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if you suppose that all the $a_i$, all the $b_i$, and all the $c_i$ are distinct, can't you do that by induction ? One can assume wlog that $a_1 > \ldots > a_d > 0$ and $b_1 > \ldots > b_d > 0$ etc.. so that taking $p \to \infty$ you can see that $a_1+b_1=c_1$. Hence $$a_1(1+\sum_2^d a_1\left(1+\sum_2^d \frac{a_k}{a_1}^p)^{\frac1p} left(\frac{a_k}{a_1}\right)^p\right)^{1/p} + b_1(1+\sum_2^d b_1\left(1+\sum_2^d \frac{b_k}{b_1}^p)^{\frac1p} left(\frac{b_k}{b_1}\right)^p\right)^{1/p} = c_1(1+\sum_2^d c_1\left(1+\sum_2^d \frac{c_k}{c_1}^p)^{\frac1p}$$ left(\frac{c_k}{c_1}\right)^p\right)^{1/p}$$and a Taylor expansion for p \to \infty tells you that \frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{c_2}{c_1}. Continuing this way, one can see that the only family of solutions is A=\lambda C and B=(1-\lambda) C. Too simple to be true ? 3 edited body; [made Community Wiki] if you suppose that all the a_i, all the b_i, and all the c_i are distinct, can't you do that by induction ? One can assume wlog that a_1 > \ldots > a_d > 0 and b_1 > \ldots > b_d > 0 etc.. so that taking p \to \infty you can see that a_1+b_1=c_1. Hence$$a_1(1+\sum_2^d \frac{a_k}{a_1}^p)^{\frac1p} + b_1(1+\sum_2^d \frac{b_k}{b_1}^p)^{\frac1p} = c_1(1+\sum_2^d \frac{c_k}{c_1}^p)^{\frac1p}$$and a Taylor expansion for p \to \infty tells you that \frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{c_2}{c_1}. Continuing this way, one can see that the only family of solutions is A=\lambda C and B=(1-\lambda C)B=(1-\lambda) C. Too simple to be true ? 2 grammar fixing if you suppose that all the a_i, all the b_i, b_i, and all the c_i are distinct, can't you do that by induction ? One can assume wlog that a_1 > \ldots > a_d > 0 and b_1 > \ldots > b_d > 0 etc.. so that taking p \to \infty you can see that a_1+b_1=c_1. Hence$$a_1(1+\sum_2^d \frac{a_k}{a_1}^p)^{\frac1p} + b_1(1+\sum_2^d \frac{b_k}{b_1}^p)^{\frac1p} = c_1(1+\sum_2^d \frac{c_k}{c_1}^p)^{\frac1p} and a Taylor expansion for $p \to \infty$ tells you that $\frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{c_2}{c_1}$. Continuing this way, one can see that the only familly family of solution solutions is $A=\lambda C$ and $B=(1-\lambda C)$. Too simple to be true ?

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