Random variable for market share

I want to probabilistically represent the following situation: There are three competitors in a market, and between them, they share 100% of the market.

So to reflect this, I want three random variables X, Y and Z that are:

1. uniformly distributed between 0% and 100%; and also that
2. sum to 100%.

Condition 1 is easy - I just define three uniformly distributed (between 0% and 100%) and independent random variables X, Y and Z.

P(x < X) = x
P(y < Y) = y
P(z < Z) = z

Then for condition 2 (to get them to sum to 100%), I take my uniform X, Y and Z, and then define X', Y' and Z' as:

X' = X / (X + Y + Z)
Y' = Y / (X + Y + Z)
Z' = Z / (X + Y + Z)

So I now have three random variables which satisfy condition 2, but they no longer satisfy condition 1. (I can prove that X' is not uniformly distributed.)

Is there any way to define X', Y' and Z' so that they satisfy both conditions? Either using X, Y and Z or some other way?