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Nowadays, breast cancer is one of the leading cancers in Vietnam, and it causes approximately 6000 deaths every year. The rate of breast cancer patients was calculated as 26.4/100000 persons in 2018. There are 21,555 new cases reported in 2020. However, these figures can be reduced with early detection and diagnosis of breast cancer disease in women through mammographic imaging. In many hospitals in Vietnam, there is a lack of experienced breast cancer radiologists. Therefore, it is helpful to develop an intelligent system to improve radiologists’ performance in breast cancer screening for Vietnamese patients. Our research aims to develop a convolutional neural network-based system for classifying breast cancer X-Ray images into three classes of BI-RADS categories as BI-RADS 1 (“normal”), BI-RADS 23 (“benign”) and BI-RADS 045 (“incomplete and malignance”). This classification system is developed based on the convolutional neural network with ResNet 50. The system is trained and tested on a breast cancer image dataset of Vietnamese patients containing 7912 images provided by Hanoi Medical University Hospital radiologists. The system accuracy uses the testing set achieved a macAUC (a macro average of the three AUCs) of 0.754. To validate our model, we performed a reader study with the breast cancer radiologists of the Hanoi Medical University Hospital, reading about 500 random images of the test set. We confirmed the efficacy of our model, which achieved performance comparable to a committee of two radiologists when presented with the same data. Additionally, the system takes only 6 seconds to interpret a breast cancer X-Ray image instead of 450 seconds interpreted by a Vietnamese radiologist. Therefore, our system can be considered as a “second radiologist,” which can improve radiologists’ performance in breast cancer screening for Vietnamese patients.