1973 AHSME Problems
1973 AHSME (Answer Key) Printable versions: • 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 |
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 See also
Problem 1
A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length
Problem 2
One thousand unit cubes are fastened together to form a large cube with edge length 10 units; this is painted and then separated into the original cubes. The number of these unit cubes which have at least one face painted is
Problem 3
The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, the largest possible difference between the two primes is
Problem 4
Two congruent 30-60-90 are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is
Problem 5
Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean),
those which are always true are
Problem 6
If 554 is the base representation of the square of the number whose base representation is 24, then , when written in base 10, equals
Problem 7
The sum of all integers between 50 and 350 which end in 1 is
Problem 8
If 1 pint of paint is needed to paint a statue 6 ft. high, then the number of pints it will take to paint (to the same thickness) 540 statues similar to the original but only 1 ft. high is
Problem 9
In with right angle at , altitude and median trisect the right angle. If the area of is , then the area of is
Problem 10
If is a real number, then the simultaneous system
has no solution if and only if is equal to
Problem 11
A circle with a circumscribed and an inscribed square centered at the origin of a rectangular coordinate system with positive and axes and is shown in each figure to below.
The inequalities
are represented geometrically* by the figure numbered
* An inequality of the form , for all and is represented geometrically by a figure showing the containment
for a typical real number .
Problem 12
The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is
Problem 13
The fraction is equal to
Problem 14
Each valve , , and , when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves and open it takes 1.5 hours, and with only valves and open it takes 2 hours. The number of hours required with only valves and open is
Problem 15
A sector with acute central angle is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is
Problem 16
If the sum of all the angles except one of a convex polygon is , then the number of sides of the polygon must be
Problem 17
If is an acute angle and , then equals
Problem 18
If is a prime number, then divides without remainder
Problem 19
Define for and positive to be
where is the greatest integer for which . Then the quotient is equal to
Problem 20
A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is
Problem 21
The number of sets of two or more consecutive positive integers whose sum is 100 is
Problem 22
The set of all real solutions of the inequality
is
[Note: I updated the notation on this problem.]
Problem 23
There are two cards; one is red on both sides and the other is red on one side and blue on the other. The cards have the same probability (1/2) of being chosen, and one is chosen and placed on the table. If the upper side of the card on the table is red, then the probability that the under-side is also red is
Problem 24
The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to . The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to
Problem 25
A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is
Problem 26
The number of terms in an A.P. (Arithmetic Progression) is even. The sum of the odd and even-numbered terms are 24 and 30, respectively. If the last term exceeds the first by 10.5, the number of terms in the A.P. is
Problem 27
Cars A and B travel the same distance. Care A travels half that distance at miles per hour and half at miles per hour. Car B travels half the time at miles per hour and half at miles per hour. The average speed of Car A is miles per hour and that of Car B is miles per hour. Then we always have
Problem 28
If , , and are in geometric progression (G.P.) with and is an integer, then , , form a sequence
Problem 29
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first me at the point A again, then the number of times they meet, excluding the start and finish, is
Problem 30
Let denote the greatest integer where and . Then we have
Problem 31
In the following equation, each of the letters represents uniquely a different digit in base ten:
The sum equals
Problem 32
The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length is
Problem 33
When one ounce of water is added to a mixture of acid and water, the new mixture is acid. When one ounce of acid is added to the new mixture, the result is acid. The percentage of acid in the original mixture is
Problem 34
A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was
Problem 35
In the unit circle shown in the figure, chords and are parallel to the unit radius of the circle with center at . Chords , , and are each units long and chord is units long.
Of the three equations
those which are necessarily true are
See also
1973 AHSME (Problems • Answer Key • Resources) | ||
Preceded by First AHSME, see 1972 AHSC |
Followed by 1974 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 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.