Forming Teams

After the first year of meetups, Dr. Hong identifies 10 students who have the best record on their assignments. He decides to run a competition. He randomly divides the students into 5 teams, with each team having at least 1 student. Student(s) of the winning team will be recognized as the spotlight student(s) of the year. How many different ways can he form the 5 teams?

(This problem is beyond Level E, so it is not counted against spotlight student eligibility.)

Counting Squares

48 CMM students have a meetup together. Each holds a square tile. They lay the tiles down to form a big 7 by 7 grid, with the center position empty. How many squares are there that are fully covered by tiles. 

Leo's Homework

Every week, Leo works on the 7 problems from CMM assignments. During the 3rd week, he recorded the time for each problem, and found the following interesting facts:

• problems 1, 2, 3 together took him 6 minutes; 
• problems 2, 3, 4 together took him 7 minutes; 
• problems 3, 4, 5 together took him 8 minutes; 
• problems 4, 5, 6 together took him 9 minutes; 
• problems 5, 6, 7 together took him 10 minutes; 
• problems 1, 6, 7 together took him 11 minutes; 
• problems 1, 2, 7 together took him 12 minutes. 

How many minutes did Leo spend on problem #5?

Dr. Hong's Magic Number

Dr. Hong has a magic number ABACDD that satisfies the following conditions:

1. ABACDD = EF × BE × GD; 
2. BE × GD = EGHF; 
3. EF, BE, and GD are prime numbers;  
4. A, B, C, D, E, F, G, H represent eight different single-digit non-zero integers. 

what is Dr. Hong's magic number ABACDD?