International Mathematical Olympiad (IMO) $2018$ , Day $1$ , Problem $3$ of $6$

$\text{Let us suppose that n = 2018 and N = (1 + 2 + 3 + ... + n)}$

For any number not in the bottom row (let us name it $d$ for easy use) draw a line from $d$ to the larger number which is directly below it (there should be two numbers below a number) This will form a shape that looks like a lightning strike (it looks cool, try it yourself with a large anti-Pascal!)

Consider the directed path starting from the top vertex $A$ . Starting from the first number, it increases by at least $1 + 2 + ··· + n$ , since the increases at each step in the path are distinct; therefore equality must hold and thus the path from the top ends at $N = 1 + 2 + ··· + n$ with all the numbers ${1, 2, ... , n}$ being close by. Let $B$ be that position.

Consider the two left/right neighbors X and Y of the endpoint B. Assume that B is to the right of the midpoint of the bottom side, and complete the equilateral triangle as shown to an apex C. Consider the lightning strike from C hitting the bottom at D.

It travels at least $(floor){n/2 - 1}$ steps, by construction. But the increases must be at least $n + 1, n + 2, ...$ since $1, 2, ... , n$ are close to the A → B lightning path. Then the number at D is at least $(n + 1) + (n + 2) + ··· + (n + (bn/2 - 1c)) > 1 + 2 + ··· + n$

for $n ≥ 2018$ , a contradiction occurs.