle with center at . Chords , , and  are each  units long and chord  is  units long.
+In the unit circle shown in the figure, chords <math>PQ</math> and <math>MN</math> are parallel to the unit radius <math>OR</math> of the circle with center at <math>O</math>. Chords <math>MP</math>, <math>PQ</math>, and <math>NR</math> are each <math>s</math> units long and chord <math>MN</math> is <math>d</math> units long.
+<asy>
+draw(Circle((0,0),10));
+draw((0,0)--(10,0)--(8.5,5.3)--(-8.5,5.3)--(-3,9.5)--(3,9.5));
+dot((0,0));
+dot((10,0));
+dot((8.5,5.3));
+dot((-8.5,5.3));
+dot((-3,9.5));
+dot((3,9.5));
+label("1", (5,0), S);
+label("s", (8,2.6));
+label("d", (0,4));
+label("s", (-5,7));
+label("s", (0,8.5));
+label("O", (0,0),W);
+label("R", (10,0), E);
+label("M", (-8.5,5.3), W);
+label("N", (8.5,5.3), E);
+label("P", (-3,9.5), NW);
+label("Q", (3,9.5), NE);
+</asy>
 Of the three equations
+<cmath> \textbf{I.}\ d-s=1, \qquad \textbf{II.}\ ds=1, \qquad \textbf{III.}\ d^2-s^2=\sqrt{5} </cmath>
 those which are necessarily true are
+<math>\textbf{(A)}\ \textbf{I}\ \text{only} \qquad\textbf{(B)}\ \textbf{II}\ \text{only} \qquad\textbf{(C)}\ \textbf{III}\ \text{only} \qquad\textbf{(D)}\ \textbf{I}\ \text{and}\ \textbf{II}\ \text{only} \qquad\textbf{(E)}\ \textbf{I, II}\ \text{and} \textbf{ III}</math>
 [[1973 AHSME Problems/Problem 35|Solution]]
@@ Line 298: / Line 485: @@
 * [[Mathematics competition resources]]
-{{AHSME box|year=1973|before=[[1972 AHSME]]|after=[[1974 AHSME]]}}
+{{AHSME 30p box|year=1973|before=First AHSME, see [[1972 AHSC]]|after=[[1974 AHSME]]}}
 {{MAA Notice}}

Page

Toolbox

Search

Difference between revisions of "1973 AHSME Problems"

Latest revision as of 18:25, 22 August 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

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

See also