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Ukrainian National Mathematical Olympiad

Algebra Difficulty 5.0 AIME, harder Find the answer Ukraine

Find the integer that is closest to the value of the expression:
$\left( (3 + \sqrt{1})^{2023} - \left( \frac{1}{3 - \sqrt{1}} \right)^{2023} \right) \cdot \left( (3 + \sqrt{2})^{2023} - \left( \frac{1}{3 - \sqrt{2}} \right)^{2023} \right) \cdot \left( (3 + \sqrt{3})^{2023} - \left( \frac{1}{3 - \sqrt{3}} \right)^{2023} \right) \dots \left( (3 + \sqrt{8})^{2023} - \left( \frac{1}{3 - \sqrt{8}} \right)^{2023} \right).$

Your answer

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

Solution

Let's consider the last factor:
$\frac{1}{3 - \sqrt{8}} = \frac{3 + \sqrt{8}}{9 - 8} = 3 + \sqrt{8} \Rightarrow (3 + \sqrt{8})^{2023} = \left(\frac{1}{3 - \sqrt{8}}\right)^{2023},$
which means that the last factor equals $0$ , and therefore the whole product equals $0$ .

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