1982 USAMO

Part of USAMO

Subcontests

(5)

1

1982 USAMO #5

A,B

C

are three interior points of a sphere

S

AB

AC

are perpendicular to the diameter of

S

A

, and so that two spheres can be constructed through

A

B

C

which are both tangent to

S

. Prove that the sum of their radii is equal to the radius of

S

1

1982 USAMO #4

Prove that there exists a positive integer

k

k\cdot2^n+1

is composite for every integer

n

1

1982 USAMO #3

A_1

is in the interior of an equilateral triangle

ABC

A_2

is in the interior of

\triangle{A_1BC}

\operatorname{I.Q.} (A_1BC) > \operatorname{I.Q.} (A_2BC),

where the isoperrimetric quotient of a figure

F

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

1

1982 USAMO #2

X_r=x^r+y^r+z^r

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)

(5,2)

. Determine all other pairs of integers

(m,n)

if any, so that

(*)

holds for all real numbers

x,y,z

x+y+z=0

1

1982 USAMO #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?