Sometimes it's great to include multicollinear terms!

Suppose than an exact relationship exists between Y and X1 and X2.
The R^2 is 1.00, by definition.

However, the correlation between X1 and X2 is 0.94. From what the
book tells me, we should be careful to neglect one of these variables
because it's so highly correlated with another variable. The model
doesn't benefit from the added independent variable. Using
multicorrelated variables has the tendency to increase F scores, or
something to that effect.

However, another part of the book says to test the R^2 using adjusted
R^2.

Suppose that Y is a function of X and X^2. The R^2 is 1.00, but the
correlation between R and R^2 is 0.94.

What are your opinions/thoughts?

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