Difference between revisions of "1978 AHSME Problems"
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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 \ \{and} \ y=2xy</cmath> |
is | is | ||
Revision as of 21:29, 21 January 2020
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:
\[x=x^2+y^2 \ \{and} \ y=2xy\] (Error compiling LaTeX. ! Extra }, or forgotten $.)
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.