Difference between revisions of "1981 USAMO Problems"
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==Problem 5== | ==Problem 5== | ||
− | Show that for any positive real <math>x</math>, < | + | Show that for any positive real <math>x</math>, <cmath>[nx]\ge \sum_{k=1}^{n}\left(\frac{[kx]}{k}\right),</cmath> |
where <math>[x]</math> denotes the greatest integer not exceeding <math>x</math> | where <math>[x]</math> denotes the greatest integer not exceeding <math>x</math> | ||
[[1981 USAMO Problems/Problem 5 | Solution]] | [[1981 USAMO Problems/Problem 5 | Solution]] |
Revision as of 16:28, 11 August 2018
Problems from the 1981 USAMO.
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 train 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 , where denotes the greatest integer not exceeding Solution
See Also
1981 USAMO (Problems • Resources) | ||
Preceded by 1980 USAMO |
Followed by 1982 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.