Tour de France Tour de France 2023, stage 16: Passy - Combloux, 22.4k (ITT)

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they don't have the small enough gear selected on the spare bike because they know the push is allowed anyway - there is absolutely no reason for push to be necessary because of big gears on TT bike as you stated. plenty of range on the TT bike too.

If you want to be able to ride smaller gears, you need a second chainring on the bike, so yes, there is a reason.

Og course they can all just ride slower, to satisfy your weird obsession with people getting a push for 3 seconds.