Difference between revisions of "1981 USAMO Problems"

Latest revision as of 08:09, 22 October 2022

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.

Solution

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?

Solution

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 $n=3$ .

Solution

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 $x$ .

Solution

@@ Line 24: / Line 24: @@
 ==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>, <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>.
 [[1981 USAMO Problems/Problem 5 | Solution]]

1981 USAMO (Problems • Resources)
Preceded by 1980 USAMO	Followed by 1982 USAMO
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Difference between revisions of "1981 USAMO Problems"

Latest revision as of 08:09, 22 October 2022

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See Also