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Irrational numbers - Colegiul Național din Iași (Iași - Roumanie)

Titre du sujet: 
Irrational numbers
Let d > 1 be an integer. Determine all numbers of the form a+b√d with a et b integers such that their inverses are of the same form (a+b√d with a et b integers). For instance, for d = 2, 1+ √2 has this property, since: (-1+√2)(1+√2)=1. One can assume d is squarefree (d ≠ 0 and 1, and d is not divisible by the square of any prime; d can be 2, 3, 5, 6,..., but not 8 or 18).
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Sujet | by Dr. Radut