Maths Olympiad Prep

Track / Stage 4 / 299 of 340 #559 of 1964

Problem 559

AMC 12 late, AIME early
Algebra Difficulty 5.0 Multiple choice

1. Calculate
2018×2021×2022×2023+20242−20242 \sqrt{2018 \times 2021 \times 2022 \times 2023+2024^{2}}-2024^{2}

The value is:

Pick one

Official solution

- 1. C.

Let x=2024x=2024.
Then the original expression
=(x−6)(x−3)(x−2)(x−1)+x2−x2=(x2−7x+6)(x2−5x+6)+x2−x2=(x2−6x+6)2−x2=6(1−x)=−12138. \begin{array}{l} =\sqrt{(x-6)(x-3)(x-2)(x-1)+x^{2}}-x^{2} \\ =\sqrt{\left(x^{2}-7 x+6\right)\left(x^{2}-5 x+6\right)+x^{2}}-x^{2} \\ =\sqrt{\left(x^{2}-6 x+6\right)^{2}}-x^{2} \\ =6(1-x)=-12138 . \end{array}

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.