Difference between revisions of "1957 AHSME Problems"
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1. If two angles of a triangle are not equal, the triangle is not isosceles. | 1. If two angles of a triangle are not equal, the triangle is not isosceles. | ||
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2. The base angles of an isosceles triangle are equal. | 2. The base angles of an isosceles triangle are equal. | ||
+ | |||
3. If a triangle is not isosceles, then two of its angles are not equal. | 3. If a triangle is not isosceles, then two of its angles are not equal. | ||
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4. A necessary condition that two angles of a triangle be equal is that the triangle be isosceles. | 4. A necessary condition that two angles of a triangle be equal is that the triangle be isosceles. | ||
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<math>\textbf{(A)}\ \text{the center of the inscribed circle} \qquad \\ | <math>\textbf{(A)}\ \text{the center of the inscribed circle} \qquad \\ | ||
\textbf{(B)}\ \text{the center of the circumscribed circle}\qquad\\ | \textbf{(B)}\ \text{the center of the circumscribed circle}\qquad\\ | ||
− | \textbf{(C)}\ \text{such that the three angles | + | \textbf{(C)}\ \text{such that the three angles formed at the point each be }{120^\circ}\qquad\\ |
\textbf{(D)}\ \text{the intersection of the altitudes of the triangle}\qquad\\ | \textbf{(D)}\ \text{the intersection of the altitudes of the triangle}\qquad\\ | ||
\textbf{(E)}\ \text{the intersection of the medians of the triangle} </math> | \textbf{(E)}\ \text{the intersection of the medians of the triangle} </math> | ||
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Given the system of equations | Given the system of equations | ||
− | <cmath>ax + (a - 1)y = 1 | + | <cmath>ax + (a - 1)y = 1</cmath> |
+ | <cmath>(a + 1)x - ay = 1</cmath> | ||
For which one of the following values of <math>a</math> is there no solution <math>x</math> and <math>y</math>? | For which one of the following values of <math>a</math> is there no solution <math>x</math> and <math>y</math>? | ||
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If two real numbers <math>x</math> and <math>y</math> satisfy the equation <math>\frac{x}{y} = x - y</math>, then: | If two real numbers <math>x</math> and <math>y</math> satisfy the equation <math>\frac{x}{y} = x - y</math>, then: | ||
− | <math>\textbf{(A)}\ {x \ge 4}\text{ | + | <math>\textbf{(A)}\ {x \ge 4}\text{ or }{x \le 0}\qquad \\ |
\textbf{(B)}\ {y}\text{ can equal }{1}\qquad \\ | \textbf{(B)}\ {y}\text{ can equal }{1}\qquad \\ | ||
\textbf{(C)}\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad\\ | \textbf{(C)}\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad\\ | ||
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* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
− | {{AHSME 50p box|year=1957|before=[[1956 AHSME|1956 | + | {{AHSME 50p box|year=1957|before=[[1956 AHSME|1956 AHSME]]|after=[[1958 AHSME|1958 AHSME]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 11:55, 27 July 2024
1957 AHSC (Answer Key) Printable version: | 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 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 |
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
- 41 Problem 41
- 42 Problem 42
- 43 Problem 43
- 44 Problem 44
- 45 Problem 45
- 46 Problem 46
- 47 Problem 47
- 48 Problem 48
- 49 Problem 49
- 50 Problem 50
- 51 See also
Problem 1
The number of distinct lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is:
Problem 2
In the equation , the sum of the roots is
and the product of the roots is
.
Then
and
have the values, respectively:
Problem 3
The simplest form of is:
Problem 4
The first step in finding the product by use of the distributive property in the form
is:
Problem 5
Through the use of theorems on logarithms
can be reduced to:
Problem 6
An open box is constructed by starting with a rectangular sheet of metal in. by
in.
and cutting a square of side
inches from each corner. The resulting projections are folded up and the seams welded.
The volume of the resulting box is:
Problem 7
The area of a circle inscribed in an equilateral triangle is . The perimeter of this triangle is:
Problem 8
The numbers are proportional to
. The sum of
, and
is
. The number y is given by the equation
. Then a is:
Problem 9
The value of when
and
is:
Problem 10
The graph of has its:
Problem 11
The angle formed by the hands of a clock at is:
Problem 12
Comparing the numbers and
we may say:
Problem 13
A rational number between and
is:
Problem 14
If , then
is:
Problem 15
The table below shows the distance in feet a ball rolls down an inclined plane in
seconds.
The distance for
is:
Problem 16
Goldfish are sold at cents each. The rectangular coordinate graph showing the cost of
to
goldfish is:
Problem 17
A cube is made by soldering twelve -inch lengths of wire properly at the vertices of the cube.
If a fly alights at one of the vertices and then walks along the edges, the greatest distance it
could travel before coming to any vertex a second time, without retracing any distance, is:
Problem 18
Circle has diameters
and
perpendicular to each other.
is any chord intersecting
at
.
Then
is equal to:
Problem 19
The base of the decimal number system is ten, meaning, for example, that .
In the binary system, which has base two, the first five positive integers are
.
The numeral
in the binary system would then be written in the decimal system as:
Problem 20
A man makes a trip by automobile at an average speed of 50 mph. He returns over the same route at an average speed of mph.
His average speed for the entire trip is:
Problem 21
Start with the theorem "If two angles of a triangle are equal, the triangle is isosceles," and the following four statements:
1. If two angles of a triangle are not equal, the triangle is not isosceles.
2. The base angles of an isosceles triangle are equal.
3. If a triangle is not isosceles, then two of its angles are not equal.
4. A necessary condition that two angles of a triangle be equal is that the triangle be isosceles.
Which combination of statements contains only those which are logically equivalent to the given theorem?
Problem 22
If , then
equals:
Problem 23
The graph of and the graph of
meet in two points. The distance between these two points is:
Problem 24
If the square of a number of two digits is decreased by the square of the number formed by reversing the digits, then the result is not always divisible by:
Problem 25
The vertices of have coordinates as follows:
, where
and
are positive.
The origin and point
lie on opposite sides of
. The area of
may be found from the expression:
Problem 26
From a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be:
Problem 27
The sum of the reciprocals of the roots of the equation is:
Problem 28
If and
are positive and
, then the value of
is:
Problem 29
The relation is true only for:
Problem 30
The sum of the squares of the first n positive integers is given by the expression ,
if
and
are, respectively:
Problem 31
A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square. If the square has sides of one unit, the leg of each of the triangles has length:
Problem 32
The largest of the following integers which divides each of the numbers of the sequence is:
Problem 33
If , then the value of
is:
Problem 34
The points that satisfy the system , constitute the following set:
Problem 35
Side of right triangle
is divided into
equal parts. Seven line segments parallel to
are drawn to
from the points of division.
If
, then the sum of the lengths of the seven line segments:
Problem 36
If , then the largest value of
is:
Problem 37
In right triangle , and
;
is on
. If
, one-half the perimeter of rectangle
, then:
Problem 38
From a two-digit number we subtract the number with the digits reversed and find that the result is a positive perfect cube. Then:
Problem 39
Two men set out at the same time to walk towards each other from and
,
miles apart.
The first man walks at the rate of
mph. The second man walks
miles the first hour,
miles the second hour,
miles the third hour, and so on in arithmetic progression. Then the men will meet:
Problem 40
If the parabola has its vertex on the
-axis, then
must be:
Problem 41
Given the system of equations
For which one of the following values of
is there no solution
and
?
Problem 42
If , where
and
is an integer, then the total number of possible distinct values for
is:
Problem 43
We define a lattice point as a point whose coordinates are integers, zero admitted.
Then the number of lattice points on the boundary and inside the region bounded by the -axis,
the line
, and the parabola
is:
Problem 44
In and
. Then
is:
Problem 45
If two real numbers and
satisfy the equation
, then:
Problem 46
Two perpendicular chords intersect in a circle. The segments of one chord are and
;
the segments of the other are
and
. Then the diameter of the circle is:
Problem 47
In circle , the midpoint of radius
is
; at
,
. The semi-circle with
as diameter intersects
in
. Line
intersects circle
in
, and line
intersects circle
in
. Line
is drawn. Then, if the radius of circle
is
,
is:
Problem 48
Let be an equilateral triangle inscribed in circle
.
is a point on arc
.
Lines
,
, and
are drawn. Then
is:
Problem 49
The parallel sides of a trapezoid are and
. The non-parallel sides are
and
.
A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters.
The ratio in which each of the non-parallel sides is divided is:
Problem 50
In circle ,
is a moving point on diameter
.
is drawn perpendicular to
and equal to
.
is drawn perpendicular to
, on the same side of diameter
as
,
and equal to
. Let
be the midpoint of
. Then, as
moves from
to
, point
:
See also
1957 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1956 AHSME |
Followed by 1958 AHSME | |
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All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.