Difference between revisions of "1982 USAMO Problems"

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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>.
 
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>.
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--[[User:Msinghal|Msinghal]] 15:05, 27 April 2013 (EDT)
  
 
[[1982 USAMO Problems/Problem 2 | Solution]]
 
[[1982 USAMO Problems/Problem 2 | Solution]]

Revision as of 15:05, 27 April 2013

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 $S_1=0$.

--Msinghal 15:05, 27 April 2013 (EDT)

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