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OMB 2014 Finale MAXI Question 3
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On considère le système d'équations suivant, d'inconnues entières et :




et sont deux paramètres entiers.

(a) Montrer que le système n'a pas de solution si et sont de parités différentes.

(b) Trouver un couple avec et distincts mais de même parité pour lequel le système n'admet pas de solution.

(c) Montrer que le système possède au plus une solution pour tout couple tel que .



Solution(s) proposée(s) :


 
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a)x est de même parité que x^2 ainsi que pour y et y^2 donc x^2+y a la même parité que y^2+x donc à et b doivent être de même parité.
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le système devient: x²=a-y ; y²=b-x
x²-y²=a-y -(b-x)-->(x-y)(x+y)=(x-y)+(a-b)
-->(x-y)(x+y-1)=(a-b)
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