Everyone shakes hands with everyone else: find the number of people from the number of handshakes

The same question, asked 2 times: 9 May 2024 S1*, 19 May 2026 S1. Every wording below, oldest first.

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Question 1
9 May 2024 S1
114 $If at the end of certain meeting, everyone had shaken hands with every one else, it was found that 45 handshakes were exchanged, then the number of members present at the meeting, are$
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Question 2
19 May 2026 S1
58 $Everyone in the room handshakes with everyone else. If the total number of handshakes is 66, then the number of persons in the room is...$
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Question 1