The Princess of Wales congratulated Arthur Fery on his run to the semi-finals and wished him a happy birthday.
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The prince is set to attend the elite boarding school from September, Kensington Palace has announced.
He will follow in the footsteps of his father, the Prince of Wales, who also attended the private school.
As the UK's head of state, King Charles has number of parliamentary and constitutional duties.
The photo shows William and Catherine lying in grass with their children, Prince George, Princess Charlotte and Prince Louis.
How the line of succession passes through generations of the Royal Family
A post shared on Kensington Palace's social media reads: "Wishing everyone a very Happy Christmas."
A new photograph of Prince George to mark his 12th birthday has been issued by Kensington Palace.
