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махать шляпой stresses
gen.	agitate hat
Makarov.	agitate one's hat
махать кому-л. шляпой
gen.	wave to smb. with one's hat (with one's stick, with one's umbrella, etc., и т.д.)
по махать шляпой
gen.	wave a hat (one's handkerchief, a flag, a branch, one's umbrella, etc., и т.д.)