# Pālua ka makahiki o ʻelua mau kānaka1017

E noʻonoʻo i nā poʻe ʻelua $$A$$ a me $$B$$ i hānau ʻole ʻia ma ka lā like a ʻo $$A$$ kaikaina ma mua o $$B$$ . Hōʻike aia aia ʻelua mau makahiki o nā hōkū $$a,b \in \mathbb{N}$$ , no nā mea e pili ana: $$2\cdot a = b$$ . Ua hoʻonohonoho mua mākou iā $$d \in \mathbb{R}^+$$ ma ke ʻano he ʻokoʻa makahiki ma waena o $$A$$ a me $$B$$ i ka hānau ʻana o $$A$$ me $$d = d_0 + d_1$$ , $$d_0 \in \mathbb{N}_0, d_1 \in \mathbb{R}, d_1 \in [0;1[$$ . E noʻonoʻo mākou i kahi manawa kūpono i ka manawa $$x \in \mathbb{R}^+$$ ma hope o ka hānau ʻana o $$A$$ me $$x = x_0 + x_1$$ , $$x_0 \in \mathbb{N}_0, x_1 \in \mathbb{R}, x_1 \in [0;1[$$ .

I kēia manawa i ka manawa, e like me ka wehewehe ʻana, $$a = \lfloor x \rfloor$$ a me $$b = \lfloor x+d \rfloor$$ . Hoʻoholo mākou i kēia manawa $$x$$ no nā mea paʻa:

$$2 \lfloor x \rfloor = \lfloor x+d \rfloor \Leftrightarrow 2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor$$

Hihia 1st: $$0 \leq x_1 + d_1 < 1$$:

A laila $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 \Leftrightarrow x_0 = d_0.$$

ʻO kēia ka manaʻo ʻo $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + x_1 \rfloor = d_0$$ a me $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ ka $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ makahiki mua a $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ .

2 hihia: $$1 \leq x_1 + d_1 < 2$$:

A laila $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 + 1 \Leftrightarrow x_0 = d_0 + 1.$$

ʻO kēia ka manaʻo ʻo $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + 1 + x_1 \rfloor = d_0 + 1$$ a me $$b = \lfloor x+d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + 1 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor = 2 d_0 + 2$$ ka ʻelua mau makahiki i makemake ʻia.

ʻO ke kikoʻī, ke ʻano o kēia, no ka laʻana: Inā hānau kou makuahine iā ʻoe i ka makahiki o $$20$$ mau makahiki, ʻelua mau manawa ʻo ia i kou makahiki i $$40$$ a me $$42$$ mau makahiki. Hoihoi nō hoʻi inā a ʻoiai ʻo ia ʻo $$n$$ manawa i ʻelemakule ai: Eia $$n \in \mathbb{N}$$ a loaʻa mākou $$x_0 = \frac{d_0}{n-1} \in \mathbb{N} \Leftrightarrow d_0 = k (n-1)$$ . Hana pololei kēia i ka manawa o ka ʻokoʻa o ka makahiki integer $$\lfloor d \rfloor = d_0$$ maha o $$n-1$$ , e laʻa me nā mea i luna aʻe nei ʻo kou makuahine he $$24$$ mau makahiki $$6$$ mau manawa i kou mau makahiki.

Hope