MathDB

A5

Part of 1984 Putnam

Problems(1)

Putnam 1984/A5

Source:

11/21/2006
Putnam 1984/A5) Let RR be the region consisting of all triples (x,y,z)(x,y,z) of nonnegative real numbers satisfying x+y+z≤1x+y+z\leq 1. Let w=1−x−y−zw=1-x-y-z. Express the value of the triple integral ∭Rxy9z8w4 dx dy dz\iiint_{R}xy^{9}z^{8}w^{4}\ dx\ dy\ dz in the form a!b!c!d!/n!a!b!c!d!/n! where a,b,c,da,b,c,d and nn are positive integers. [hide="A solution"]\iiint_{R}xy^{9}z^{8}w^{4}\ dx dy dz = 4\iiint_{R}\int_{0}^{1-x-y-z}xy^{9}z^{8}w^{3}\ dw dx dy dz = 4\iiiint_{Q}xy^{9}z^{8}w^{3}\ dw dx dy dz
where Q={(x,y,z,w)∈R4∣ x,y,z,w≥0,x+y+z+w≤1}Q=\left\{ (x,y,z,w)\in\mathbb{R}^{4}|\ x,y,z,w\geq 0, x+y+z+w\leq 1\right\}, which is a Dirichlet integral giving
4\iiiint_{Q}x^{1}y^{9}z^{8}w^{3}\ dw dx dy dz = 4\cdot\frac{1!9!8!3!}{(2+10+9+4)!}= \frac{1!9!8!4!}{25!}
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