Berkeley Math Circle #0hvh

Berkeley Math CircleHard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Problem

Problem: A bulldozer is touring Pascal's triangle. It starts at the top of the triangle, at $\left(\begin{array}{l}0 \\ 0\end{array}\right)=1$ . Each move, it travels to an adjacent positive integer, but can never return to a spot it has already visited. Moreover, if it has visited two numbers $a>b$ , it may not visit $a+b$ or $a-b$ . Finally, the bulldozer is confined to the first 140 rows of Pascal's triangle. Prove that the bulldozer may visit at least 2017 numbers. (By convention, the $n$ th row contains the entries $\left(\begin{array}{c}n-1 \\ k\end{array}\right)$ for $k=0, \ldots, n-1$ , hence the $n$ th row has $n$ entries.)