We consider a quantum dot coupled to both superconducting and spin-polarized electrodes, and study the triad interplay of the Kondo effect, superconductivity, and ferromagnetism, any pair of which compete with and suppress each other. We find that the interplay leads to a mixed-valence quantum phase transition, which for other typical sysmstems is merely a crossover rather than a true transition. At the transition, the system changes from the spin doublet to singlet state. The singlet phase is adiabatically connected (through crossovers) to the so-called charge Kondo state and to the superconducting state. We analyze in detail the physical characteristics of different states and propose that the measurement of the cross-current correlation and the charge relaxation resistance can clearly distinguish between them.
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