HMMT 二月 2024 · 团队赛 · 第 4 题

HMMT February 2024 — Team Round — Problem 4

[30] Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such a way that the point (0 , 0) is labeled by 0 , and for every x, y ≥ 0 , the set of numbers labeled on the points ( x, y ) , ( x, y + 1) , and ( x + 1 , y ) is { n, n + 1 , n + 2 } for some nonnegative integer n . Determine, with proof, all possible labels for the point (2000 , 2024) . 2 2

解析

[30] Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such a way that the point (0 , 0) is labeled by 0 , and for every x, y ≥ 0 , the set of numbers labeled on the points ( x, y ) , ( x, y + 1) , and ( x + 1 , y ) is { n, n + 1 , n + 2 } for some nonnegative integer n . Determine, with proof, all possible labels for the point (2000 , 2024) . Proposed by: Nithid Anchaleenukoon Answer: 0 , 3 , 6 , 9 , . . . , 6048 Solution: We claim the answer is all multiples of 3 from 0 to 2000 + 2 · 2024 = 6048 . First, we prove no other values are possible. Let ℓ ( x, y ) denote the label of cell ( x, y ) . The label is divisible by 3. Observe that for any x and y , ℓ ( x, y ) , ℓ ( x, y + 1) , and ℓ ( x + 1 , y ) are all distinct mod 3 . Thus, for any a and b , ℓ ( a + 1 , b + 1) cannot match ℓ ( a + 1 , b ) or ℓ ( a, b + 1) mod 3 , so it must be equivalent to ℓ ( a, b ) modulo 3. Since ℓ ( a, b + 1) , ℓ ( a, b + 2) , ℓ ( a + 1 , b + 1) are all distinct mod 3 , and ℓ ( a + 1 , b + 1) and ℓ ( a, b ) are equivalent mod 3 , then ℓ ( a, b ) , ℓ ( a, b + 1) , ℓ ( a, b + 2) are all distinct mod 3 , and thus similarly ℓ ( a, b +