The number of clinical prediction models sharing the same prediction task has increased in the medical literature. However, evidence synthesis methodologies that use the results of these prediction models have not been sufficiently studied, particularly in the context of meta-analysis settings where only summary statistics are available. In particular, we consider the following situation: we want to predict an outcome Y, that is not included in our current data, while the covariate data are fully available. In addition, the summary statistics from prior studies, which share the same prediction task (i.e., the prediction of Y), are available. This study introduces a new method for synthesizing the summary results of binary prediction models reported in the prior studies using a linear predictor under a distributional assumption between the current and prior studies. The method provides an integrated predictor combining all predictors reported in the prior studies with weights. The vector of the weights is designed to achieve the hypothetical improvement of area under the receiver operating characteristic curve (AUC) on the current available data under a practical situation where there are different sets of covariates in the prior studies. We observe a counterintuitive aspect in typical situations where a part of weight components in the proposed method becomes negative. It implies that flipping the sign of the prediction results reported in each individual study would improve the overall prediction performance. Finally, numerical and real-world data analysis were conducted and showed that our method outperformed conventional methods in terms of AUC.