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Can anyone solve this?This is a 2015 AMTI Question

Note by Saran Vijai
11 months ago

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Using the first equation and AM - GM inequality, we have

\( \frac{a+b}{2} \geq \sqrt{ab} \)

\( 4 \geq ab \)

From the second eq.

\( c^2 - 2\sqrt{3}c + 3 = \frac{ab}{2}- 2 \)

\( (c - \sqrt{3})^2 = \frac{ab}{2} - 2 \)

Since the left side is a square and thus positive, the right side must be positive, that is we must have \( \frac{ab}{2} \geq 2 \implies ab \geq 4 \)

Therefore, we see that \( ab = 4 \). Which implies \( a = b = 2 \) and \( c = \sqrt{3} \) Siddhartha Srivastava · 11 months ago

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@Siddhartha Srivastava Could you please tell me what is AM-GM inequality?I am just 13. Saran Vijai · 10 months, 3 weeks ago

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@Saran Vijai Try the wiki. Siddhartha Srivastava · 10 months, 3 weeks ago

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