1991 USAMO

Part of USAMO

Subcontests

(4)

1

Interior tracing

\, D \,

be an arbitrary point on side

\, AB \,

of a given triangle

\, ABC, \,

\, E \,

be the interior point where

\, CD \,

intersects the external common tangent to the incircles of triangles

\, ACD \,

\, BCD

\, D \,

assumes all positions between

\, A \,

\, B \,

, prove that the point

\, E \,

traces the arc of a circle.

1

Sums

For any nonempty set

\,S\,

of numbers, let

\,\sigma(S)\,

\,\pi(S)\,

denote the sum and product, respectively, of the elements of

\,S\,

\sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1),

\Sigma

'' denotes a sum involving all nonempty subsets

S

\{1,2,3, \ldots,n\}

1

Minimum perimeter of a triangle

\, ABC, \,

\,A\,

\,B,\,

\,C\,

is obtuse, and the three side lengths

\,a,b,c\,

are integers. Determine, with proof, the minimum possible perimeter.

1

Bernoulli's inequality

a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n}

m

n

are positive integers. Prove that

a^m + a^n \geq m^m + n^n