Difference between revisions of "1983 AHSME Problems"
Sevenoptimus (talk | contribs) m (Fixed problem statement and formatting of Problem 23) |
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+ | {{AHSME Problems | ||
+ | |year = 1983 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
Line 13: | Line 16: | ||
== Problem 2 == | == Problem 2 == | ||
− | Point <math>P</math> is outside circle <math>C</math> on the plane. At most how many points on <math>C</math> are <math>3 | + | Point <math>P</math> is outside circle <math>C</math> on the plane. At most how many points on <math>C</math> are <math>3</math> cm from P? |
<math>\textbf{(A)} \ 1 \qquad | <math>\textbf{(A)} \ 1 \qquad | ||
Line 119: | Line 122: | ||
Segment <math>AB</math> is both a diameter of a circle of radius <math>1</math> and a side of an equilateral triangle <math>ABC</math>. | Segment <math>AB</math> is both a diameter of a circle of radius <math>1</math> and a side of an equilateral triangle <math>ABC</math>. | ||
− | The circle also intersects <math>AC</math> and <math> | + | The circle also intersects <math>AC</math> and <math>BC</math> at points <math>D</math> and <math>E</math>, respectively. The length of <math>AE</math> is |
<math> | <math> | ||
Line 319: | Line 322: | ||
\textbf{(E)} \ 14 </math> | \textbf{(E)} \ 14 </math> | ||
− | [[1983 AHSME Problems/Problem | + | [[1983 AHSME Problems/Problem 23|Solution]] |
== Problem 24 == | == Problem 24 == | ||
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How many non-congruent right triangles are there such that the perimeter in <math>\text{cm}</math> and the area in <math>\text{cm}^2</math> are numerically equal? | How many non-congruent right triangles are there such that the perimeter in <math>\text{cm}</math> and the area in <math>\text{cm}^2</math> are numerically equal? | ||
− | <math>\ | + | <math>\textbf{(A)} \ \text{none} \qquad |
− | \ | + | \textbf{(B)} \ 1 \qquad |
− | \ | + | \textbf{(C)} \ 2 \qquad |
− | \ | + | \textbf{(D)} \ 4 \qquad |
− | \ | + | \textbf{(E)} \ \text{infinitely many}</math> |
[[1983 AHSME Problems/Problem 24|Solution]] | [[1983 AHSME Problems/Problem 24|Solution]] | ||
Line 335: | Line 338: | ||
== Problem 25 == | == Problem 25 == | ||
− | If <math>60^a = 3</math> and <math>60^b = 5</math>, then <math>12^{ | + | If <math>60^a = 3</math> and <math>60^b = 5</math>, then <math>12^{(1-a-b)/\left(2\left(1-b\right)\right)}</math> is |
− | <math>\ | + | <math>\textbf{(A)} \ \sqrt{3} \qquad |
− | \ | + | \textbf{(B)} \ 2 \qquad |
− | \ | + | \textbf{(C)} \ \sqrt{5} \qquad |
− | \ | + | \textbf{(D)} \ 3 \qquad |
− | \ | + | \textbf{(E)} \ \sqrt{12} </math> |
[[1983 AHSME Problems/Problem 25|Solution]] | [[1983 AHSME Problems/Problem 25|Solution]] | ||
Line 421: | Line 424: | ||
<asy> | <asy> | ||
− | + | import geometry; | |
− | + | import graph; | |
− | draw(M--N | + | |
− | + | unitsize(2 cm); | |
− | draw( | + | |
− | draw( | + | pair A, B, C, M, N, P; |
− | + | ||
− | label("$A$", A, | + | M = (-1,0); |
− | label("$B$", B, | + | N = (1,0); |
+ | C = (0,0); | ||
+ | A = dir(140); | ||
+ | B = dir(20); | ||
+ | P = extension(A, A + rotate(10)*(C - A), B, B + rotate(10)*(C - B)); | ||
+ | |||
+ | draw(M--N); | ||
+ | draw(arc(C,1,0,180)); | ||
+ | draw(A--C--B); | ||
+ | draw(A--P--B); | ||
+ | |||
+ | label("$A$", A, NW); | ||
+ | label("$B$", B, E); | ||
label("$C$", C, S); | label("$C$", C, S); | ||
− | label("$M$", M, | + | label("$M$", M, SW); |
− | label("$N$", N, | + | label("$N$", N, SE); |
label("$P$", P, S); | label("$P$", P, S); | ||
− | + | </asy> | |
− | |||
− | |||
<math>\textbf{(A)}\ 10^{\circ}\qquad | <math>\textbf{(A)}\ 10^{\circ}\qquad | ||
Line 443: | Line 456: | ||
\textbf{(D)}\ 25^{\circ}\qquad | \textbf{(D)}\ 25^{\circ}\qquad | ||
\textbf{(E)}\ 30^{\circ} </math> | \textbf{(E)}\ 30^{\circ} </math> | ||
+ | |||
+ | [[1983 AHSME Problems/Problem 30|Solution]] | ||
== See also == | == See also == |
Latest revision as of 23:43, 2 December 2021
1983 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 and
, then
equals
Problem 2
Point is outside circle
on the plane. At most how many points on
are
cm from P?
Problem 3
Three primes and
satisfy
and
. Then
equals
Problem 4
In the adjoining plane figure, sides and
are parallel, as are sides
and
,
and sides
and
. Each side has length
. Also,
.
The area of the figure is
Problem 5
Triangle has a right angle at
. If
, then
is
Problem 6
When and
are multiplied, the product is a polynomial of degree
Problem 7
Alice sells an item at less than the list price and receives
of her selling price as her commission.
Bob sells the same item at
less than the list price and receives
of his selling price as his commission.
If they both get the same commission, then the list price is
Problem 8
Let . Then for
is
Problem 9
In a certain population the ratio of the number of women to the number of men is to
.
If the average (arithmetic mean) age of the women is
and the average age of the men is
,
then the average age of the population is
Problem 10
Segment is both a diameter of a circle of radius
and a side of an equilateral triangle
.
The circle also intersects
and
at points
and
, respectively. The length of
is
Problem 11
Simplify .
Problem 12
If , then
equals
Problem 13
If and
, and none of these quantities is
, then
equals
Problem 14
The units digit of is
Problem 15
Three balls marked and
are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is
, what is the probability that the ball numbered
was drawn all three times?
Problem 16
Let , where the digits are obtained by writing the integers
through
in order.
The
rd digit to the right of the decimal point is
Problem 17
The diagram above shows several numbers in the complex plane. The circle is the unit circle centered at the origin.
One of these numbers is the reciprocal of . Which one?
Problem 18
Let be a polynomial function such that, for all real
,
.
For all real
is
Problem 19
Point is on side
of triangle
. If
,
then the length of
is
Problem 20
If and
are the roots of
, and
and
are the roots of
, then
is necessarily
Problem 21
Find the smallest positive number from the numbers below.
Problem 22
Consider the two functions and
, where the variable
and the constants
and
are real numbers.
Each such pair of constants
and
may be considered as a point
in an
-plane.
Let
be the set of such points
for which the graphs of
and
do not intersect (in the
-plane). The area of
is
Problem 23
In the adjoining figure the five circles are tangent to one another consecutively and to the lines
and
.
If the radius of the largest circle is
and that of the smallest one is
, then the radius of the middle circle is
Problem 24
How many non-congruent right triangles are there such that the perimeter in and the area in
are numerically equal?
Problem 25
If and
, then
is
Problem 26
The probability that event occurs is
; the probability that event B occurs is
.
Let
be the probability that both
and
occur. The smallest interval necessarily containing
is the interval
Problem 27
A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance
of m from the point where the sphere touches the ground. At the same instant a meter stick
(held vertically with one end on the ground) casts a shadow of length
m. What is the radius of the sphere in meters?
(Assume the sun's rays are parallel and the meter stick is a line segment.)
Problem 28
Triangle in the figure has area
. Points
and
, all distinct from
and
,
are on sides
and
respectively, and
. If triangle
and quadrilateral
have equal areas, then that area is
Problem 29
A point lies in the same plane as a given square of side
. Let the vertices of the square,
taken counterclockwise, be
and
. Also, let the distances from
to
and
, respectively, be
and
.
What is the greatest distance that
can be from
if
?
Problem 30
Distinct points and
are on a semicircle with diameter
and center
.
The point
is on
and
. If
, then
equals
See also
1983 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1982 AHSME |
Followed by 1984 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.