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[*45] However, fraud is not proven when a court is left with only a suspicion of fraud, and even a strong suspicion is not sufficient to establish a taxpayer’s liability for the fraud penalty. See Olinger v. Commissioner, 234 F.2d 823, 824 (5th Cir. 1956), aff’g in part, rev’g in part on another ground T.C. Memo. 1955-9; Davis v. Commissioner, 184 F.2d 86, 87 (10th Cir. 1950); Green v. Commissioner, 66 T.C. 538, 550 (1976). 2. Badges of Fraud Because it is difficult to prove fraudulent intent by direct evidence, the Commissioner may establish fraud by circumstantial evidence, which includes various “badges of fraud” (hereinafter, factors) on which the courts often rely. See Bradford v. Commissioner, 796 F.2d 303, 307 (9th Cir. 1986), aff’g T.C. Memo. 1984-601; DiLeo v. Commissioner, 96 T.C. at 875. The factors focus on whether the taxpayer engaged in certain conduct that is indicative of fraudulent intent, such as: (1) understating income; (2) failing to maintain adequate records; (3) offering implausible or inconsistent explanations; (4) concealing income or assets; (5) failing to cooperate with tax authorities; (6) engaging in illegal activities; (7) providing incomplete or misleading information to the taxpayer’s tax return preparer; (8) offering false or incredible testimony; (9) filing false documents, including filing false income tax returns; (10) failing to file tax returns; and (11) ___________