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Least value using x + 1/x >= 2 (AM-GM)

This type of question, asked 3 times: CDS 2 2019, CDS 1 2024, CDS 2 2024. All 3 questions below, oldest first.

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  1. Question 1

    CDS 2 2019

    8
    For x>0x > 0, what is the minimum value of x+x+22xx + \frac{x + 2}{2x} ?

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  2. Question 2

    CDS 1 2024

    60
    What is the minimum value of (a2+3a+1a)(b2+3b+1b)\left(\frac{a^2 + 3a + 1}{a}\right)\left(\frac{b^2 + 3b + 1}{b}\right) for a,b>0a, b > 0 ?

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  3. Question 3

    CDS 2 2024

    33
    What is the minimum value of (a8+a4+1)(b8+b4+1)a4b4\dfrac{(a^8 + a^4 + 1)(b^8 + b^4 + 1)}{a^4 b^4}, where a>0a > 0, b>0b > 0 ?

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