Difference between revisions of "1972 AHSME Problems"
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[[1972 AHSME Problems/Problem 23|Solution]] | [[1972 AHSME Problems/Problem 23|Solution]] | ||
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== Problem 24 == | == Problem 24 == | ||
Latest revision as of 23:21, 25 May 2024
1972 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 |
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 See also
Problem 1
The lengths in inches of the three sides of each of four triangles , and
are as follows:
Of these four given triangles, the only right triangles are
Problem 2
If a dealer could get his goods for % less while keeping his selling price fixed,
his profit, based on cost, would be increased to
% from his present profit of
%, which is
Problem 3
If where
, then
is equal to
Problem 4
The number of solutions to , where
is a subset of
is
Problem 5
From among those which have the greatest and the next to the greatest values, in that order, are
Problem 6
If , then the value of
is
Problem 7
If , then
is equal to
Problem 8
If where
and
are real, then
Problem 9
Ann and Sue bought identical boxes of stationery. Ann used hers to write -sheet letters and Sue used hers to write
-sheet letters.
Ann used all the envelopes and had
sheets of paper left, while Sue used all of the sheets of paper and had
envelopes left.
The number of sheets of paper in each box was
Problem 10
For real, the inequality
is equivalent to
Problem 11
The value(s) of for which the following pair of equations
may have a real common solution, are
Problem 12
The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge expressed as a number of feet is
Problem 13
Inside square (See figure) with sides of length
inches, segment
is drawn where
is the point on
which is
inches from
.
The perpendicular bisector of
is drawn and intersects
, and
at points
, and
respectively. The ratio of segment
to
is
Problem 14
A triangle has angles of and
. If the side opposite the
angle has length
, then the side opposite the
angle has length
Problem 15
A contractor estimated that one of his two bricklayers would take hours to build a certain wall and the other
hours.
However, he knew from experience that when they worked together, their combined output fell by
bricks per hour.
Being in a hurry, he put both men on the job and found that it took exactly 5 hours to build the wall. The number of bricks in the wall was
Problem 16
There are two positive numbers that may be inserted between and
such that the first three are in geometric progression
while the last three are in arithmetic progression. The sum of those two positive numbers is
Problem 17
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is
Problem 18
Let be a trapezoid with the measure of base
twice that of base
, and let
be the point of intersection of the diagonals.
If the measure of diagonal
is
, then that of segment
is equal to
Problem 19
The sum of the first terms of the sequence
in terms of
is
Problem 20
If where
and
, then
is equal to
Problem 21
If the sum of the measures in degrees of angles and
in the figure above is
, then
is equal to
Problem 22
If are imaginary roots of the equation
where
, and
are real numbers, then
in terms of
and
is
Problem 23
The radius of the smallest circle containing the symmetric figure composed of the 3 unit squares shown above is
Problem 24
A man walked a certain distance at a constant rate. If he had gone mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone
mile per hour slower, he would have been
hours longer on the road. The distance in miles he walked was
Problem 25
Inscribed in a circle is a quadrilateral having sides of lengths , and
taken consecutively. The diameter of this circle has length
Problem 26
In the circle above, is the midpoint of arc
and segment
is perpendicular to chord
at
.
If the measure of chord
is
and that of segment
is
, then segment
has measure equal to
Problem 27
If the area of is
square units and the geometric mean (mean proportional)
between sides
and
is
inches, then
is equal to
Problem 28
A circular disc with diameter is placed on an
checkerboard with width
so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is
Problem 29
If for
, then
in terms of
is
Problem 30
A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle is
Problem 31
When the number is divided by
, the remainder in the division is
Problem 32
Chords and
in the circle above intersect at E and are perpendicular to each other.
If segments
, and
have measures
, and
respectively, then the length of the diameter of the circle is
Problem 33
The minimum value of the quotient of a (base ten) number of three different non-zero digits divided by the sum of its digits is
Problem 34
Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is
Problem 35
Equilateral triangle (see figure) with side
of length
inches is placed inside square
with side of length
inches so that
is on side
. The triangle is rotated clockwise about
, then
, and so on along the sides of the square
until
returns to its original position. The length of the path in inches traversed by vertex
is equal to
See also
1972 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1971 AHSC |
Followed by Last AHSC, see 1973 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 • 31 • 32 • 33 • 34 • 35 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.