7

Part of 2018 South East Mathematical Olympiad

Problems(2)

Minimum number of K_3 under some constraints.

Source: 2018 China Southeast MO Grade 10 P7

24

participants attended a meeting. Each two of them shook hands once or not. A total of

216

handshakes occured in the meeting. For any two participants who have shaken hands, at most

10

22

participants have shaken hands with exactly one of these two persons. Define a friend circle to be a group of

3

participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.

combinatoricsgraph theory

Does there exist positive integers m,n such that f(m,n)=2018?

Source: 2018 China Southeast MO Grade 11 P7

For positive integers

m,n,

f(m,n)

as the number of ordered triples

(x,y,z)

of integers such that

\begin{cases} xyz=x+y+z+m, \\ \max\{|x|,|y|,|z|\} \leq n \end{cases}

Does there exist positive integers

m,n,

f(m,n)=2018?

Please prove your conclusion.