Difference between revisions of "1973 AHSME Problems"
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| + | ==Problem 1== | ||
| + | |||
1 A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length | 1 A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length | ||
<math> \textbf{(A)}\ 3\sqrt3\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 6\sqrt3\qquad\textbf{(D)}\ 12\sqrt3\qquad\textbf{(E)}\ \text{ none of these} </math> | <math> \textbf{(A)}\ 3\sqrt3\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 6\sqrt3\qquad\textbf{(D)}\ 12\sqrt3\qquad\textbf{(E)}\ \text{ none of these} </math> | ||
| + | [[1973 AHSME Problems/Problem 1|Solution]] | ||
| + | ==Problem 2== | ||
| + | |||
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 | 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 | ||
<math> \textbf{(A)}\ 600\qquad\textbf{(B)}\ 520\qquad\textbf{(C)}\ 488\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 400 </math> | <math> \textbf{(A)}\ 600\qquad\textbf{(B)}\ 520\qquad\textbf{(C)}\ 488\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 400 </math> | ||
| + | [[1973 AHSME Problems/Problem 2|Solution]] | ||
| + | ==Problem 3== | ||
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 | 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 | ||
<math> \textbf{(A)}\ 112\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 88\qquad\textbf{(E)}\ 80 </math> | <math> \textbf{(A)}\ 112\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 88\qquad\textbf{(E)}\ 80 </math> | ||
| − | + | ||
| + | [[1973 AHSME Problems/Problem 3|Solution]] | ||
| + | ==Problem 4== | ||
4 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 | 4 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 | ||
<math> \textbf{(A)}\ 6\sqrt3\qquad\textbf{(B)}\ 8\sqrt3\qquad\textbf{(C)}\ 9\sqrt3\qquad\textbf{(D)}\ 12\sqrt3\qquad\textbf{(E)}\ 24 </math> | <math> \textbf{(A)}\ 6\sqrt3\qquad\textbf{(B)}\ 8\sqrt3\qquad\textbf{(C)}\ 9\sqrt3\qquad\textbf{(D)}\ 12\sqrt3\qquad\textbf{(E)}\ 24 </math> | ||
| + | [[1973 AHSME Problems/Problem 4|Solution]] | ||
| + | ==Problem 5== | ||
| + | |||
5 Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean), | 5 Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean), | ||
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<math> \textbf{(A)}\ \text{All}\qquad\textbf{(B)}\ \text{I and II only}\qquad\textbf{(C)}\ \text{II and III only}\qquad\textbf{(D)}\ \text{II and IV only}\qquad\textbf{(E)}\ \text{II and V only} </math> | <math> \textbf{(A)}\ \text{All}\qquad\textbf{(B)}\ \text{I and II only}\qquad\textbf{(C)}\ \text{II and III only}\qquad\textbf{(D)}\ \text{II and IV only}\qquad\textbf{(E)}\ \text{II and V only} </math> | ||
| − | + | [[1973 AHSME Problems/Problem 5|Solution]] | |
| + | ==Problem 6== | ||
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 | 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 | ||
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<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 16 </math> | <math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 16 </math> | ||
| − | + | [[1973 AHSME Problems/Problem 6|Solution]] | |
| + | ==Problem 7== | ||
7 The sum of all integers between 50 and 350 which end in 1 is | 7 The sum of all integers between 50 and 350 which end in 1 is | ||
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<math> \textbf{(A)}\ 5880\qquad\textbf{(B)}\ 5539\qquad\textbf{(C)}\ 5208\qquad\textbf{(D)}\ 4877\qquad\textbf{(E)}\ 4566 </math> | <math> \textbf{(A)}\ 5880\qquad\textbf{(B)}\ 5539\qquad\textbf{(C)}\ 5208\qquad\textbf{(D)}\ 4877\qquad\textbf{(E)}\ 4566 </math> | ||
| + | [[1973 AHSME Problems/Problem 7|Solution]] | ||
| + | ==Problem 8== | ||
| + | |||
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 | 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 | ||
<math> \textbf{(A)}\ 90\qquad\textbf{(B)}\ 72\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 15 </math> | <math> \textbf{(A)}\ 90\qquad\textbf{(B)}\ 72\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 15 </math> | ||
| − | + | [[1973 AHSME Problems/Problem 8|Solution]] | |
| + | ==Problem 9== | ||
| + | |||
9 In with right angle at , altitude and median trisect the right angle. If the area of is , then the area of is | 9 In with right angle at , altitude and median trisect the right angle. If the area of is , then the area of is | ||
<math> \textbf{(A)}\ 6K\qquad\textbf{(B)}\ 4\sqrt3\ K\qquad\textbf{(C)}\ 3\sqrt3\ K\qquad\textbf{(D)}\ 3K\qquad\textbf{(E)}\ 4K </math> | <math> \textbf{(A)}\ 6K\qquad\textbf{(B)}\ 4\sqrt3\ K\qquad\textbf{(C)}\ 3\sqrt3\ K\qquad\textbf{(D)}\ 3K\qquad\textbf{(E)}\ 4K </math> | ||
| + | [[1973 AHSME Problems/Problem 9|Solution]] | ||
| + | ==Problem 10== | ||
| + | |||
10 If is a real number, then the simultaneous system | 10 If is a real number, then the simultaneous system | ||
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<math> \textbf{(A)}\ -1\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 0\text{ or }1\qquad\textbf{(E)}\ \frac{1}2 </math> | <math> \textbf{(A)}\ -1\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 0\text{ or }1\qquad\textbf{(E)}\ \frac{1}2 </math> | ||
| + | [[1973 AHSME Problems/Problem 10|Solution]] | ||
| + | ==Problem 11== | ||
| + | |||
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. | 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. | ||
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for a typical real number . | for a typical real number . | ||
| + | [[1973 AHSME Problems/Problem 11|Solution]] | ||
| + | ==Problem 12== | ||
| + | |||
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 | 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 | ||
<math> \textbf{(A)}\ 3: 2\qquad\textbf{(B)}\ 3: 1\qquad\textbf{(C)}\ 2: 3\qquad\textbf{(D)}\ 2: 1\qquad\textbf{(E)}\ 1: 2 </math> | <math> \textbf{(A)}\ 3: 2\qquad\textbf{(B)}\ 3: 1\qquad\textbf{(C)}\ 2: 3\qquad\textbf{(D)}\ 2: 1\qquad\textbf{(E)}\ 1: 2 </math> | ||
| − | + | [[1973 AHSME Problems/Problem 12|Solution]] | |
| + | ==Problem 13== | ||
| + | |||
13 The fraction is equal to | 13 The fraction is equal to | ||
<math> \frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}} </math> | <math> \frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}} </math> | ||
| + | [[1973 AHSME Problems/Problem 13|Solution]] | ||
| + | ==Problem 14== | ||
| + | |||
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 | 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 | ||
<math> \textbf{(A)}\ 1.1\qquad\textbf{(B)}\ 1.15\qquad\textbf{(C)}\ 1.2\qquad\textbf{(D)}\ 1.25\qquad\textbf{(E)}\ 1.75 </math> | <math> \textbf{(A)}\ 1.1\qquad\textbf{(B)}\ 1.15\qquad\textbf{(C)}\ 1.2\qquad\textbf{(D)}\ 1.25\qquad\textbf{(E)}\ 1.75 </math> | ||
| + | [[1973 AHSME Problems/Problem 14|Solution]] | ||
| + | ==Problem 15== | ||
| + | |||
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 | 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 | ||
| + | [[1973 AHSME Problems/Problem 15|Solution]] | ||
| + | ==Problem 16== | ||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 16|Solution]] | |
| + | ==Problem 17== | ||
17 If is an acute angle and , then equals | 17 If is an acute angle and , then equals | ||
| + | [[1973 AHSME Problems/Problem 17|Solution]] | ||
| + | ==Problem 18== | ||
18 If is a prime number, then divides without remainder | 18 If is a prime number, then divides without remainder | ||
| − | + | [[1973 AHSME Problems/Problem 18|Solution]] | |
| + | ==Problem 19== | ||
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where is the greatest integer for which . Then the quotient is equal to | where is the greatest integer for which . Then the quotient is equal to | ||
| − | + | [[1973 AHSME Problems/Problem 19|Solution]] | |
| + | ==Problem 20== | ||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 20|Solution]] | |
| + | ==Problem 21== | ||
21 The number of sets of two or more consecutive positive integers whose sum is 100 is | 21 The number of sets of two or more consecutive positive integers whose sum is 100 is | ||
| − | + | [[1973 AHSME Problems/Problem 21|Solution]] | |
| + | ==Problem 22== | ||
22 The set of all real solutions of the inequality | 22 The set of all real solutions of the inequality | ||
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[Note: I updated the notation on this problem.] | [Note: I updated the notation on this problem.] | ||
| + | [[1973 AHSME Problems/Problem 22|Solution]] | ||
| + | ==Problem 23== | ||
| + | |||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 23|Solution]] | |
| + | ==Problem 24== | ||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 24|Solution]] | |
| + | ==Problem 25== | ||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 25|Solution]] | |
| + | ==Problem 26== | ||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 26|Solution]] | |
| + | ==Problem 27== | ||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 27|Solution]] | |
| + | ==Problem 28== | ||
28 If , , and are in geometric progression (G.P.) with and is an integer, then , , form a sequence | 28 If , , and are in geometric progression (G.P.) with and is an integer, then , , form a sequence | ||
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| − | + | [[1973 AHSME Problems/Problem 28|Solution]] | |
| + | ==Problem 29== | ||
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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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 29|Solution]] | |
| + | ==Problem 30== | ||
30 Let denote the greatest integer where and . Then we have | 30 Let denote the greatest integer where and . Then we have | ||
| − | + | [[1973 AHSME Problems/Problem 30|Solution]] | |
| − | + | ||
| − | + | ==Problem 31== | |
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The sum equals | The sum equals | ||
| − | + | [[1973 AHSME Problems/Problem 31|Solution]] | |
| + | ==Problem 32== | ||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 32|Solution]] | |
| + | ==Problem 33== | ||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 33|Solution]] | |
| + | ==Problem 34== | ||
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 | 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 | ||
| − | + | [[1973 AHSME Problems/Problem 34|Solution]] | |
| + | ==Problem 35== | ||
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. | 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. | ||
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Of the three equations | Of the three equations | ||
those which are necessarily true are | those which are necessarily true are | ||
| + | |||
| + | [[1973 AHSME Problems/Problem 35|Solution]] | ||
| + | |||
| + | ==See Also== | ||
| + | * [[American Invitational Mathematics Examination]] | ||
| + | * [[AIME Problems and Solutions]] | ||
| + | * [[Mathematics competition resources]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 14:42, 27 September 2014
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
1 A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length
Problem 2
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
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
4 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
Problem 5
5 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
Problem 6
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
7 The sum of all integers between 50 and 350 which end in 1 is
Problem 8
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
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
10 If is a real number, then the simultaneous system
has no solution if and only if is equal to
Problem 11
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.
![[asy] size((400)); draw((0,0)--(22,0), EndArrow); draw((10,-10)--(10,12), EndArrow); draw((25,0)--(47,0), EndArrow); draw((35,-10)--(35,12), EndArrow); draw((-25,0)--(-3,0), EndArrow); draw((-15,-10)--(-15,12), EndArrow); draw((-50,0)--(-28,0), EndArrow); draw((-40,-10)--(-40,12), EndArrow); draw(Circle((-40,0),6)); draw(Circle((-15,0),6)); draw(Circle((10,0),6)); draw(Circle((35,0),6)); draw((-34,0)--(-40,6)--(-46,0)--(-40,-6)--(-34,0)--(-34,6)--(-46,6)--(-46,-6)--(-34,-6)--cycle); draw((-6.5,0)--(-15,8.5)--(-23.5,0)--(-15,-8.5)--cycle); draw((-10.8,4.2)--(-19.2,4.2)--(-19.2,-4.2)--(-10.8,-4.2)--cycle); draw((14.2,4.2)--(5.8,4.2)--(5.8,-4.2)--(14.2,-4.2)--cycle); draw((16,6)--(4,6)--(4,-6)--(16,-6)--cycle); draw((41,0)--(35,6)--(29,0)--(35,-6)--cycle); draw((43.5,0)--(35,8.5)--(26.5,0)--(35,-8.5)--cycle); label("I", (-49,9)); label("II", (-24,9)); label("III", (1,9)); label("IV", (26,9)); label("X", (-28,0), S); label("X", (-3,0), S); label("X", (22,0), S); label("X", (47,0), S); label("Y", (-40,12), E); label("Y", (-15,12), E); label("Y", (10,12), E); label("Y", (35,12), E);[/asy]](http://latex.artofproblemsolving.com/d/4/1/d414580296d8ae8eb80ac3c416e1f59836c40113.png)
The inequalities
![\[|x|+|y|\leq\sqrt{2(x^{2}+y^{2})}\leq 2\mbox{Max}(|x|, |y|)\]](http://latex.artofproblemsolving.com/2/3/6/236c6813feb8c51840f249e3b66e496d069c4402.png)
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
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
13 The fraction is equal to
Problem 14
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
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
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
17 If is an acute angle and , then equals
Problem 18
18 If is a prime number, then divides without remainder
Problem 19
19 Define for and positive to be
where is the greatest integer for which . Then the quotient is equal to
Problem 20
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
21 The number of sets of two or more consecutive positive integers whose sum is 100 is
Problem 22
22 The set of all real solutions of the inequality
is
[Note: I updated the notation on this problem.]
Problem 23
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
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
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
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
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
28 If , , and are in geometric progression (G.P.) with and is an integer, then , , form a sequence
Problem 29
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
30 Let denote the greatest integer where and . Then we have
Problem 31
31 In the following equation, each of the letters represents uniquely a different digit in base ten:
The sum equals
Problem 32
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
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
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
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
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
