〜たまえ
(do it), go ahead and (said by a man to a junior)
Verb ます-stem + たまえ
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たまえ is a command used by men speaking to someone of lower status, such as a boss to a subordinate or a professor to a student: 遠慮せずに、座りたまえ, "don't be shy, have a seat".
It attaches to the ます-stem: 座り + たまえ, 飲み + たまえ, 考え + たまえ. It sounds authoritative but not rude, a bit old-fashioned, and very common for company presidents, teachers and aristocrats in fiction.
It comes from the classical honorific 給う ("to bestow"), so it was originally respectful. Today it's the opposite: only a superior uses it.
In everyday speech, you'd use なさい (N4) or てください.
Examples
遠慮せずに、座りたまえ。
Don't be shy, have a seat.
君も一杯飲みたまえ。
Have a drink yourself.
早く答えたまえ。
Answer me, quickly.
まあ、落ち着きたまえ。
Now, calm down.
自分の頭で考えたまえ。
Think for yourself.