1981 USAMO Problems

Problems from the 1981 USAMO.

Problem 1

Prove that if $n$ is not a multiple of $3$, then the angle $\frac{\pi}{n}$ 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, $A, B, C$ are such that the links between $AB, AC$ and $BC$ are all air, all bus or all train.


Problem 3

Show that for any triangle, $\frac{3\sqrt{3}}{2}\ge \sin(3A) + \sin(3B) + \sin (3C) \ge -2$.

When does the equality hold?


Problem 4

A convex polygon has $n$ sides. Each vertex is joined to a point $P$ not in the same plane. If $A, B, C$ are adjacent vertices of the polygon take the angle between the planes $PAB$ and $PBC$. The sum of the $n$ such angles equals the sum of the $n$ angles in the polygon. Show that $n=3$.


Problem 5

Show that for any positive real $x$, \[[nx]\ge \sum_{k=1}^{n}\left(\frac{[kx]}{k}\right),\] where $[x]$ denotes the greatest integer not exceeding $x$.


See Also

1981 USAMO (ProblemsResources)
Preceded by
1980 USAMO
Followed by
1982 USAMO
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All USAMO Problems and Solutions

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