Difference between revisions of "1982 USAMO Problems"

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==Problem 1==
 
==Problem 1==
 
A graph has <math>1982</math> points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to <math>1981</math> points?
 
A graph has <math>1982</math> points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to <math>1981</math> points?
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[[1982 USAMO Problems/Problem 1 | Solution]]
  
 
==Problem 2==
 
==Problem 2==
 
Show that if <math>m, n</math> are positive integers such that <math>\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{(m+n)} =\frac{ (x^m + y^m + z^m)}{\frac{m \left(x^n + y^n + z^n\right)}{n}}</math> for all real <math>x, y, z</math> with sum <math>0</math>, then <math>(m, n) = (2, 3) </math> or <math>(2, 5)</math>.
 
Show that if <math>m, n</math> are positive integers such that <math>\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{(m+n)} =\frac{ (x^m + y^m + z^m)}{\frac{m \left(x^n + y^n + z^n\right)}{n}}</math> for all real <math>x, y, z</math> with sum <math>0</math>, then <math>(m, n) = (2, 3) </math> or <math>(2, 5)</math>.
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[[1982 USAMO Problems/Problem 2 | Solution]]
  
 
==Problem 3==
 
==Problem 3==
 
<math>D</math> is a point inside the equilateral triangle <math>ABC</math>. <math>E</math> is a point inside <math>DBC</math>. Show that <math>\frac{\text{area}DBC}{\text{perimeter} DBC^2} > \frac{\text{area} EBC}{\text{perimeter} EBC^2}.</math>
 
<math>D</math> is a point inside the equilateral triangle <math>ABC</math>. <math>E</math> is a point inside <math>DBC</math>. Show that <math>\frac{\text{area}DBC}{\text{perimeter} DBC^2} > \frac{\text{area} EBC}{\text{perimeter} EBC^2}.</math>
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[[1982 USAMO Problems/Problem 3 | Solution]]
  
 
==Problem 4==
 
==Problem 4==
 
Show that there is a positive integer <math>k</math> such that, for every positive integer <math>n</math>, <math>k 2^n+1</math> is composite.
 
Show that there is a positive integer <math>k</math> such that, for every positive integer <math>n</math>, <math>k 2^n+1</math> is composite.
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[[1982 USAMO Problems/Problem 4 | Solution]]
  
 
==Problem 5==
 
==Problem 5==
 
<math>O</math> is the center of a sphere <math>S</math>. Points <math>A, B, C</math> are inside <math>S</math>, <math>OA</math> is perpendicular to <math>AB</math> and <math>AC</math>, and there are two spheres through <math>A, B</math>, and <math>C</math> which touch <math>S</math>.  Show that the sum of their radii equals the radius of <math>S</math>.
 
<math>O</math> is the center of a sphere <math>S</math>. Points <math>A, B, C</math> are inside <math>S</math>, <math>OA</math> is perpendicular to <math>AB</math> and <math>AC</math>, and there are two spheres through <math>A, B</math>, and <math>C</math> which touch <math>S</math>.  Show that the sum of their radii equals the radius of <math>S</math>.
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[[1982 USAMO Problems/Problem 5 | Solution]]
  
 
== See Also ==
 
== See Also ==
 
{{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}}
 
{{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}}

Revision as of 14:38, 17 September 2012

Problem 1

A graph has $1982$ points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to $1981$ points?

Solution

Problem 2

Show that if $m, n$ are positive integers such that $\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{(m+n)} =\frac{ (x^m + y^m + z^m)}{\frac{m \left(x^n + y^n + z^n\right)}{n}}$ for all real $x, y, z$ with sum $0$, then $(m, n) = (2, 3)$ or $(2, 5)$.

Solution

Problem 3

$D$ is a point inside the equilateral triangle $ABC$. $E$ is a point inside $DBC$. Show that $\frac{\text{area}DBC}{\text{perimeter} DBC^2} > \frac{\text{area} EBC}{\text{perimeter} EBC^2}.$

Solution

Problem 4

Show that there is a positive integer $k$ such that, for every positive integer $n$, $k 2^n+1$ is composite.

Solution

Problem 5

$O$ is the center of a sphere $S$. Points $A, B, C$ are inside $S$, $OA$ is perpendicular to $AB$ and $AC$, and there are two spheres through $A, B$, and $C$ which touch $S$. Show that the sum of their radii equals the radius of $S$.

Solution

See Also

1982 USAMO (ProblemsResources)
Preceded by
1981 USAMO
Followed by
1983 USAMO
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All USAMO Problems and Solutions
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