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Multi-contrast Magnetic Resonance Imaging (MRI) generates multiple medical images with rich and complementary information for routine clinical use; however, it suffers from a long acquisition time. Recent works for accelerating MRI, mainly designed for single contrast, may not be optimal for multi-contrast scenario since the inherent correlations among the multi-contrast images are not exploited. In addition, independent reconstruction of each contrast usually does not translate to optimal performance of downstream tasks. Motivated by these aspects, in this paper we design an end-to-end framework for accelerating multi-contrast MRI which simultaneously optimizes the entire MR imaging workflow including sampling, reconstruction and downstream tasks to achieve the best overall outcomes. The proposed framework consists of a sampling mask generator for each image contrast and a reconstructor exploiting the inter-contrast correlations with a recurrent structure which enables the information sharing in a holistic way. The sampling mask generator and the reconstructor are trained jointly across the multiple image contrasts. The acceleration ratio of each image contrast is also learnable and can be driven by a downstream task performance. We validate our approach on a multi-contrast brain dataset and a multi-contrast knee dataset. Experiments show that (1) our framework consistently outperforms the baselines designed for single contrast on both datasets; (2) our newly designed recurrent reconstruction network effectively improves the reconstruction quality for multi-contrast images; (3) the learnable acceleration ratio improves the downstream task performance significantly. Overall, this work has potentials to open up new avenues for optimizing the entire multi-contrast MR imaging workflow.