Opposites of hackneyed are new, fresh, original...
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The opposite of tired (fatigued) would be energetic or enthusastic. The opposite of tired (starting the day) would be "rested." The opposite of tired (old, hackneyed) would be fresh, or innovative.
The opposite (antonym) of opposite is same.
opposite of opposite to opposite from
If the opposite is meant to be the additive opposite and not the multiplicative opposite, then their sum is zero. The reason is that is what defines an additive opposite!
It depends on what you mean by an opposite. Every real number has an additive opposite but 0 does not have a multiplicative opposite.