Page-based DSM systems suffer from false sharing since they use a large page as a coherence unit. The optimal page size is dynamically affected by application characteristics. Therefore, a fixed-size page cannot satisfy various applications even if it is small as a cache line size. In this paper we present a software-only coherence protocol called BCP (Buddy Coherence Protocol) to support multiple page sizes that vary adaptively according to the behavior of each application during run time. In BCP, the address of a remote access and the address of the most recent local access is compared. If they are to the different halves of a page, BCP considers it as false sharing and demotes the page to two subpages of equal size. If two contiguous pages belong to the same node, BCP promotes two pages to a superpage to reduce the number of the following coherence activities. We also suggest a mechanism to detect data sharing patterns to optimize the protocol. It detects and keeps the sharing pattern for each page by a state transition mechanism. By referring to those patterns, BCP selectively demotes the page and increases the effectiveness of a demotion. Self-invalidation of the migratorily shared page is also employed to reduce the number of invalidations. Our simulations show that the optimized BCP outperforms almost all the best cases of the write-invalidate protocols using fixed-size pages. BCP improves performance by 42.2% for some applications when compared against the case of the fixed-size page.