Question

Vă rog mult, nu am nicio idee...

Answer 1

Răspuns:

Explicație pas cu pas:

$\frac{\overline{abc}}{bc} + \frac{\overline{bca}}{ac} + \frac{\overline{cab}}{ab} = 111*(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} )$

$\implies \frac{\overline{abc}}{bc} + \frac{\overline{bca}}{ac} + \frac{\overline{cab}}{ab} - 111*(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} ) = 0$

$\frac{a(100a+10b+c)}{abc} + \frac{b(100b+10c+a)}{abc} + \frac{c(100c+10a+b)}{abc} - \frac{111(bc + ac + ab)}{abc} = 0$

$\frac{a(100a+10b+c) + b(100b+10c+a) + c(100c+10a+b)}{abc} - \frac{111(bc + ac + ab)}{abc} = 0$

$100a^{2}+10ab+ac + 100b^{2}+10bc+ab + 100c^{2}+10ac+bc - 111bc - 111ac -111ab = 0$

$100a^{2} + 100b^{2} + 100c^{2} + 11bc + 11ac + 11ab - 111bc - 111ac -111ab = 0$

$100(a^{2} + b^{2} + c^{2}) - 100(bc + ac + ab) = 0$

$\iff 2(a^{2} + b^{2} + c^{2}) - 2(bc + ac + ab) = 0 \\ \iff (a - b)^{2} + (b - c)^{2} + (c - a)^{2} = 0$

este o sumă de pătrate => fiecare termen este 0

$a - b = 0 \iff a = b\\b - c = 0 \iff b = c\\c - a = 0 \iff c = a\\\implies a = b = c$

q.e.d.