Difference between revisions of "1952 AHSME Problems"
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== Problem 46 == | == Problem 46 == | ||
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− | <math>\ | + | The base of a new rectangle equals the sum of the diagonal and the greater side of a given rectangle, while the altitude of the new rectangle equals the difference of the diagonal and the greater side of the given rectangle. The area of the new rectangle is: |
− | \ | + | |
− | \ | + | <math>\text{(A) greater than the area of the given rectangle} \quad\\ |
− | \ | + | \text{(B) equal to the area of the given rectangle} \quad\\ |
− | \ | + | \text{(C) equal to the area of a square with its side equal to the smaller side of the given rectangle} \quad\\ |
+ | \text{(D) equal to the area of a square with its side equal to the greater side of the given rectangle} \quad\\ | ||
+ | \text{(E) equal to the area of a rectangle whose dimensions are the diagonal and the shorter side of the given rectangle}</math> | ||
+ | |||
[[1952 AHSME Problems/Problem 46|Solution]] | [[1952 AHSME Problems/Problem 46|Solution]] |
Revision as of 16:24, 26 March 2017
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
If the radius of a circle is a rational number, its area is given by a number which is:
Problem 2
Two high school classes took the same test. One class of students made an average grade of ; the other class of students made an average grade of . The average grade for all students in both classes is:
Problem 3
The expression equals:
Problem 4
The cost of sending a parcel post package weighing pounds, an integer, is cents for the first pound and cents for each additional pound. The formula for the cost is:
Problem 5
The points and are connected by a straight line. Another point on this line is:
Problem 6
The difference of the roots of is:
Problem 7
When simplified, is equal to:
Problem 8
Two equal circles in the same plane cannot have the following number of common tangents.
Problem 9
If , then equals:
Problem 10
An automobile went up a hill at a speed of miles an hour and down the same distance at a speed of miles an hour. The average speed for the round trip was:
Problem 11
If , then it is incorrect to say:
Problem 12
The sum to infinity of the terms of an infinite geometric progression is . The sum of the first two terms is . The first term of the progression is:
Problem 13
The function with and greater than zero has its minimum value when:
Problem 14
A house and store were sold for each. The house was sold at a loss of of the cost, and the store at a gain of of the cost. The entire transaction resulted in:
Problem 15
The sides of a triangle are in the ratio . Then:
Problem 16
If the base of a rectangle is increased by and the area is unchanged, then the altitude is decreased by:
Problem 17
A merchant bought some goods at a discount of of the list price. He wants to mark them at such a price that he can give a discount of of the marked price and still make a profit of of the selling price.. The per cent of the list price at which he should mark them is:
Problem 18
only if:
Problem 19
Angle of triangle is trisected by and which meet at and respectively. Then:
Problem 20
If , then the incorrect expression in the following is:
Problem 21
The sides of a regular polygon of sides, , are extended to form a star. The number of degrees at each point of the star is:
Problem 22
On hypotenuse of a right triangle a second right triangle is constructed with hypotenuse . If , , and , then equals:
Problem 23
If has roots which are numerically equal but of opposite signs, the value of must be:
Problem 24
In the figure, it is given that angle ,,, , and . The area of quadrilateral is:
Problem 25
A powderman set a fuse for a blast to take place in seconds. He ran away at a rate of yards per second. Sound travels at the rate of feet per second. When the powderman heard the blast, he had run approximately:
Problem 26
If , then equals
Problem 27
The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is:
Problem 28
In the table shown, the formula relating and is:
Problem 29
In a circle of radius units, and are perpendicular diameters. A chord cutting cutting at is units long. The diameter is divided into two segments whose dimensions are:
Problem 30
When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:
Problem 31
Given points in a plane no three of which are collinear, the number of lines they determine is:
Problem 32
takes minutes less time than to travel a distance of miles. travels mile per hour faster than . If is 's rate of speed in miles per hours, then 's time for the distance is:
Problem 33
A circle and a square have the same perimeter. Then:
Problem 34
The price of an article was increased . Later the new price was decreased . If the last price was one dollar, the original price was:
Problem 35
With a rational denominator, the expression is equivalent to:
Problem 36
To be continuous at , the value of is taken to be:
Problem 37
Two equal parallel chords are drawn inches apart in a circle of radius inches. The area of that part of the circle that lies between the chords is:
Problem 38
The area of a trapezoidal field is square yards. Its altitude is yards. Find the two bases, if the number of yards in each base is an integer divisible by . The number of solutions to this problem is:
Problem 39
If the perimeter of a rectangle is and its diagonal is , the difference between the length and width of the rectangle is:
Problem 40
In order to draw a graph of , a table of values was constructed. These values of the function for a set of equally spaced increasing values of were , and . The one which is incorrect is:
Problem 41
Increasing the radius of a cylinder by units increased the volume by cubic units. Increasing the altitude of the cylinder by units also increases the volume by cubic units. If the original altitude is , then the original radius is:
Problem 42
Let represent a repeating decimal. If denotes the figures of which do not repeat themselves, and denotes the figures of which do repeat themselves, then the incorrect expression is:
Problem 43
The diameter of a circle is divided into equal parts. On each part a semicircle is constructed. As becomes very large, the sum of the lengths of the arcs of the semicircles approaches a length:
equal to the semi-circumference of the original circle equal to the diameter of the original circle greater than the diameter, but less than the semi-circumference of the original circle that is infinite greater than the semi-circumference
Problem 44
If an integer of two digits is times the sum of its digits, the number formed by interchanging the the digits is the sum of the digits multiplied by
Problem 45
If and are two unequal positive numbers, then:
Problem 46
The base of a new rectangle equals the sum of the diagonal and the greater side of a given rectangle, while the altitude of the new rectangle equals the difference of the diagonal and the greater side of the given rectangle. The area of the new rectangle is:
Problem 47
In the set of equations , the integral roots in the order are:
Problem 48
Two cyclists, miles apart, and starting at the same time, would be together in hours if they traveled in the same direction, but would pass each other in hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is:
Problem 49
In the figure, , and are one-third of their respective sides. It follows that , and similarly for lines BE and CF. Then the area of triangle is:
Problem 50
A line initially 1 inch long grows according to the following law, where the first term is the initial length.
If the growth process continues forever, the limit of the length of the line is:
See also
1952 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1951 AHSME |
Followed by 1953 AHSME | |
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 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.