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HMMT February

Combinatorics Difficulty 5.0 AIME Find the answer

How many elements are in the set obtained by transforming {(0,0),(2,0)}14\{(0,0),(2,0)\} 14 times?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Transforming it k≥1k \geq 1 times yields the diamond {(n,m):∣n−1∣+∣m∣≤k+1}\{(n, m):|n-1|+|m| \leq k+1\} with the points (1,k),(1,k+1),(1,−k),(1,−k−1)(1, k),(1, k+1),(1,-k),(1,-k-1) removed (this can be seen inductively). So we get (k+1)2+k2−4(k+1)^{2}+k^{2}-4 lattice points, making the answer 477.

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