Difference between revisions of "1981 USAMO Problems"
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==Problem 1== | ==Problem 1== | ||
Prove that if <math>n</math> is not a multiple of <math>3</math>, then the angle <math>\frac{\pi}{n}</math> can be trisected with ruler and compasses. | Prove that if <math>n</math> is not a multiple of <math>3</math>, then the angle <math>\frac{\pi}{n}</math> can be trisected with ruler and compasses. | ||
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| + | [[1981 USAMO Problems/Problem 1 | Solution]] | ||
==Problem 2== | ==Problem 2== | ||
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least one pair by train. No town has an air link, a bus link and a trian link. No three towns, <math>A, B, C </math> | least one pair by train. No town has an air link, a bus link and a trian link. No three towns, <math>A, B, C </math> | ||
are such that the links between <math>AB, AC</math> and <math>BC</math> are all air, all bus or all train. | are such that the links between <math>AB, AC</math> and <math>BC</math> are all air, all bus or all train. | ||
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| + | [[1981 USAMO Problems/Problem 1 | Solution]] | ||
==Problem 3== | ==Problem 3== | ||
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When does the equality hold? | When does the equality hold? | ||
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| + | [[1981 USAMO Problems/Problem 1 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
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are adjacent vertices of the polygon take the angle between the planes <math>PAB</math> and <math>PBC</math>. The sum of | are adjacent vertices of the polygon take the angle between the planes <math>PAB</math> and <math>PBC</math>. The sum of | ||
the <math>n</math> such angles equals the sum of the <math>n</math> angles in the polygon. Show that <math>n=3</math> | the <math>n</math> such angles equals the sum of the <math>n</math> angles in the polygon. Show that <math>n=3</math> | ||
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| + | [[1981 USAMO Problems/Problem 1 | Solution]] | ||
==Problem 5== | ==Problem 5== | ||
Show that for any positive real <math>x</math>, <math>[nx]\ge \sum_{1}^{n}\left(\frac{[kx]}{k}\right)</math> | Show that for any positive real <math>x</math>, <math>[nx]\ge \sum_{1}^{n}\left(\frac{[kx]}{k}\right)</math> | ||
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| + | [[1981 USAMO Problems/Problem 1 | Solution]] | ||
== See Also == | == See Also == | ||
{{USAMO box|year=1981|before=[[1980 USAMO]]|after=[[1982 USAMO]]}} | {{USAMO box|year=1981|before=[[1980 USAMO]]|after=[[1982 USAMO]]}} | ||
Revision as of 13:38, 17 September 2012
Problem 1
Prove that if 
is not a multiple of 
, then the angle 
can be trisected with ruler and compasses.
Problem 2
What is the largest number of towns that can meet the following criteria. Each pair is directly
linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at
least one pair by train. No town has an air link, a bus link and a trian link. No three towns, 
are such that the links between 
and 
are all air, all bus or all train.
Problem 3
Show that for any triangle, 
.
When does the equality hold?
Problem 4
A convex polygon has sides. Each vertex is joined to a point 
not in the same plane. If
are adjacent vertices of the polygon take the angle between the planes 
and 
. The sum of
the such angles equals the sum of the angles in the polygon. Show that 
Problem 5
Show that for any positive real 
, ![$[nx]\ge \sum_{1}^{n}\left(\frac{[kx]}{k}\right)$](http://latex.artofproblemsolving.com/8/9/0/890ce23466abfd1bcb2eb253ac571925c0362163.png)
See Also
| 1981 USAMO (Problems • Resources) | ||
| Preceded by 1980 USAMO |
Followed by 1982 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
