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一道不等式證明問題的N種解法

2022-03-25 17:20 作者:桐輝數(shù)學(xué)  | 我要投稿


一道不等式證明題,一不小心琢磨出10個(gè)解法。


%E5%B7%B2%E7%9F%A5%3A%20a%EF%BC%9E2%2C%20b%EF%BC%9E2.%20%E8%AF%81%E6%98%8E%3Aab%EF%BC%9Ea%2Bb

證法1:

? ? ??%E8%AE%BEa%3D2%2Bx%2Cb%3D2%2By%2Cx%2Cy%EF%BC%9E0%2C%20%E5%88%99

? ? ??ab%3D4%2B2(x%2By)%2Bxy%2Ca%2Bb%3D4%2Bx%2By

? ? ?那么,ab-(a%2Bb)%3Dx%2By%2Bxy%EF%BC%9E0%2C%20%E2%88%B4ab%EF%BC%9Ea%2Bb

證法2:

? ? ???%E2%88%B5a%EF%BC%9E2%2Cb%EF%BC%9E2%2C%E2%88%B4a%C3%97b%3Dab%EF%BC%9E2b%2C%20b%C3%97a%3Dab%EF%BC%9E2b

? ? ? ??%E2%88%B4ab%EF%BC%9E2a%2B2b%2C%20%E2%88%B4ab%EF%BC%9Ea%2Bb

證法3: ?

??%E2%88%B5a%EF%BC%9E2%2Cb%EF%BC%9E2%2C%E2%88%B4%5Cfrac%7B1%7D%7Ba%7D%20%20%EF%BC%9C%5Cfrac%7B1%7D%7B2%7D%2C%20%5Cfrac%7B1%7D%7Bb%7D%20%20%EF%BC%9C%5Cfrac%7B1%7D%7B2%7D%20,則

%5Cfrac%7B1%7D%7Ba%7D%20%2B%5Cfrac%7B1%7D%7Bb%7D%EF%BC%9C%20%5Cfrac%7B1%7D%7B2%7D%2B%20%5Cfrac%7B1%7D%7B2%7D%20%3D1,

%5Cfrac%7Ba%2Bb%7D%7Bab%7D%20%EF%BC%9C1%2C%20%E5%8D%B3ab%EF%BC%9Ea%2Bb

證法4:

? %E2%88%B5%20a%EF%BC%9E2%2Cb%EF%BC%9E2%2C? ?%E2%88%B4%20a-1%EF%BC%9E1%2Cb-1%EF%BC%9E1

%E2%88%B4(a-1)(b-1)%3Dab-a-b%2B1%EF%BC%9E1

??%E5%8D%B3%20ab-a-b%EF%BC%9E0%2C%20%E6%95%85ab%EF%BC%9Ea%2Bb

證法5:

? ? ? ???%E2%88%B5a%EF%BC%9E2%2C%20b%EF%BC%9E2%2C%20%20%E2%88%B4a-1%EF%BC%9E1%2C%20b-1%EF%BC%9E1%2C(a-1)(b-1)%EF%BC%9E1

? ? ? ? ??ab-(a%2Bb)%3Dab-a-b%0A%3Da(b-1)-(b-1)-1

? ? ? ?%3D(a-1)(b-1)-1%EF%BC%9E0%2C%20%E2%88%B4ab%EF%BC%9Ea%2Bb

證法6:

? ? ? ???%E2%88%B5%20a%EF%BC%9E2%2C%20b%EF%BC%9E2%2C%20%20%E2%88%B4a%2Bb%EF%BC%9E4%2C%20(a-2)(b-2)%EF%BC%9E0

? ? ? ? ? ?%E2%88%B4%20ab-2(a%2Bb)%2B4%EF%BC%9E0

? ? ? ? ?ab%EF%BC%9E2(a%2Bb)-4%2C%20ab%EF%BC%9E(a%2Bb)%2B(a%2Bb)-4

? ? ? ? ?%E2%88%B4%20ab%EF%BC%9E(a%2Bb)%2B4-4%2C%20%E6%95%85ab%EF%BC%9Ea%2Bb

證法7: ?

%E4%BB%A4b%3D2%2Bm(m%EF%BC%9E0)%2C%20%E5%88%99

%20%20ab-(a%2Bb)%3Da(2%2Bm)-(a%2B2%2Bm)%20%20

? ? ??%3Da%2Bam-2-m

? ? ? %3D(a-2)%2B(a-1)m%EF%BC%9E0

? ????%E5%8D%B3%20ab%EF%BC%9Ea%2Bb

證法8:

? ?%E2%88%B5%20a%EF%BC%9E2%2C%20b%EF%BC%9E2%2C%20%E2%88%B4%20a-2%EF%BC%9E0

? ? ??b-2%EF%BC%9E0%2C(a-2)(b-2)%EF%BC%9E0 ?

%E5%8D%B3%20ab-2a-2b%2B4%EF%BC%9E0

%E5%88%99%20ab%EF%BC%9Ea%2Bb%2B(a-2)%2B(b-2)%EF%BC%9Ea%2Bb

??%E5%8D%B3%20%20ab%EF%BC%9Ea%2Bb

證法9:

2ab-2(a%2Bb)%3D(ab-2a)%2B(ab-2b)

? ???%3Da(b-2)%2Bb(a-2)%EF%BC%9E0

? ? ?%E5%8D%B3%202ab-2(a%2Bb)%EF%BC%9E0%2C%20%E6%95%85ab%EF%BC%9Ea%2Bb

證法10:

? ? ??不妨設(shè)a%EF%BC%9Eb%EF%BC%9E2.%20%E4%BB%A4a%3Dkb%2C%20k%EF%BC%9E1%2C%E5%88%99

? ? ?ab-(a%2Bb)%3Dkb%C3%97b-(kb%2Bb)

? ??%3Db(kb-k-1)%EF%BC%9Eb(k-1)%EF%BC%9E0

? ? ?即ab%EF%BC%9Ea%2Bb

??

? ?如果你有其他解法,歡迎在評(píng)論區(qū)交流哦?。?/p>

? ?

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一道不等式證明問題的N種解法的評(píng)論 (共 條)

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