1960 AHSME Problems
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
Problem 1
If is a solution (root) of
, then
equals:
Problem 2
It takes seconds for a clock to strike
o'clock beginning at
o'clock precisely.
If the strikings are uniformly spaced, how long, in seconds, does it take to strike
o'clock?
Problem 3
Applied to a bill for the difference between a discount of
% and two successive discounts of
% and
%,
expressed in dollars, is:
Problem 4
Each of two angles of a triangle is and the included side is
inches. The area of the triangle, in square inches, is:
Problem 5
The number of distinct points common to the graphs of and
is:
Problem 6
The circumference of a circle is inches. The side of a square inscribed in this circle, expressed in inches, is:
Problem 7
Circle passes through the center of, and is tangent to, circle
. The area of circle
is
square inches.
Then the area of circle
, in square inches, is:
Problem 8
The number can be written as a fraction.
When reduced to lowest terms the sum of the numerator and denominator of this fraction is:
Problem 9
The fraction is (with suitable restrictions of the values of a, b, and c):
Problem 10
Given the following six statements:
The statement that negates statement is:
Problem 11
For a given value of the product of the roots of
is
. The roots may be characterized as:
Problem 12
The locus of the centers of all circles of given radius , in the same plane, passing through a fixed point, is:
Problem 13
The polygon(s) formed by , and
, is (are):
Problem 14
If and
are real numbers, the equation
has a unique solution
[The symbol
means that
is different from zero]:
Problem 15
Triangle is equilateral with side
, perimeter
, area
, and circumradius
(radius of the circumscribed circle).
Triangle
is equilateral with side
, perimeter
, area
, and circumradius
. If
is different from
, then:
Problem 16
In the numeration system with base , counting is as follows:
.
The number whose description in the decimal system is
, when described in the base
system, is a number with:
Problem 17
The formula gives, for a certain group, the number of individuals whose income exceeds
dollars.
The lowest income, in dollars, of the wealthiest
individuals is at least:
Problem 18
The pair of equations and
has:
Problem 19
Consider equation where
, and
are positive integers, and equation
,
where
, and
are positive integers. Then
Problem 20
The coefficient of in the expansion of
is:
Problem 21
The diagonal of square is
. The perimeter of square
with twice the area of
is:
Problem 22
The equality , where
and
are unequal non-zero constants, is satisfied by
, where:
Problem 23
The radius of a cylindrical box is
inches, the height
is
inches.
The volume
is to be increased by the same fixed positive amount when
is increased by
inches as when
is increased by
inches. This condition is satisfied by:
Problem 24
If , where
is real, then
is:
Problem 25
Let and
be any two odd numbers, with
less than
.
The largest integer which divides all possible numbers of the form
is:
Problem 26
Find the set of -values satisfying the inequality
. [The symbol
means
if
is positive,
if
is negative,
if
is zero. The notation
means that a can have any value between
and
, excluding
and
. ]
Problem 27
Let be the sum of the interior angles of a polygon
for which each interior angle is
times the
exterior angle at the same vertex. Then
Problem 28
The equation has:
Problem 29
Five times 's money added to
's money is more than $\texdollar{51.00}$ (Error compiling LaTeX. Unknown error_msg). Three times
's money minus
's money is
.
If
represents
's money in dollars and
represents
's money in dollars, then:
Problem 30
Given the line and a point on this line equidistant from the coordinate axes. Such a point exists in:
Problem 31
For to be a factor of
, the values of
and
must be, respectively:
Problem 32
In this figure the center of the circle is .
,
is a straight line,
, and
has a length twice the radius. Then:
Problem 33
You are given a sequence of terms; each term has the form
where
stands for the product
of all prime numbers less than or equal to
, and
takes, successively, the values
.
Let
be the number of primes appearing in this sequence. Then
is:
Problem 34
Two swimmers, at opposite ends of a -foot pool, start to swim the length of the pool,
one at the rate of
feet per second, the other at
feet per second.
They swim back and forth for
minutes. Allowing no loss of times at the turns, find the number of times they pass each other.
Problem 35
From point outside a circle, with a circumference of
units, a tangent is drawn.
Also from
a secant is drawn dividing the circle into unequal arcs with lengths
and
.
It is found that
, the length of the tangent, is the mean proportional between
and
.
If
and
are integers, then
may have the following number of values:
Problem 36
Let be the respective sums of
terms of the same arithmetic progression with
as the first term and
as the common difference. Let
. Then
is dependent on:
Problem 37
The base of a triangle is of length , and the latitude is of length
.
A rectangle of height
is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:
Problem 38
In this diagram and
are the equal sides of an isosceles
, in which is inscribed equilateral
.
Designate
by
,
by
, and
by
. Then:
Problem 39
To satisfy the equation ,
and
must be:
Problem 40
Given right with legs
. Find the length of the shorter angle trisector from
to the hypotenuse: