Difference between revisions of "1981 USAMO Problems"

Revision as of 13:35, 17 September 2012

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 trian 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 $x$ , $[nx]\ge \sum_{1}^{n}\left(\frac{[kx]}{k}\right)$

@@ Line 20: / Line 20: @@
 ==Problem 5==
 Show that for any positive real <math>x</math>, <math>[nx]\ge \sum_{1}^{n}\left(\frac{[kx]}{k}\right)</math>
+== See Also ==
+{{USAMO box|year=1981|before=[[1980 USAMO]]|after=[[1982 USAMO]]}}

1981 USAMO (Problems • Resources)
Preceded by 1980 USAMO	Followed by 1982 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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