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The current COVID-19 pandemic overloads healthcare systems, including radiology departments. Though several deep learning approaches were developed to assist in CT analysis, nobody considered study triage directly as a computer science problem. We describe two basic setups: Identification of COVID-19 to prioritize studies of potentially infected patients to isolate them as early as possible; Severity quantification to highlight studies of severe patients and direct them to a hospital or provide emergency medical care. We formalize these tasks as binary classification and estimation of affected lung percentage. Though similar problems were well-studied separately, we show that existing methods provide reasonable quality only for one of these setups. We employ a multitask approach to consolidate both triage approaches and propose a convolutional neural network to combine all available labels within a single model. In contrast with the most popular multitask approaches, we add classification layers to the most spatially detailed upper part of U-Net instead of the bottom, less detailed latent representation. We train our model on approximately 2000 publicly available CT studies and test it with a carefully designed set consisting of 32 COVID-19 studies, 30 cases with bacterial pneumonia, 31 healthy patients, and 30 patients with other lung pathologies to emulate a typical patient flow in an out-patient hospital. The proposed multitask model outperforms the latent-based one and achieves ROC AUC scores ranging from 0.87+-01 (bacterial pneumonia) to 0.97+-01 (healthy controls) for Identification of COVID-19 and 0.97+-01 Spearman Correlation for Severity quantification. We release all the code and create a public leaderboard, where other community members can test their models on our test dataset.