Handshakes

When two people shake hands with each another, that counts as one handshake. After the COVID-19 quarantine, all students from Dr. Hong's Charlotte Math Meetup have a big party in person. Everyone shakes hands with each other exactly once. There are 1225 handshakes! How many students are in the party?

Terminal Zeros

Terminal zeros of a number are the zeros to the right of the last nonzero digit. For instance, 506,400,000 has 5 terminal zeros. During a math meetup, students play a game. They form a queue. The first student yells 1, the second student yells 2, the third student yells 6, the fourth student yells 24, the nth student yells the product of 1, 2, 3, ..., n. Leo is the last student to yell his number. He forgets to bring the calculator, but he says there are 7 terminal zeros in his number. If Leo's statement is true, what's the largest possible number of students in the queue?

Counting Game

Leo, Kyle, Issaic, and several other Charlotte Math Meetup students form a circle to play a game of counting numbers clockwise. Each round starts from Leo, with each student on the circle adds the same prime number continuously. In Round 1, Leo counts 2. When it's Leo's turn to begin Round 6, he counts 107. Then three more students join the circle to continue the same game. What is the number Leo counts in Round 10?

DDD

A, B, C, D represent four different non-zero digits. AB × CD = DDD. What's the largest possible value of D?