Combinatorics | Number Theory Answers

Let us denote S_n = aⁿ + bⁿ + cⁿ for arbitrary numbers a, b, c. It is known that S₁ = 8, 5, S₂ = 74, 25, S₃ = 639, 625 for some values of a, b, c. What is the largest possible value of S²₈₁₁ — S₈₁₀S₈₁₂? 

Kate drew a 41 x 41 checkered square (lattice) on the asphalt with white chalk (i.e. there are 42 horizontal segments and 42 vertical segments drawn).

By one move it is allowed to pick out an arbitrary square (of any size) and repaint its boundary using a chalk of blue colour. In different moves it is allowed to repaint any segment more than once. What is the smallest number of such moves required to repaint all the initial lines in blue colour?

36 students are members of a sports club. Every two of them are either friends or enemies. (Friendship and enmity are reciprocal, i.e. if A is a friend to B then B is a friend to A, and the same applies to being enemies.) It has turned out that each of the students has exactly 8 enemies. Let us call a group of three students concurrent if they are either pairwise enemies or pairwise friends to each other. What is the maximum possible quantity of concurrent student triples in this sports club? (Two distinct concurrent student triples may have mutual students in them.)

Find a counter example to the statement that every even positive integer can be written as the sum of the squares of three integers.