What is the greatest possible number of rays in space emanating from a single point and forming obtuse angles with each other

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Answer: 4. We will consider vectors instead of rays. We can assume that the first vector has coordinates (1, $0,0)$. Then the other vectors have coordinates ( $x_{i}, y_{i}, z_{i}$ ), where $x_{i}0$, and thus, $y_{\mathrm{i}}2$ the inequality $x_{\mathrm{i}} x_{\mathrm{j}}+y_{\mathrm{i}} y_{\mathrm{j}}+z_{\mathrm{i}} z_{\mathrm{j}}0$ and $y_{\mathrm{i}} y_{\mathrm{j}}>0$ holds, so $z_{\mathrm{i}} z_{\mathrm{j}}<0$, i.e., the numbers $z_{\mathrm{i}}$ and $z_{\mathrm{j}}$ have different signs. There cannot be more than two such numbers.