Difference between revisions of "1973 AHSME"

Latest revision as of 12:52, 20 February 2020

1973 AHSME problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

Entire Exam
Answer Key
- Problem 1
- Problem 2
- Problem 3
- Problem 4
- Problem 5
- Problem 6
- Problem 7
- Problem 8
- Problem 9
- Problem 10
- Problem 11
- Problem 12
- Problem 13
- Problem 14
- Problem 15
- Problem 16
- Problem 17
- Problem 18
- Problem 19
- Problem 20
- Problem 21
- Problem 22
- Problem 23
- Problem 24
- Problem 25
- Problem 26
- Problem 27
- Problem 28
- Problem 29
- Problem 30
- Problem 31
- Problem 32
- Problem 33
- Problem 34
- Problem 35

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

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Difference between revisions of "1973 AHSME"

Latest revision as of 12:52, 20 February 2020

See Also

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-       A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length
+'''1973 [[AHSME]]''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.
+*[[1973 AHSME Problems|Entire Exam]]
+*[[1973 AHSME/Answer Key|Answer Key]]
-	 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
+**[[1973 AHSME Problems/Problem 1|Problem 1]]
+**[[1973 AHSME Problems/Problem 2|Problem 2]]
+**[[1973 AHSME Problems/Problem 3|Problem 3]]
+**[[1973 AHSME Problems/Problem 4|Problem 4]]
+**[[1973 AHSME Problems/Problem 5|Problem 5]]
+**[[1973 AHSME Problems/Problem 6|Problem 6]]
+**[[1973 AHSME Problems/Problem 7|Problem 7]]
+**[[1973 AHSME Problems/Problem 8|Problem 8]]
+**[[1973 AHSME Problems/Problem 9|Problem 9]]
+**[[1973 AHSME Problems/Problem 10|Problem 10]]
+**[[1973 AHSME Problems/Problem 11|Problem 11]]
+**[[1973 AHSME Problems/Problem 12|Problem 12]]
+**[[1973 AHSME Problems/Problem 13|Problem 13]]
+**[[1973 AHSME Problems/Problem 14|Problem 14]]
+**[[1973 AHSME Problems/Problem 15|Problem 15]]
+**[[1973 AHSME Problems/Problem 16|Problem 16]]
+**[[1973 AHSME Problems/Problem 17|Problem 17]]
+**[[1973 AHSME Problems/Problem 18|Problem 18]]
+**[[1973 AHSME Problems/Problem 19|Problem 19]]
+**[[1973 AHSME Problems/Problem 20|Problem 20]]
+**[[1973 AHSME Problems/Problem 21|Problem 21]]
+**[[1973 AHSME Problems/Problem 22|Problem 22]]
+**[[1973 AHSME Problems/Problem 23|Problem 23]]
+**[[1973 AHSME Problems/Problem 24|Problem 24]]
+**[[1973 AHSME Problems/Problem 25|Problem 25]]
+**[[1973 AHSME Problems/Problem 26|Problem 26]]
+**[[1973 AHSME Problems/Problem 27|Problem 27]]
+**[[1973 AHSME Problems/Problem 28|Problem 28]]
+**[[1973 AHSME Problems/Problem 29|Problem 29]]
+**[[1973 AHSME Problems/Problem 30|Problem 30]]
+**[[1973 AHSME Problems/Problem 31|Problem 31]]
+**[[1973 AHSME Problems/Problem 32|Problem 32]]
+**[[1973 AHSME Problems/Problem 33|Problem 33]]
+**[[1973 AHSME Problems/Problem 34|Problem 34]]
+**[[1973 AHSME Problems/Problem 35|Problem 35]]
+==See Also==
+{{AHSME 30p box|year=1973|before=First AHSME, see [[1972 AHSME|1972 AHSC]]|after=[[1974 AHSME]]}}
-	 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
+* [[AHSME Problems and Solutions]]
+* [[AMC 12 Problems and Solutions]]
-	 Two congruent -- 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
+* [[Mathematics competitions]]
+* [[Mathematics competition resources]]
+* [[Math books]]
-	 Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean),
-I. Averaging is associative
-II. Averaging is commutative
-III. Averaging distributes over addition
-IV. Addition distributes over averaging
-V. Averaging has an identity element
-those which are always true are
-	 If 554 is the base  representation of the square of the number whose base  representation is 24, then , when written in base 10, equals
-	 The sum of all integers between 50 and 350 which end in 1 is
-	 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
-	 In  with right angle at , altitude  and median  trisect the right angle. If the area of  is , then the area of  is
-	 If  is a real number, then the simultaneous system
-has no solution if and only if  is equal to
-	 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 .
-	 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
-	 The fraction  is equal to
-	 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
-	 A sector with acute central angle  is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is
-	 If the sum of all the angles except one of a convex polygon is , then the number of sides of the polygon must be
-	 If  is an acute angle and , then  equals
-	 If  is a prime number, then  divides  without remainder
-	 Define  for  and  positive to be
-where  is the greatest integer for which . Then the quotient  is equal to
-	 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
-	 The number of sets of two or more consecutive positive integers whose sum is 100 is
-	 The set of all real solutions of the inequality
-is
-[Note: I updated the notation on this problem.]
-	 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
-	 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
-	 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
-	 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
-	 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
-	 If , , and  are in geometric progression (G.P.) with  and  is an integer, then , ,  form a sequence
-	 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
-	 Let  denote the greatest integer  where  and . Then we have
-	 In the following equation, each of the letters represents uniquely a different digit in base ten:
-The sum  equals
-	 The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length  is
-	 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
-	 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
-	 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