Difference between revisions of "1978 USAMO Problems"
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| + | Problems from the '''1978 [[United States of America Mathematical Olympiad | USAMO]]'''. | ||
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==Problem 1== | ==Problem 1== | ||
| + | The sum of 5 real numbers is 8 and the sum of their squares is 16. What is the largest possible value for one of the numbers? | ||
| − | + | [[1978 USAMO Problems/Problem 1 | Solution]] | |
| − | |||
==Problem 2== | ==Problem 2== | ||
| − | Two square maps cover exactly the same area of terrain on different scales. The smaller map is | + | Two square maps cover exactly the same area of terrain on different scales. The smaller map is placed on top of the larger map and inside its borders. Show that there is a unique point on the top map which lies exactly above the corresponding point on the lower map. How can this point be constructed? |
| − | placed on top of the larger map and inside its borders. Show that there is a unique point on the top | + | |
| − | map which lies exactly above the corresponding point on the lower map. How can this point be | + | [[1978 USAMO Problems/Problem 2 | Solution]] |
| − | constructed? | ||
==Problem 3== | ==Problem 3== | ||
| − | You are told that all integers from <math>33</math> to <math>73</math> inclusive can be expressed as a sum of positive | + | You are told that all integers from <math>33</math> to <math>73</math> inclusive can be expressed as a sum of positive integers whose reciprocals sum to 1. Show that the same is true for all integers greater than <math>73</math>. |
| − | integers whose reciprocals sum to 1. Show that the same is true for all integers greater than <math>73</math>. | + | |
| + | [[1978 USAMO Problems/Problem 3 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
Show that if the angle between each pair of faces of a tetrahedron is equal, then the tetrahedron is regular. Does a tetrahedron have to be regular if five of the angles are equal? | Show that if the angle between each pair of faces of a tetrahedron is equal, then the tetrahedron is regular. Does a tetrahedron have to be regular if five of the angles are equal? | ||
| + | |||
| + | [[1978 USAMO Problems/Problem 4 | Solution]] | ||
==Problem 5== | ==Problem 5== | ||
There are 9 delegates at a conference, each speaking at most three languages. Given any three delegates, at least 2 speak a common language. Show that there are three delegates with a common language. | There are 9 delegates at a conference, each speaking at most three languages. Given any three delegates, at least 2 speak a common language. Show that there are three delegates with a common language. | ||
| + | |||
| + | [[1978 USAMO Problems/Problem 5 | Solution]] | ||
| + | |||
| + | == See Also == | ||
| + | {{USAMO box|year=1978|before=[[1977 USAMO]]|after=[[1979 USAMO]]}} | ||
Revision as of 14:33, 17 September 2012
Problems from the 1978 USAMO.
Problem 1
The sum of 5 real numbers is 8 and the sum of their squares is 16. What is the largest possible value for one of the numbers?
Problem 2
Two square maps cover exactly the same area of terrain on different scales. The smaller map is placed on top of the larger map and inside its borders. Show that there is a unique point on the top map which lies exactly above the corresponding point on the lower map. How can this point be constructed?
Problem 3
You are told that all integers from 
to 
inclusive can be expressed as a sum of positive integers whose reciprocals sum to 1. Show that the same is true for all integers greater than .
Problem 4
Show that if the angle between each pair of faces of a tetrahedron is equal, then the tetrahedron is regular. Does a tetrahedron have to be regular if five of the angles are equal?
Problem 5
There are 9 delegates at a conference, each speaking at most three languages. Given any three delegates, at least 2 speak a common language. Show that there are three delegates with a common language.
See Also
| 1978 USAMO (Problems • Resources) | ||
| Preceded by 1977 USAMO |
Followed by 1979 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
