Difference between revisions of "1984 AHSME Problems"
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*[[1984 AHSME Answer Key]] | *[[1984 AHSME Answer Key]] | ||
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Revision as of 12:48, 5 July 2013
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
equals
Problem 2
If , and
are not
, then
equals
Problem 3
Let be the smallest nonprime integer greater than
with no prime factor less than
. Then
Problem 4
Points are picked on a circle such that
. When
is extended to the left, point
is marked outside the circle such that
and
. When
is extended to the left, point
is marked outside the circle such that
.
is perpendicular to both
and
. Find the length of
.
Problem 5
The largest integer for which
is
Problem 6
In a certain school, there are times as many boys as girls and
times as many girls as teachers. Using the letters
to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the expression
Problem 7
When Dave walks to school, he averages steps per minute, and each of his steps is
cm long. It takes him
minutes to get to school. His brother, Jack, going to the same school by the same route, averages
steps per minute, but his steps are only
cm long. How long does it take Jack to get to school?
Problem 8
Figure is a trapezoid with
,
,
,
, and
. The length of
is
Problem 9
The number of digits in (when written in the usual base
form) is
Problem 10
Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are , and
. The fourth number is
Problem 11
A calculator has a key that replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let be the final result when one starts with a number
and alternately squares and reciprocates
times each. Assuming the calculator is completely accurate (e.g. no roundoff or overflow), then
equals
Problem 12
If the sequence is defined by
where .
Then equals
Problem 13
equals
Problem 14
The product of all real roots of the equation is
Problem 15
If , then one value for
is
Problem 16
The function satisfies
for all real numbers
. If the equation
has exactly four distinct real roots, then the sum of these roots is
Problem 17
A right triangle with hypotenuse
has side
. Altitude
divides
into segments
and
, with
. The area of
is:
Problem 18
A point is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line
. Then
is
Problem 19
A box contains balls, numbered
. If
balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd?
Problem 20
The number of the distinct solutions to the equation
is
Problem 21
The number of triples of positive integers which satisfy the simultaneous equations
is
Problem 22
Let and
be fixed positive numbers. For each real number
let
be the vertex of the parabola
. If the set of the vertices
for all real numbers of
is graphed on the plane, the graph is
Problem 23
equals
Problem 24
If and
are positive real numbers and each of the equations
and
has real roots, then the smallest possible value of
is
Problem 25
The total area of all the faces of a rectangular solid is , and the total length of all its edges is
. Then the length in cm of any one of its interior diagonals is
Problem 26
In the obtuse triangle with
,
,
, and
(
is on
,
is on
, and
is on
). If the area of
is
, then the area of
is
Problem 27
In ,
is on
and
is on
. Also,
,
, and
. Find
.
Problem 28
The number of distinct pairs of integers such that
and
is
Problem 29
Find the largest value for for pairs of real numbers
which satisfy
.
Problem 30
For any complex number ,
is defined to be the real number
. If
, then
equals
See Also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.