ELMO 2011 Problem 2

Wanda the Worm likes to eat Pascal's triangle. One day, she starts at the top of the triangle and eats $\binom{0}{0}=1$ . Each move, she travels to an adjacent positive integer and eats it, but she can never return to a spot that she has previously eaten. If Wanda can never eat numbers $a,b,c$ such that $a+b=c$ , prove that it is possible for her to eat 100,000 numbers in the first 2011 rows given that she is not restricted to traveling only in the first 2011 rows.

(Here, the $n+1$ st row of Pascal's triangle consists of entries of the form $\binom{n}{k}$ for integers $0\le k\le n$ . Thus, the entry $\binom{n}{k}$ is considered adjacent to the entries $\binom{n-1}{k-1}$ , $\binom{n-1}{k}$ , $\binom{n}{k-1}$ , $\binom{n}{k+1}$ , $\binom{n+1}{k}$ , $\binom{n+1}{k+1}$ .)

Linus Hamilton.