numbers 1 to 21 on circle with sum at least 33

Question

if we write 1,2,3,....,21 on a circle. there are at least one three consecutive set with sum at least 33.

obviously the main idea is strictly increasing on sum of numbers. but i wanna know how to organize the solution.

Answer 1

You have $21$ consecutive $3$ sets. Each number appears $3$ times in these $21$ sets. Moreover, the sum of numbers $1$ to $21$ is $ \dfrac{21 \times 22}{2} = 11 \times 21 $. So the sum of numbers of these sets will be: $$ 3 \times \dfrac{21 \times 22}{2} = 33 \times 21. $$ Now, if all these sets have sum less than $33$, then the sum of numbers of these $21$ sets will be less than $33 \times 21$, a contradiction.