If all quantities produced rise by 10 percent, and all prices fall by 10 percent, which of the following occurs?
It helps to write out things as (simple but) proper equations. Denote production as $Y$ and prices as $P$

It is given that

$$Y_{t+1} = 1.1 Y_t \\
P_{t+1} = 0.9 P_t$$

Nominal GDP is given by $P_t Y_t$

$$ P_{t+1}Y_{t+1} = 1.1 Y_t \cdot 0.9 P_t  \\
= 1.1\cdot 0.9 \cdot Y_t P_t$$

So nominal GDP changed by $1.1 \cdot 0.9 = 0.99$ - it dropped by one percent. 

So which answer is it?

I assumed that the initial value is base value for the description of the relative changes. If, instead, we have that the larger value is base for the relative change, 

$$0.9 Y_{t+1} = Y_t \\
P_{t+1} = 0.9 P_t$$

we get that 
$$ P_{t+1}Y_{t+1} = \frac{1}{0.9} Y_t \cdot 0.9 P_t = Y_t P_t  $$


nominal GDP does not change.