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Level 2

Evaluate 1999 ⋅ 2000 ⋅ 2001 ⋅ 2002 + 1 . \sqrt{1999 \cdot 2000 \cdot 2001 \cdot 2002 + 1}.


The answer is 4001999.

Oon Han by Oon Han , 16, Singapore

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1 solution

Oon Han
Sep 10, 2019

Let x = 2000 x = 2000 . 1999 ⋅ 2000 ⋅ 2001 ⋅ 2002 + 1 = ( x − 1 ) ( x ) ( x + 1 ) ( x + 2 ) + 1 = x 4 + 2 x 3 − x 2 − 2 x + 1 = x 4 + x 3 − x 2 + x 3 + x 2 − x − x 2 − x + 1 = ( x 2 + x − 1 ) ( x 2 + x − 1 ) = ( x 2 + x − 1 ) 2 = ∣ x 2 + x − 1 ∣ = x 2 + x − 1 = 200 0 2 + 2000 − 1 = 4001999 \begin{aligned} \sqrt{1999 \cdot 2000 \cdot 2001 \cdot 2002 + 1} &= \sqrt{(x-1)(x)(x+1)(x+2) + 1} \\ &= \sqrt{x^4 + 2x^3 - x^2 - 2x + 1} \\ &= \sqrt{x^4 + x^3 - x^2 + x^3 + x^2 - x - x^2 - x + 1} \\ &= \sqrt{(x^2 + x - 1)(x^2 + x - 1)} \\ &= \sqrt{(x^2 + x - 1)^2} \\ &= |x^2 + x - 1| \\ &= x^2 + x - 1 \\ &= 2000^2 + 2000 - 1 \\ &= \boxed{4001999} \end{aligned}

Note: ∣ x 2 + x − 1 ∣ = x 2 + x − 1 |x^2 + x - 1| = x^2 + x - 1 since we are dealing with only non-negative numbers.

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