Cabanga ngabantu ababili \(A\) kanye \(B\) abangenalo usuku lokuzalwa olufanayo, lapho \(A\) emncane kuno \(B\) . Bonisa: Kukhona imilaza yeminyaka emibili ncamashi \(a,b \in \mathbb{N}\) lapho lokhu okulandelayo kusebenza khona: \(2\cdot a = b\) . Siqale sibeke \(d \in \mathbb{R}^+\) njengomehluko weminyaka phakathi kuka \(A\) kanye \(B\) ekuzalweni kuka \(A\) nge \( d = d_0 + d_1 \) , \( d_0 \in \mathbb{N}_0, d_1 \in \mathbb{R}, d_1 \in [0;1[\) . Manje sicabangela isikhathi esinqunyiwe \(x \in \mathbb{R}^+\) ngemva kokuzalwa kuka \(A\) no- \(x = x_0 + x_1\) , \(x_0 \in \mathbb{N}_0, x_1 \in \mathbb{R}, x_1 \in [0;1[\) .
Ngalesi sikhathi ngesikhathi, ngencazelo, \(a = \lfloor x \rfloor \) kanye \(b = \lfloor x+d \rfloor\) . Manje sinquma konke \(x\) okuphethe:
$$2 \lfloor x \rfloor = \lfloor x+d \rfloor \Leftrightarrow 2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor$$
Icala lokuqala: \(0 \leq x_1 + d_1 < 1\):
Ngemuva kwalokho i- $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 \Leftrightarrow x_0 = d_0.$$
Lokhu kusho ukuthi $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + x_1 \rfloor = d_0$$ futhi $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ lokuqala lobudala $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ .
Icala lesibili: \( 1 \leq x_1 + d_1 < 2 \):
Ngemuva kwalokho i- $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 + 1 \Leftrightarrow x_0 = d_0 + 1.$$
Lokhu kusho ukuthi $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + 1 + x_1 \rfloor = d_0 + 1$$ futhi $$b = \lfloor x+d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + 1 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor = 2 d_0 + 2$$ lesibili leminyaka eliyifunayo.
Ngokwezinto ezibonakalayo, lokhu kusho, isibonelo: Uma umama wakho ekuzala uneminyaka \(20\) , umdala ngokuphindwe kabili kunawe ku \(40\) kanye \(42\) . Indaba yokuthi noma nini umama wakho \(n\) unezikhathi ezidala futhi iyathakazelisa: Lapha usetha \(n \in \mathbb{N}\) ngokungenasisekelo futhi uthole \(x_0 = \frac{d_0}{n-1} \in \mathbb{N} \Leftrightarrow d_0 = k (n-1)\) . Lokhu kusebenza uma futhi kuphela uma umehluko wenombolo yobudala \( \lfloor d \rfloor = d_0 \) uyimpinda ye- \(n-1\) . Isibonelo, esimweni esingenhla, umama wakho \(24\) uneminyaka engu \(6\) ephindwe ngokweminyaka yakho.