Difference between revisions of "1955 AHSME Problems"
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If <math>a, b</math>, and <math>c</math> are positive integers, the radicals <math>\sqrt{a+\frac{b}{c}}</math> and <math>a\sqrt{\frac{b}{c}}</math> are equal when and only when: | If <math>a, b</math>, and <math>c</math> are positive integers, the radicals <math>\sqrt{a+\frac{b}{c}}</math> and <math>a\sqrt{\frac{b}{c}}</math> are equal when and only when: | ||
− | <math> \textbf{(A)}\ a=b=c=1\qquad\textbf{(B)}\ a=b\text{ and }c=a=1\qquad\textbf{(C)}\ c=\frac{b(a^2-1)}{ | + | <math> \textbf{(A)}\ a=b=c=1\qquad\textbf{(B)}\ a=b\text{ and }c=a=1\qquad\textbf{(C)}\ c=\frac{b(a^2-1)}{a}\\ \textbf{(D)}\ a=b\text{ and }c\text{ is any value}\qquad\textbf{(E)}\ a=b\text{ and }c=a-1 </math> |
[[1955 AHSME Problems/Problem 42|Solution]] | [[1955 AHSME Problems/Problem 42|Solution]] | ||
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== Problem 43== | == Problem 43== | ||
Revision as of 07:24, 16 June 2016
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
Which one of the following is not equivalent to ?
Problem 2
The smaller angle between the hands of a clock at p.m. is:
Problem 3
If each number in a set of ten numbers is increased by , the arithmetic mean (average) of the original ten numbers:
Problem 4
The equality is satisfied by:
Problem 5
varies inversely as the square of . When . When equals:
Problem 6
A merchant buys a number of oranges at for cents and an equal number at for cents. To "break even" he must sell all at:
Problem 7
If a worker receives a % cut in wages, he may regain his original pay exactly by obtaining a raise of:
Problem 8
The graph of :
Problem 9
A circle is inscribed in a triangle with sides , and . The radius of the circle is:
Problem 10
How many hours does it take a train traveling at an average rate of 40 mph between stops to travel a miles it makes n stops of m minutes each?
Problem 11
The negation of the statement "No slow learners attend this school" is:
Problem 12
The solution of is:
Problem 13
The fraction is equal to:
Problem 14
The length of rectangle is % more than the side of square . The width of the rectangle is % less than the side of the square. The ratio of the areas, , is:
Problem 15
The ratio of the areas of two concentric circles is . If the radius of the smaller is , then the difference between the radii is best approximated by:
Problem 16
The value of when and is:
Problem 17
If , then equals:
Problem 18
The discriminant of the equation is zero. Hence, its roots are:
Problem 19
Two numbers whose sum is and the absolute value of whose difference is are roots of the equation:
Problem 20
The expression equals zero for:
Problem 21
Represent the hypotenuse of a right triangle by and the area by . The altitude on the hypotenuse is:
Problem 22
On a \texdollar{10000} order a merchant has a choice between three successive discounts of %, %, and % and three successive discounts of %, %, and %. By choosing the better offer, he can save:
Problem 23
In checking the petty cash a clerk counts quarters, dimes, nickels, and cents. Later he discovers that of the nickels were counted as quarters and of the dimes were counted as cents. To correct the total obtained the clerk must:
Problem 24
The function :
Problem 25
One of the factors of is:
Problem 26
Mr. A owns a house worth \textdollar{10000}. He sells it to Mr. at % profit. Mr. sells the house back to Mr. at a % loss. Then:
Problem 27
If and are the roots of , then equals:
Problem 28
On the same set of axes are drawn the graph of and the graph of the equation obtained by replacing by in the given equation. If and these two graphs intersect:
Problem 29
In the figure, is tangent to semicircle ; is tangent to semicircle ; is a straight line; the arcs are indicated in the figure. is measured by:
Problem 30
Each of the equations has:
Problem 31
An equilateral triangle whose side is is divided into a triangle and a trapezoid by a line drawn parallel to one of its sides. If the area of the trapezoid equals one-half of the area of the original triangle, the length of the median of the trapezoid is:
Problem 32
If the discriminant of is zero, then another true statement about , and is that:
Problem 33
Henry starts a trip when the hands of the clock are together between a.m. and a.m. He arrives at his destination between p.m. and p.m. when the hands of the clock are exactly apart. The trip takes:
Problem 34
A -inch and -inch diameter pole are placed together and bound together with wire. The length of the shortest wire that will go around them is:
Problem 35
Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles:
Problem 36
A cylindrical oil tank, lying horizontally, has an interior length of feet and an interior diameter of feet. If the rectangular surface of the oil has an area of square feet, the depth of the oil is:
Problem 37
A three-digit number has, from left to right, the digits , and , with . When the number with the digits reversed is subtracted from the original number, the units' digit in the difference of r. The next two digits, from right to left, are:
Problem 38
Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus the numbers , and are obtained. One of the original integers is:
Problem 39
If , then if the least possible value of is zero is equal to:
Problem 40
The fractions and are unequal if:
Problem 41
A train traveling from Aytown to Beetown meets with an accident after hr. It is stopped for hr., after which it proceeds at four-fifths of its usual rate, arriving at Beetown hr. late. If the train had covered miles more before the accident, it would have been just hr. late. The usual rate of the train is:
Problem 42
If , and are positive integers, the radicals and are equal when and only when:
Problem 43
The pairs of values of and that are the common solutions of the equations and are:
Problem 44
In circle chord is produced so that equals a radius of the circle. is drawn and extended to . is drawn. Which of the following expresses the relationship between and ?
Problem 45
Given a geometric sequence with the first term and and an arithmetic sequence with the first term . A third sequence is formed by adding corresponding terms of the two given sequences. The sum of the first ten terms of the third sequence is:
Problem 46
The graphs of , and intersect in:
Problem 47
The expressions and are:
Problem 48
Given with medians ; parallel and equal to ; are drawn; extended meets in . Which one of the following statements is not necessarily correct?
Problem 49
The graphs of and intersect in:
Problem 50
In order to pass going mph on a two-lane highway, , going mph, must gain feet. Meantime, feet from , is headed toward him at mph. If and maintain their speeds, then, in order to pass safely, must increase his speed by:
See also
1955 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1954 AHSME |
Followed by 1956 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.