1963 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 12
- 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
Which one of the following points is not on the graph of ?
$\textbf{(A)}\(0,0)\qquad \textbf{(B)}\ \left(-\frac{1}{2},-1\right)\qquad \textbf{(C)}\ \left(\frac{1}{2},\frac{1}{3}\right)\qquad \textbf{(D)}\ (-1,1)\qquad \textbf{(E)}\ (-2,2)$ (Error compiling LaTeX. Unknown error_msg)
Problem 2
let . Find
when
and
.
Problem 3
If the reciprocal of is
, then
equals:
Problem 4
For what value(s) of does the pair of equations
and
have two identical solutions?
Problem 5
If and
are real numbers and
, then:
Problem 6
is right-angled at
. On
there is a point
for which
and
. The magnitude of
is:
Problem 7
Given the four equations:
The pair representing the perpendicular lines is:
Problem 8
The smallest positive integer for which
, where
is an integer, is:
Problem 9
In the expansion of the coefficient of
is:
Problem 10
Point is taken interior to a square with side-length
and such that is it equally distant from two
consecutive vertices and from the side opposite these vertices. If
represents the common distance, then
equals:
Problem 12
The arithmetic mean of a set of numbers is
. If two numbers of the set, namely
and
, are discarded,
the arithmetic mean of the remaining set of numbers is:
Problem 12
Three vertices of parallelogram are
with
and
diagonally opposite.
The sum of the coordinates of vertex
is:
Problem 13
If , the number of integers
which can possibly be negative, is, at most:
Problem 14
Given the equations and
. If, when the roots of the equation are suitably listed,
each root of the second equation is
more than the corresponding root of the first equation, then
equals:
Problem 15
A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is:
Problem 16
Three numbers , none zero, form an arithmetic progression. Increasing
by
or increasing
by
results
in a geometric progression. Then
equals:
Problem 17
The expression ,
real,
, has the value
for:
Problem 18
Chord is the perpendicular bisector of chord
, intersecting it in
. Between
and
point
is taken,
and
A
U
\triangle EUM$ is similar to triangle:
Problem 19
In counting colored balls, some red and some black, it was found that
of the first
counted were red.
Thereafter,
out of every
counted were red. If, in all,
% or more of the balls counted were red, the maximum value of
is:
Problem 20
Two men at points and
,
miles apart, set out at the same time to walk towards each other.
The man at
walks uniformly at the rate of
miles per hour; the man at
walks at the constant
rate of
miles per hour for the first hour, at
miles per hour for the second hour,
and so on, in arithmetic progression. If the men meet
miles nearer
than
in an integral number of hours, then
is:
Problem 21
The expression has:
Problem 22
Acute-angled is inscribed in a circle with center at
;
and
.
point
is taken in minor arc
such that
is perpendicular to
. Then the ratio of the magnitudes of
and
is:
Problem 23
A gives as many cents as
has and
as many cents as
has. Similarly,
then gives
and
as many cents as each then has.
, similarly, then gives
and
as many cents as each then has. If each finally has
cents, with how many cents does
start?
Problem 24
Consider equations of the form . How many such equations have real roots and have coefficients
and
selected
from the set of integers
?
Problem 25
Point is taken in side
of square
. At
a perpendicular is drawn to
, meeting
extended at
.
The area of
is
square inches and the area of
is
square inches. Then the number of inches in
is:
Problem 26
Version 1 Consider the statements:
where , and
are propositions. How many of these imply the truth of
?
Version 2
Consider the statements (1) and
are true and
is false (2)
is true and
and
are false (3)
is true and
and
are false (4)
and
are true and
is false.
How many of these imply the truth of the statement
"
is implied by the statement that
implies
"?
Problem 27
Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is:
Problem 28
Given the equation with real roots. The value of
for which the product of the roots of the equation is a maximum is:
Problem 29
A particle projected vertically upward reaches, at the end of seconds, an elevation of
feet where
. The highest elevation is:
Problem 30
Let . Find a new function
by replacing each
in
by
, and simplify.
The simplified expression
is equal to:
Problem 31
The number of solutions in positive integers of is:
Problem 32
The dimensions of a rectangle are
and
,
. It is required to obtain a rectangle with dimensions
and
,
,
so that its perimeter is one-third that of
, and its area is one-third that of
. The number of such (different) rectangles is:
Problem 33
Given the line and a line
parallel to the given line and
units from it. A possible equation for
is:
Problem 34
In , side
, side
, and side
. Let
be the largest number such that the magnitude,
in degrees, of the angle opposite side
exceeds
. Then
equals:
Problem 35
The lengths of the sides of a triangle are integers, and its area is also an integer.
One side is and the perimeter is
. The shortest side is:
Problem 36
A person starting with $64 and making 6 bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is:
$\textbf{(A)}\ \text{a loss of $27} \qquad \textbf{(B)}\ \text{a gain of $27} \qquad \textbf{(C)}\ \text{a loss of $37} \qquad \textbf{(D)}\ \text{neither a gain nor a loss}\qquad \textbf{(E)}\ \text{a gain or a loss depending upon the order in which the wins and losses occur}$ (Error compiling LaTeX. Unknown error_msg)
Problem 37
Given points on a straight line, in the order stated (not necessarily evenly spaced).
Let
be an arbitrarily selected point on the line and let
be the sum of the undirected lengths
. Then
is smallest if and only if the point
is:
Problem 38
Point is taken on the extension of side
of parallelogram
.
intersects diagonal
at
and side
at
.
If
and
, then
equals:
Problem 39
In lines
and
are drawn so that
and
. Let
where
is the intersection point of
and
. Then
equals:
Problem 40
If is a number satisfying the equation
, then
is between:
Solution
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.