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Datasets with missing values are frequent in real-world classification problems. It seems obvious that imputation of missing values can be considered as a series of secondary tasks, while classification is the main purpose of any machine dealing with these datasets. Consequently, Multi-Task Learning (MTL) schemes offer an interesting alternative approach to solve missing data problems. In this paper, we propose an MTL-based method for training and operating a modified Multi-Layer Perceptron (MLP) architecture to work in incomplete data contexts. The proposed approach achieves a balance between both classification and imputation by exploiting the advantages of MTL. Extensive experimental comparisons with well-known imputation algorithms show that this approach provides excellent results. The method is never worse than the traditional algorithms - an important robustness property - and, also, it clearly outperforms them in several problems.