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Question Number 155144 by mnjuly1970 last updated on 26/Sep/21

    how many integer solutions       are there :      ⌊ (1/(2+(√x)))⌋ = 3         ■

$$ \\ $$$$\:\:{how}\:{many}\:{integer}\:{solutions}\: \\ $$$$\:\:\:\:{are}\:{there}\:: \\ $$$$\:\:\:\:\lfloor\:\frac{\mathrm{1}}{\mathrm{2}+\sqrt{{x}}}\rfloor\:=\:\mathrm{3}\:\:\:\:\:\:\:\:\:\blacksquare \\ $$$$ \\ $$

Commented by mr W last updated on 26/Sep/21

no integer solution and even  no real solution!  since x≥0 ⇒2+(√x)≥2  ⇒0<(1/(2+(√x)))≤(1/2)  ⇒⌊(1/(2+(√x)))⌋=0 ≠3

$${no}\:{integer}\:{solution}\:{and}\:{even} \\ $$$${no}\:{real}\:{solution}! \\ $$$${since}\:{x}\geqslant\mathrm{0}\:\Rightarrow\mathrm{2}+\sqrt{{x}}\geqslant\mathrm{2} \\ $$$$\Rightarrow\mathrm{0}<\frac{\mathrm{1}}{\mathrm{2}+\sqrt{{x}}}\leqslant\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow\lfloor\frac{\mathrm{1}}{\mathrm{2}+\sqrt{{x}}}\rfloor=\mathrm{0}\:\neq\mathrm{3} \\ $$

Commented by mnjuly1970 last updated on 26/Sep/21

grateful...mr W

$${grateful}...{mr}\:{W} \\ $$

Commented by talminator2856791 last updated on 26/Sep/21

    is the question supposed to be     ⌊(1/(2−(√x)))⌋ = 3      ?

$$\: \\ $$$$\:\mathrm{is}\:\mathrm{the}\:\mathrm{question}\:\mathrm{supposed}\:\mathrm{to}\:\mathrm{be} \\ $$$$\:\:\:\lfloor\frac{\mathrm{1}}{\mathrm{2}−\sqrt{{x}}}\rfloor\:=\:\mathrm{3} \\ $$$$\:\:\:\:? \\ $$$$\: \\ $$

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