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Geometry Difficulty 4.9 AIME Prove it Ukraine

In the coordinate plane xOyxOy, depict the set of all points such that y[x]=2y[x]|y - [x]| = 2 - y - [x], where [x][x] is the integer part of xx, i.e. the largest integer not exceeding xx.

Solution

Вихідне співвідношення рівносильне системі
{y=1,[x]=1;y2[x]. \begin{cases} y = 1, \\ [x] = 1; \\ y \le 2 - [x]. \end{cases}
Слід зобразити множину точок {(x;y):1x<2,y<1}{(x;1):x<2}\{(x; y) : 1 \le x < 2, y < 1\} \cup \{(x; 1) : x < 2\}.

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