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Difference between revisions of "1981 USAMO Problems"

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==Problem 2==
 
==Problem 2==
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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, <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.
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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, <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.
  
 
[[1981 USAMO Problems/Problem 2 | Solution]]
 
[[1981 USAMO Problems/Problem 2 | Solution]]
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==Problem 4==
 
==Problem 4==
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A convex polygon has <math>n</math> sides. Each vertex is joined to a point <math>P</math> not in the same plane. If <math>A, B, C</math>  
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A convex polygon has <math>n</math> sides. Each vertex is joined to a point <math>P</math> not in the same plane. If <math>A, B, C</math> 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>.
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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  
 
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the <math>n</math> such angles equals the sum of the <math>n</math> angles in the polygon. Show that <math>n=3</math>
 
  
 
[[1981 USAMO Problems/Problem 4 | Solution]]
 
[[1981 USAMO Problems/Problem 4 | Solution]]
  
 
==Problem 5==
 
==Problem 5==
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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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Show that for any positive real <math>x</math>, <cmath>[nx]\ge \sum_{k=1}^{n}\left(\frac{[kx]}{k}\right),</cmath>
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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]]
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== See Also ==
 
== See Also ==
 
{{USAMO box|year=1981|before=[[1980 USAMO]]|after=[[1982 USAMO]]}}
 
{{USAMO box|year=1981|before=[[1980 USAMO]]|after=[[1982 USAMO]]}}
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{{MAA Notice}}

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.

Solution

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

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. AMC logo.png

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