2023 SSMO Accuracy Round Problems/Problem 6

Problem

Let the roots of $P(x) = x^3 - 2023x^2 + 2023^{2023}$ be $\alpha, \beta, \gamma.$ . Find $\frac{\alpha^2 + \beta^2}{\alpha + \beta} + \frac{\beta^2 + \gamma^2}{\beta+\gamma} + \frac{\gamma^2 + \alpha^2}{\gamma + \alpha}$

Solution

By Vieta's Relation we get, $\sum_{cyc}{}\alpha=2023,$ $\sum_{cyc}{}\alpha\beta=0$ and $\prod_{cyc}{}\alpha=-2023^{2023}$ Therefore we have to find the value of $\sum_{cyc}{}\left(\frac{\alpha^2+\beta^2}{\alpha+\beta}\right)\implies$

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2023 SSMO Accuracy Round Problems/Problem 6

Problem

Solution