1957 AHSME Problems
1957 AHSC (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
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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 AHSC |
Followed by 1958 AHSC | |
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.