Maths Olympiad Prep

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Algebra Difficulty 5.8 AIME, harder Prove it

(5) 若 nn 是正整数,则
xn−yn=(x−y)(xn−1+xn−2y+⋯+xyn−2+yn−1).x^{n}-y^{n}=(x-y)\left(x^{n-1}+x^{n-2} y+\cdots+x y^{n-2}+y^{n-1}\right) .
（6）若 nn 是正奇数,则 (在上式中用 −y-y 代换 yy )
xn+yn=(x+y)(xn−1−xn−2y+⋯−xyn−2+yn−1).x^{n}+y^{n}=(x+y)\left(x^{n-1}-x^{n-2} y+\cdots-x y^{n-2}+y^{n-1}\right) .

Solution

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