Difference between revisions of "1982 USAMO Problems"

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==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}=\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>X_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 [i]all[/i] 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>x+y+z=0</math>.
  
 
[[1982 USAMO Problems/Problem 2 | Solution]]
 
[[1982 USAMO Problems/Problem 2 | Solution]]

Revision as of 11:32, 6 March 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 $X_r=x^r+y^r+z^r$ 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 [i]all[/i] other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$.

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
1 2 3 4 5
All USAMO Problems and Solutions
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