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Difference between revisions of "1982 USAMO Problems"

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Problems from the '''1982 [[United States of America Mathematical Olympiad | USAMO]]'''.
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==Problem 1==
 
==Problem 1==
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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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In a party with <math>1982</math> persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?
  
 
[[1982 USAMO Problems/Problem 1 | Solution]]
 
[[1982 USAMO Problems/Problem 1 | Solution]]
  
 
==Problem 2==
 
==Problem 2==
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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}=\left(\frac{x^m + y^m + z^m}{m}\right) \left(\dfrac{x^n + y^n + z^n}{n}}\right)</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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Let <math>S_r=x^r+y^r+z^r</math> with <math>x,y,z</math> real. It is known that if <math>S_1=0</math>,
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<math>(*) </math>   <math>\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}</math>
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for <math>(m,n)=(2,3),(3,2),(2,5)</math>, or <math>(5,2)</math>. Determine ''all'' other pairs of integers <math>(m,n)</math> if any, so that <math>(*)</math> holds for all real numbers <math>x,y,z</math> such that <math>S_1=0</math>.
  
 
[[1982 USAMO Problems/Problem 2 | Solution]]
 
[[1982 USAMO Problems/Problem 2 | Solution]]
  
 
==Problem 3==
 
==Problem 3==
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<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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If a point <math>A_1</math> is in the interior of an equilateral triangle <math>ABC</math> and point <math>A_2</math> is in the interior of <math>\triangle{A_1BC}</math>, prove that
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<math>I.Q. (A_1BC) > I.Q.(A_2BC)</math>,
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where the ''isoperimetric quotient'' of a figure <math>F</math> is defined by
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<math>I.Q.(F) = \frac{\text{Area (F)}}{\text{[Perimeter (F)]}^2}</math>
  
 
[[1982 USAMO Problems/Problem 3 | Solution]]
 
[[1982 USAMO Problems/Problem 3 | Solution]]
  
 
==Problem 4==
 
==Problem 4==
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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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Prove that there exists a positive integer <math>k</math> such that <math>k\cdot2^n+1</math> is composite for every positive integer <math>n</math>.
  
 
[[1982 USAMO Problems/Problem 4 | Solution]]
 
[[1982 USAMO Problems/Problem 4 | Solution]]
  
 
==Problem 5==
 
==Problem 5==
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<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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<math>A,B</math>, and <math>C</math> are three interior points of a sphere <math>S</math> such that <math>AB</math> and <math>AC</math> are perpendicular to the diameter of <math>S</math> through <math>A</math>, and so that two spheres can be constructed through <math>A</math>, <math>B</math>, and <math>C</math> which are both tangent to <math>S</math>. Prove that the sum of their radii is equal to the radius of <math>S</math>.
  
 
[[1982 USAMO Problems/Problem 5 | Solution]]
 
[[1982 USAMO Problems/Problem 5 | Solution]]
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== See Also ==
 
== See Also ==
 
{{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}}
 
{{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 21:36, 19 March 2020

Problems from the 1982 USAMO.

Problem 1

In a party with $1982$ persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?

Solution

Problem 2

Let $S_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$,

$(*)$ $\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}$

for $(m,n)=(2,3),(3,2),(2,5)$, or $(5,2)$. Determine all other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $S_1=0$.

Solution

Problem 3

If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that

$I.Q. (A_1BC) > I.Q.(A_2BC)$,

where the isoperimetric quotient of a figure $F$ is defined by

$I.Q.(F) = \frac{\text{Area (F)}}{\text{[Perimeter (F)]}^2}$

Solution

Problem 4

Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every positive integer $n$.

Solution

Problem 5

$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.

Solution

See Also

1982 USAMO (Problems • Resources)
Preceded by
1981 USAMO 		Followed by
1983 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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