Difference between revisions of "1978 AHSME Problems"
m (→Problem 21) |
m (extra answer choice) |
||
(16 intermediate revisions by 8 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{AHSME Problems | ||
+ | |year = 1978 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
Line 65: | Line 68: | ||
The number of distinct pairs <math>(x,y)</math> of real numbers satisfying both of the following equations: | 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 | is | ||
Line 83: | Line 86: | ||
\textbf{(B) }6\sqrt{2}\qquad | \textbf{(B) }6\sqrt{2}\qquad | ||
\textbf{(C) }5\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> | \textbf{(E) }4\sqrt{3} </math> | ||
Line 91: | Line 93: | ||
== Problem 8 == | == 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 | <math>\textbf{(A) }\frac{2}{3}\qquad | ||
Line 247: | Line 249: | ||
== Problem 20 == | == Problem 20 == | ||
− | If <math>a,b,c</math> are non-zero real numbers such that < | + | 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>\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]] | [[1978 AHSME Problems/Problem 20|Solution]] | ||
Line 262: | Line 263: | ||
For all positive numbers <math>x</math> distinct from <math>1</math>, | 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> | + | <cmath>\frac{1}{\log_3(x)}+\frac{1}{\log_4(x)}+\frac{1}{\log_5(x)}</cmath> |
equals | equals | ||
− | <math>\text{(A) }\frac{1}{log_{60}(x)}\qquad\\ | + | <math>\text{(A) }\frac{1}{\log_{60}(x)}\qquad\\ |
− | \text{(B) }\frac{1}{log_{x}(60)}\qquad\\ | + | \text{(B) }\frac{1}{\log_{x}(60)}\qquad\\ |
− | \text{(C) }\frac{1}{(log_{3}(x))(log_{4}(x))(log_{5}(x))}\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{(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> | + | \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]] | [[1978 AHSME Problems/Problem 21|Solution]] | ||
Line 345: | Line 346: | ||
== Problem 25 == | == Problem 25 == | ||
− | Let <math> | + | 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 | <math> \text{(i) }\frac{a}{2}\le x\le 2a\qquad | ||
Line 433: | Line 434: | ||
== Problem 29 == | == 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 | 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 459: | Line 460: | ||
== See also == | == See also == | ||
− | + | ||
− | |||
* [[AMC 12 Problems and Solutions]] | * [[AMC 12 Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{AHSME box|year=1978|before=[[1977 AHSME]]|after=[[1979 AHSME]]}} | ||
+ | |||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 01:06, 22 February 2024
1978 AHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
If , then
equals
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
Problem 3
For all non-zero numbers and
such that
,
equals
Problem 4
If , and
, then
is equal to
Problem 5
Four boys bought a boat for . 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?
Problem 6
The number of distinct pairs of real numbers satisfying both of the following equations:
is
Problem 7
Opposite sides of a regular hexagon are inches apart. The length of each side, in inches, is
Problem 8
If and the sequences
and
each are in arithmetic progression, then
equals
Problem 9
If , then
equals
Problem 10
If is a point on circle
with center
, then the set of all points
in the plane of circle
such that the distance between
and
is less than or equal to the distance between
and any other point on circle
is
Problem 11
If is positive and the line whose equation is
is tangent to the circle whose equation is
, then
equals
Problem 12
In ,
, points
and
lie on sides
and
,
respectively, and points
are distinct.* If lengths
, and
are all equal,
then the measure of
is
- The specification that points
be distinct was not included in the original statement of the problem.
If , then
and
.
Problem 13
If , and
are non-zero numbers such that
and
are the solutions of
and
and
are
the solutions of
, then
equals
Problem 14
If an integer is a solution of the equation
and the representation of
in the base-
number system is
,
then the base-n representation of
is
Problem 15
If and
, then
is
Problem 16
In a room containing people,
, 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?
Problem 17
If is a positive number and
is a function such that, for every positive number
,
;
then, for every positive number
,
is equal to
Problem 18
What is the smallest positive integer such that
?
Problem 19
A positive integer not exceeding
is chosen in such a way that if
, then the probability of choosing
is
, and if
, then the probability of choosing
is
. The probability that a perfect square is chosen is
Problem 20
If are non-zero real numbers such that
and
and
then
equals
Problem 21
For all positive numbers distinct from
,
equals
Problem 22
The following four statements, and only these are found on a card:
(Assume each statement is either true or false.) Among them the number of false statements is exactly
Problem 23
Vertex of equilateral
is in the interior of square
, and
is the point of intersection
of diagonal
and line segment
. If length
is
then the area of
is
Problem 24
If the distinct non-zero numbers form a geometric progression with common ratio
,
then
satisfies the equation
Problem 25
Let be a positive number. Consider the set
of all points whose rectangular coordinates
satisfy all of the following conditions:
The boundary of set S is a polygon with
Problem 26
In and
. Circle
is the circle with smallest radius which passes through
and is tangent to
. Let
and
be the points of intersection, distinct from
, of circle
with sides
and
, respectively. The length of segment
is
Problem 27
There is more than one integer greater than which, when divided by any integer
such that
, has a remainder of
.
What is the difference between the two smallest such integers?
Problem 28
If is equilateral and
is the midpoint of line segment
for all positive integers
,
then the measure of
equals
Problem 29
Sides and
, respectively, of convex quadrilateral
are extended past
and
to points
and
.
Also,
and
; and the area of
is
. The area of
is
Problem 30
In a tennis tournament, women and
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
, then
equals
See also
1978 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1977 AHSME |
Followed by 1979 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 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.