1954 AHSME Problems
1954 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 square of is:
Problem 2
The equation can be transformed by eliminating fractions to the equation . The roots of the latter equation are and . Then the roots of the first equation are:
Problem 3
If varies as the cube of , and varies as the fifth root of , then varies as the nth power of , where n is:
Problem 4
If the Highest Common Divisor of and is diminished by , it will equal:
Problem 5
A regular hexagon is inscribed in a circle of radius inches. Its area is:
Problem 6
The value of is:
Problem 7
A housewife saved in buying a dress on sale. If she spent for the dress, she saved about:
Problem 8
The base of a triangle is twice as long as a side of a square and their areas are the same. Then the ratio of the altitude of the triangle to the side of the square is:
Problem 9
A point is outside a circle and is inches from the center. A secant from cuts the circle at and so that the external segment of the secant is inches and is inches. The radius of the circle is:
Problem 10
The sum of the numerical coefficients in the expansion of the binomial is:
Problem 11
A merchant placed on display some dresses, each with a marked price. He then posted a sign “ off on these dresses.” The cost of the dresses was of the price at which he actually sold them. Then the ratio of the cost to the marked price was:
Problem 12
The solution of the equations
is:
Problem 13
A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, their sum will be:
Problem 14
When simplified equals:
Problem 15
equals:
Problem 16
If , then equals:
Problem 17
The graph of the function goes:
Problem 18
Of the following sets, the one that includes all values of which will satisfy is:
Problem 19
If the three points of contact of a circle inscribed in a triangle are joined, the angles of the resulting triangle:
Problem 20
The equation has:
Problem 21
The roots of the equation can be found by solving:
Problem 22
The expression cannot be evaluated for or , since division by zero is not allowed. For other values of :
Problem 23
If the margin made on an article costing dollars and selling for dollars is , then the margin is given by:
Problem 24
The values of for which the equation will have real and equal roots are:
Problem 25
The two roots of the equation are and:
Problem 26
The straight line is divided at so that . Circles are described on and as diameters and a common tangent meets produced at . Then equals:
Problem 27
A right circular cone has for its base a circle having the same radius as a given sphere. The volume of the cone is one-half that of the sphere. The ratio of the altitude of the cone to the radius of its base is:
Problem 28
If and , the value of is:
Problem 29
If the ratio of the legs of a right triangle is , then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:
Problem 30
and together can do a job in days; and can do it in four days; and and in days. The number of days required for A to do the job alone is:
Problem 31
In , , . Point is within the triangle with . The number of degrees in is:
Problem 32
The factors of are:
Problem 33
A bank charges for a loan of . The borrower receives and repays the loan in easy installments of a month. The interest rate is approximately:
Problem 34
The fraction :
Problem 35
In the right triangle shown the sum of the distances and is equal to the sum of the distances and . If , and , then equals:
Problem 36
A boat has a speed of mph in still water. In a stream that has a current of mph it travels a certain distance downstream and returns. The ratio of the average speed for the round trip to the speed in still water is:
Problem 37
Given with bisecting , extended to and a right angle, then:
Problem 38
If and , the value of when is approximately:
Problem 39
The locus of the midpoint of a line segment that is drawn from a given external point to a given circle with center and radius , is:
Problem 40
If , then equals:
Problem 41
The sum of all the roots of is:
Problem 42
Consider the graphs of and on the same set of axis. These parabolas are exactly the same shape. Then:
Problem 43
The hypotenuse of a right triangle is inches and the radius of the inscribed circle is inch. The perimeter of the triangle in inches is:
Problem 44
A man born in the first half of the nineteenth century was years old in the year . He was born in:
Problem 45
In a rhombus, , line segments are drawn within the rhombus, parallel to diagonal , and terminated in the sides of the rhombus. A graph is drawn showing the length of a segment as a function of its distance from vertex . The graph is:
Problem 46
In the diagram, if points and are points of tangency, then equals:
Problem 47
At the midpoint of line segment which is units long, a perpendicular is erected with length units. An arc is described from with a radius equal to , meeting at . Then and are the roots of:
Problem 48
A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at of its former rate and arrives hours late. Had the accident happened miles farther along the line, it would have arrived only hours late. The length of the trip in miles was:
Problem 49
The difference of the squares of two odd numbers is always divisible by . If , and and are the odd numbers, to prove the given statement we put the difference of the squares in the form:
Problem 50
The times between and o'clock, correct to the nearest minute, when the hands of a clock will form an angle of are:
See also
1954 AHSC (Problems • Answer Key • Resources) | ||
Preceded by 1953 AHSC |
Followed by 1955 AHSC | |
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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.