1982 USAMO Problems

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 all 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

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

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

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

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See Also