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Ukrainian Mathematical Olympiad

Geometry Difficulty 4.9 AIME Prove it Ukraine

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

Solution

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

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