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Prove (or disprove): If \(a,b,c,d\) are positive real numbers with \(a\times b = c\times d\), then the solutions for \(x\) in the equation \( \frac {a^x+b^x}{c^x+d^x} = \frac{a+b}{c+d} \) is only \(\pm 1\).

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Note by Pi Han Goh
2 years, 5 months ago

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\(a=b=c=d=1\) gives \(1 = 1\) that means the \(x\) has infinitely many solutions! Samuraiwarm Tsunayoshi · 2 years, 5 months ago

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@Samuraiwarm Tsunayoshi Oh. You found a loophole. Let me fix that. Pi Han Goh · 2 years, 5 months ago

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@Pi Han Goh I think you should make \(a,b,c,d\) distinct. Prasun Biswas · 2 years, 5 months ago

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@Pi Han Goh \(a = b = c = d \neq 0\) also satifies the condition. Krishna Sharma · 2 years, 5 months ago

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@Krishna Sharma Note that he already mentioned that \(a,b,c,d\) are positive reals, so you don't need to consider non-negative reals. Prasun Biswas · 2 years, 5 months ago

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