mat 2025 Q27X(v)

mat · Uk Proof Proof Involving Combinatorial or Number-Theoretic Structure
Suppose that we want to find a nice set of four numbers that are all the same colour. By considering the possibilities for the colours of the numbers from 1 to 27 inclusive, prove such a set always exists. [Hint: consider $\{ 1,2,3 \} , \{ 4,5,6 \} , \ldots , \{ 25,26,27 \}$.] 
Suppose that we want to find a nice set of four numbers that are all the same colour. By considering the possibilities for the colours of the numbers from 1 to 27 inclusive, prove such a set always exists.
[Hint: consider $\{ 1,2,3 \} , \{ 4,5,6 \} , \ldots , \{ 25,26,27 \}$.]
Paper Questions
Q26X(i) Q26X(ii) Q26X(iii) Q26X(iv) Q26X(v) Q26Y(i)(a) Q26Y(i)(b) Q26Y(ii) Q26Y(iii) Q26Y(iv) Q26Y(v) Q26Y(vi) Q27X(i) Q27X(ii) Q27X(iii) Q27X(iv) Q27X(v) Q27Y(i) Q27Y(ii) Q27Y(iii) Q27Y(iv) Q27Y(v) Q27Y(vi)