Unlike traditional software, smart contracts have the unique organization in which a sequence of transactions shares internal states. Unfortunately, such a characteristic makes existing fuzzing tools fail to discern critical transaction sequences. To tackle this challenge, we employ a combined static and dynamic analysis for fuzzing smart contracts. First, we statically analyze smart contract binaries to predict which transaction sequences will lead to effective testing, and figure out if there is a certain constraint that each transaction should satisfy. Such information is then passed to the traditional fuzzing phase and used to construct an initial seed corpus. Furthermore, we perform a light-weight dynamic data-flow analysis to collect data-flow-based feedback to effectively guide fuzzing. We implement our technique on a practical open-source fuzzer, named SMARTIAN. SMARTIAN can discover bugs in real-world smart contracts without the need for the source code. Our experimental results show that SMARTIAN is more effective than existing state-of-the-art tools in finding known CVEs from real-world contracts, and it also outperforms other tools in terms of code coverage.