The top right corner says that that the winner also used a Royal typewriter, but I haven't found the Royal-produced postcard celebrating her triumph.
I could ask if anyone out there knows who won (tell me here, please, if you do), but a more reasonable question would be: where can I go to find out?
The postcard collectors might know, but I'm going to start by searching in the French newspapers from 1922, available online via Gallica -I'll report back if I get news.
With the Autopsies Research group I share a love of dead objects such as typewriters (see e.g. here), and had cause to investigate the identity of a Royal typewriter when posting about Rohmer's film Triple Agent (see here). For that I had help from Richard Polt and his highly informative Classic Typewriter Page.
For typewriters in films, go to Kevin Ferguson's brilliant tumblr of Typewriters in Films.
These are my favourites from among the posts so far (because they also have maps in them, of course - I yam what I yam):