Difference between revisions of "1965 AHSME Problems"
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Lines <math>AC</math> and <math>CE</math> are drawn to form <math>\angle{ACE}</math>. For this angle to be a right angle it is necessary that quadrilateral <math>ABCD</math> have: | Lines <math>AC</math> and <math>CE</math> are drawn to form <math>\angle{ACE}</math>. For this angle to be a right angle it is necessary that quadrilateral <math>ABCD</math> have: | ||
− | <math>\textbf{(A)}\ \text{all angles equal} | + | <math>\textbf{(A)}\ \text{all angles equal} \qquad |
\textbf{(B) }\ \text{all sides equal} \\ | \textbf{(B) }\ \text{all sides equal} \\ | ||
− | \textbf{(C) }\ \text{two pairs of equal sides} | + | \textbf{(C) }\ \text{two pairs of equal sides} \qquad |
\textbf{(D) }\ \text{one pair of equal sides} \\ | \textbf{(D) }\ \text{one pair of equal sides} \\ | ||
\textbf{(E) }\ \text{one pair of equal angles} </math> | \textbf{(E) }\ \text{one pair of equal angles} </math> |
Revision as of 19:27, 18 July 2024
1965 AHSC (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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 |
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 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
The number of real values of satisfying the equation
is:
Problem 2
A regular hexagon is inscribed in a circle. The ratio of the length of a side of the hexagon to the length of the shorter of the arcs intercepted by the side, is:
Problem 3
The expression has the same value as:
Problem 4
Line intersects line
and line
is parallel to
. The three lines are distinct and lie in a plane.
The number of points equidistant from all three lines is:
Problem 5
When the repeating decimal is written in simplest fractional form, the sum of the numerator and denominator is:
Problem 6
If then
equals:
Problem 7
The sum of the reciprocals of the roots of the equation is:
Problem 8
One side of a given triangle is 18 inches. Inside the triangle a line segment is drawn parallel to this side forming a trapezoid whose area is one-third of that of the triangle. The length of this segment, in inches, is:
Problem 9
The vertex of the parabola will be a point on the
-axis if the value of
is:
Problem 10
The statement is equivalent to the statement:
Problem 11
Consider the statements:
,
,
.
Of these the following are incorrect.
Problem 12
A rhombus is inscribed in in such a way that one of its vertices is
and two of its sides lie along
and
.
If
inches,
inches, and
inches, the side of the rhombus, in inches, is:
Problem 13
Let be the number of number-pairs
which satisfy
and
. Then
is:
Problem 14
The sum of the numerical coefficients in the complete expansion of is:
Problem 15
The symbol represents a two-digit number in the base
. If the number
is double the number
, then
is:
Problem 16
Let line be perpendicular to line
. Connect
to
, the midpoint of
, and connect
to
,
the midpoint of
. If
and
intersect in point
, and
inches,
then the area of triangle
, in square inches, is:
Problem 17
Given the true statement: The picnic on Sunday will not be held only if the weather is not fair. We can then conclude that:
Problem 18
If is used as an approximation to the value of
, the ratio of the error made to the correct value is:
Problem 19
If is exactly divisible by
, the value of
is:
Problem 20
For every the sum of n terms of an arithmetic progression is
. The
th term is:
Problem 21
It is possible to choose in such a way that the value of
is
Problem 22
If and
and
are the roots of
, then the equality
holds:
Problem 23
If we write for all
such that
, the smallest value we can use for
is:
Problem 24
Given the sequence ,
the smallest value of n such that the product of the first
members of this sequence exceeds
is:
Problem 25
Let be a quadrilateral with
extended to
so that
.
Lines
and
are drawn to form
. For this angle to be a right angle it is necessary that quadrilateral
have:
Problem 26
For the numbers define
to be the arithmetic mean of all five numbers;
to be the arithmetic mean of
and
;
to be the arithmetic mean of
, and
;
and
to be the arithmetic mean of
and
. Then, no matter how
, and
are chosen, we shall always have:
Problem 27
When is divided by
the quotient is
and the remainder is
.
When
is divided by
the quotient is
and the remainder is
. If
then
is:
Problem 28
An escalator (moving staircase) of uniform steps visible at all times descends at constant speed.
Two boys,
and
, walk down the escalator steadily as it moves, A negotiating twice as many escalator
steps per minute as
.
reaches the bottom after taking
steps while
reaches the bottom after taking
steps. Then
is:
Problem 29
Of students taking at least one subject the number taking Mathematics and English only equals the number
taking Mathematics only. No student takes English only or History only, and six students take Mathematics and
History, but not English. The number taking English and History only is five times the number taking all three subjects.
If the number taking all three subjects is even and non-zero, the number taking English and Mathematics only is:
Problem 30
Let of right triangle
be the diameter of a circle intersecting hypotenuse
in
.
At
a tangent is drawn cutting leg
in
. This information is not sufficient to prove that
Problem 31
The number of real values of satisfying the equality
, where
, is:
Problem 32
An article costing dollars is sold for $100 at a loss of
percent of the selling price.
It is then resold at a profit of
percent of the new selling price
.
If the difference between
and
is
dollars, then x is:
Problem 33
If the number , that is,
, ends with
zeros when given to the base
and ends with
zeros
when given to the base
, then
equals:
Problem 34
For the smallest value of
is:
Problem 35
The length of a rectangle is inches and its width is less than
inches. The rectangle is folded so that two
diagonally opposite vertices coincide. If the length of the crease is
, then the width is:
Problem 36
Given distinct straight lines and
. From a point in
a perpendicular is drawn to
;
from the foot of this perpendicular a line is drawn perpendicular to
.
From the foot of this second perpendicular a line is drawn perpendicular to
;
and so on indefinitely. The lengths of the first and second perpendiculars are
and
, respectively.
Then the sum of the lengths of the perpendiculars approaches a limit as the number of perpendiculars grows beyond all bounds. This limit is:
Problem 37
Point is selected on side
of
in such a way that
and point
is selected on side
such that
. The point of intersection of
and
is
. Then
is:
Problem 38
A takes times as long to do a piece of work as
and
together;
takes
times as long as
and
together;
and
takes
times as long as
and
together. Then
, in terms of
and
, is:
Problem 39
A foreman noticed an inspector checking a "-hole with a
"-plug and a
"-plug and suggested that two more gauges
be inserted to be sure that the fit was snug. If the new gauges are alike, then the diameter,
, of each, to the nearest hundredth of an inch, is:
Problem 40
Let be the number of integer values of
such that
is the square of an integer. Then
is:
See also
1965 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1964 AHSC |
Followed by 1966 AHSC | |
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 • 36 • 37 • 38 • 39 • 40 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.