## 1 Introduction  
As the world's highest-level international sports event, the Olympic Games have evolved into a top-tier sporting extravaganza encompassing over 200 participating countries and more than 40 major sports disciplines [1; 2] It is not only a concentrated manifestation of the highest levels of competitive sport but also serves as a crucial platform for countries to showcase their comprehensive national strength and resource allocation strategies. With the continuous expansion of the event scale and the increase in the variety of events--for instance, the 2020 Tokyo Olympics added five new events such as skateboarding and surfing--sports management departments of various countries have attached greater importance to using scientific methods to predict the distribution of medals [3]. The aim is to maintain competitiveness in dominant events and achieve breakthroughs in emerging ones. Accurate medal prediction not only helps optimize the allocation of training resources, but also provides a quantitative basis for the formulation of national sports policies and strategic planning, which holds significant practical importance for enhancing comprehensive sports competitiveness. Machine learning (ML) excels at uncovering nonlinear relationships within vast historical datasets, overcoming the limitations of traditional statistical models that rely on predefined equations [4; 5], and has been widely applied in fields such as finance, healthcare, and advertising [6; 7] As a core ML technique, neural networks exhibit powerful adaptive learning capabilities, excelling in handling nonlinear relationships and high-dimensional data with exceptional generalization and fault tolerance. By comparing multiple ML models, precise predictions can be achieved for complex dynamic systems such as Olympic medal outcomes [8; 9; 10]. Additionally, K-means clustering effectively segments data by maximizing intra-cluster similarity and inter-cluster differences [11; 12], while factor analysis reduces highly correlated variables into a few independent principal components, revealing latent data patterns and mitigating multicollinearity [13; 14]. By integrating K-means clustering and factor analysis into the ML pipeline, this study constructs an innovative framework that significantly enhances the efficiency of analyzing global medal performance trends. Furthermore, a Fully Connected Neural Network (FCNN) is employed to predict the probability of non-medal-winning nations achieving their first medal at the 2028 Olympic Games, achieving an impressive 85.5% accuracy on the test set. The impacts of the host effect and the great coach effect on athletes medal wins cannot be underestimated. The host effect significantly enhances the medal competitiveness of the host country through various mechanisms, such as increasing the number of participants, optimizing venue adaptability, and boosting athletes' morale [15]. The introduction of renowned coaches can not only improve the strength of domestic athletes through technical guidance and psychological support but can also weaken the competitiveness of competitors through strategic analysis, resulting in a situation of relative gain and loss [16; 17]. This study quantifies the host effect and the great coach effect separately. It proposes a formula for the host effect and evaluates the coach effect factors by segments. Moreover, this study designs a coach effect scoring system, which assesses the extent to which the introduction of coaches enhances the overall strength of athletes in a project through Bayesian linear regression, providing data support for the optimal allocation of coaching teams. Although prior studies have established a theoretical foundation for Olympic medal prediction, challenges such as limited research volume, overly simplistic models, and insufficient data diversity persist [18; 19; 20; 21]. These issues result in significant gaps in predicting medal distributions at the national or individual level.