Difference between revisions of "1976 AHSME Problems"
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== Problem 28 == | == Problem 28 == | ||
− | Lines <math> | + | Lines <math>L_1,L_2,\dots,L_{100}</math> are distinct. All lines <math>L_{4n}, n</math> a positive integer, are parallel to each other. |
− | All lines <math> | + | All lines <math>L_{4n-3}, n</math> a positive integer, pass through a given point <math>A.</math> The maximum number of points of intersection of pairs of lines from the complete set <math>\{L_1,L_2,\dots,L_{100}\}</math> is |
− | The maximum number of points of intersection of pairs of lines from the complete set <math>\{ | ||
<math>\textbf{(A) }4350\qquad | <math>\textbf{(A) }4350\qquad |
Latest revision as of 14:36, 8 September 2021
1976 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 one minus the reciprocal of equals the reciprocal of
, then
equals
Problem 2
For how many real numbers is
a real number?
Problem 3
The sum of the distances from one vertex of a square with sides of length to the midpoints of each of the sides of the square is
Problem 4
Let a geometric progression with n terms have first term one, common ratio and sum
, where
and
are not zero.
The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is
Problem 5
How many integers greater than and less than
, written in base-
notation,
are increased by
when their digits are reversed?
Problem 6
If is a real number and the negative of one of the solutions of
is a solution of
, then the solutions of
are
Problem 7
If is a real number, then the quantity
is positive if and only if
Problem 8
A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units?
Problem 9
In triangle ,
is the midpoint of
;
is the midpoint of
; and
is the midpoint of
.
If the area of
is
, then the area of
is
Problem 10
If , and
are real numbers and
and
, then the equation
has a solution
Problem 11
Which of the following statements is (are) equivalent to the statement "If the pink elephant on planet alpha has purple eyes, then the wild pig on planet beta does not have a long nose"?
Problem 12
A supermarket has crates of apples. Each crate contains at least
apples and at most
apples.
What is the largest integer
such that there must be at least
crates containing the same number of apples?
Problem 13
If cows give
cans of milk in
days, how many days will it take
cows to give
cans of milk?
Problem 14
The measures of the interior angles of a convex polygon are in arithmetic progression.
If the smallest angle is , and the largest is
, then the number of sides the polygon has is
Problem 15
If is the remainder when each of the numbers
, and
is divided by
, where
is an integer greater than
, then
equals
Problem 16
In triangles and
, lengths
, and
are all equal. Length
is twice the length of the altitude of
from
to
. Which of the following statements is (are) true?
Problem 17
If is an acute angle, and
, then
equals
Problem 18
In the adjoining figure, is tangent at
to the circle with center
; point
is interior to the circle;
and
intersects the circle at
. If
,
, and
, then the radius of the circle is
Problem 19
A polynomial has remainder three when divided by
and remainder five when divided by
.
The remainder when
is divided by
is
Problem 20
Let , and
be positive real numbers distinct from one. Then
Problem 21
What is the smallest positive odd integer such that the product
is greater than
?
(In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from
to
.)
Problem 22
Given an equilateral triangle with side of length , consider the locus of all points
in the plane of the
triangle such that the sum of the squares of the distances from
to the vertices of the triangle is a fixed number
. This locus
Problem 23
For integers and
such that
, let
. Then
is an integer
Problem 24
In the adjoining figure, circle has diameter
; circle
is tangent to circle
and to
at the center of circle
; and circle
tangent to circle
, to circle
and
. The ratio of the area of circle
to the area of circle
is
Problem 25
For a sequence , define
and, for all integer
.
If
, then
for all
Problem 26
In the adjoining figure, every point of circle is exterior to circle
.
Let
and
be the points of intersection of an internal common tangent with the two external common tangents.
Then the length of
is
Problem 27
If then
equals
Problem 28
Lines are distinct. All lines
a positive integer, are parallel to each other.
All lines
a positive integer, pass through a given point
The maximum number of points of intersection of pairs of lines from the complete set
is
Problem 29
Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as
Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is years, then Ann's age is
Problem 30
How many distinct ordered triples satisfy the following equations?
See also
1976 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1975 AHSME |
Followed by 1977 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.