Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME, harder Prove it Ukraine

On the coordinate space XOYXOY draw the set of points, whose coordinates satisfy the following equality:
1−x2+1−y2=2−x2−y2. \sqrt{1-x^2} + \sqrt{1-y^2} = 2 - x^2 - y^2.

Solution

Since 0≤(1−x2)≤10 \leq (1 - x^2) \leq 1, 0≤(1−y2)≤10 \leq (1 - y^2) \leq 1, we get that

1−x2+1−y2≥\sqrt{1-x^2} + \sqrt{1-y^2} \geq (1-x^2) + (1-y^2), \text{} and so the equality holds if }
{1−x2=1−x21−y2=1−y2\begin{cases} \sqrt{1-x^2} = 1-x^2 \\ \sqrt{1-y^2} = 1-y^2 \end{cases} ⇒\Rightarrow
{x2∈{0;1}y2∈{0;1}\begin{cases} x^2 \in \{0;1\} \\ y^2 \in \{0;1\} \end{cases}.
\text{} Thus we get the points from the answer.}

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