CUET UG 2025 — General Test Quantitative Reasoning

Let the 6 positive numbers be $x_1, x_2, x_3, x_4, x_5, x_6$x1​,x2​,x3​,x4​,x5​,x6​.

Given: $x_1 \times x_2 \times x_3 \times x_4 \times x_5 \times x_6 = 6^6$x1​×x2​×x3​×x4​×x5​×x6​=66

The average is: $\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6}$6x1​+x2​+x3​+x4​+x5​+x6​​

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By the AM-GM inequality:

$$\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq \sqrt[6]{x_1 \times x_2 \times x_3 \times x_4 \times x_5 \times x_6}$$6x1​+x2​+x3​+x4​+x5​+x6​​≥6x1​×x2​×x3​×x4​×x5​×x6​​

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Substituting the product value:

$$\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq \sqrt[6]{6^6}$$6x1​+x2​+x3​+x4​+x5​+x6​​≥666​

$$\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq 6^{6/6}$$6x1​+x2​+x3​+x4​+x5​+x6​​≥66/6

$$\dfrac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6}{6} \geq 6$$6x1​+x2​+x3​+x4​+x5​+x6​​≥6

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The equality holds when all numbers are equal: $x_1 = x_2 = x_3 = x_4 = x_5 = x_6 = 6$x1​=x2​=x3​=x4​=x5​=x6​=6

Therefore, the minimum value of the average is $6$6.