1966 AHSME Problems
Contents
[hide]- 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 Problem 31
- 32 Problem 32
- 33 Problem 33
- 34 Problem 34
- 35 Problem 35
- 36 Problem 36
- 37 Problem 37
- 38 Problem 38
- 39 Problem 39
- 40 Problem 40
- 41 See also
Problem 1
Given that the ratio of to
is constant, and
when
, then, when
,
equals:
Problem 2
When the base of a triangle is increased 10% and the altitude to this base is decreased 10%, the change in area is
Problem 3
If the arithmetic mean of two numbers is and their geometric mean is
, then an equation with the given two numbers as roots is:
Problem 4
Circle I is circumscribed about a given square and circle II is inscribed in the given square. If is the ratio of the area of circle
to that of circle
, then
equals:
Problem 5
The number of values of satisfying the equation
is:
Problem 6
is the diameter of a circle centered at
.
is a point on the circle such that angle
is
. If the diameter of the circle is
inches, the length of chord
, expressed in inches, is:
Problem 7
Let be an identity in
. The numerical value of
is:
Problem 8
The length of the common chord of two intersecting circles is feet. If the radii are
feet and
feet, a possible value for the distance between the centers of teh circles, expressed in feet, is:
Problem 9
If , then
equals:
Problem 10
If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is:
Problem 11
The sides of triangle are in the ratio
.
is the angle-bisector drawn to the shortest side
, dividing it into segments
and
. If the length of
is
, then the length of the longer segment of
is:
Problem 12
The number of real values of that satisfy the equation
is:
Problem 13
The number of points with positive rational coordinates selected from the set of points in the -plane such that
, is:
Problem 14
The length of rectangle is 5 inches and its width is 3 inches. Diagonal
is divided into three equal segments by points
and
. The area of triangle
, expressed in square inches, is:
Problem 15
If and
, then
Problem 16
If and
,
and
real numbers, then
equals:
Problem 17
The number of distinct points common to the curves and
is:
Problem 18
In a given arithmetic sequence the first term is , the last term is
, and the sum of all the terms is
. The common difference is:
Problem 19
Let be the sum of the first
terms of the arithmetic sequence
and let
be the sum of the first
terms of the arithmetic sequence
. Assume
. Then
for:
Problem 20
The negation of the proposition "For all pairs of real numbers , if
, then
" is: There are real numbers
such that
Problem 21
An "-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively
; for all
values of
, sides
and
are non-parallel, sides
and
being respectively identical with sides
and
; prolong the
pairs of sides numbered
and
until they meet. (A figure is shown for the case
).
Let be the degree-sum of the interior angles at the
points of the star; then
equals:
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Five points are taken in order on a straight line with distances
,
,
, and
.
is a point on the line between
and
and such that
. Then
equals:
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.