Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 3.9 AMC 10/12 Find the answer China

Let $m$ be a real number. If the real and imaginary parts of complex $z = 1 + i + \frac{m}{1+i}$ , with $i$ being the imaginary unit, are greater than zero, then the range of $m$ is ______.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

After calculation, we get $z = 1 + i + \frac{(1-i)m}{2} = \frac{2+m}{2} + \frac{2-m}{2}i$ .

By the condition, it follows that $\frac{2+m}{2} > 0$ , $\frac{2-m}{2} > 0$ , and then we find the solution is $-2 < m < 2$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.