HMMT 二月 2024 · 团队赛 · 第 1 题
HMMT February 2024 — Team Round — Problem 1
题目详情
- [20] Let a , a , a , . . . , a be integers such that 1 2 3 100 2 2 2 2 a + a + a + · · · + a 1 2 3 100 = 100 . a + a + a + · · · + a 1 2 3 100 Determine, with proof, the maximum possible value of a . 1
解析
- , ℓ ( a, b + 2) , ℓ ( a, b + 3) are all distinct mod 3 , which means that ℓ ( a, b + 3) must be neither ℓ ( a, b + 1) or ℓ ( a, b + 2) mod 3 , and thus must be equal to ℓ ( a, b ) mod 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · · · · · · · · · · · · → → → · · · · · · · · · · · · 2 2 0 2 0 · · · · · · · · · · · · 0 0 1 0 1 0 1 2 These together imply that ℓ ( w, x ) ≡ ℓ ( y, z ) mod 3 ⇐⇒ w − x ≡ y − z mod 3 . It follows that ℓ (2000 , 2024) must be equivalent to ℓ (0 , 0) mod 3 , which is a multiple of 3 . The label is at most 6048. Note that since ℓ ( x + 1 , y ) , ℓ ( x, y + 1) , and ℓ ( x, y ) are 3 consecutive numbers, ℓ ( x + 1 , y ) − ℓ ( x, y ) and ℓ ( x, y + 1) − ℓ ( x, y ) are both ≤ 2 . Moreover, since ℓ ( x + 1 , y + 1) ≤ ℓ ( x, y ) + 4 , since it is also the same mod 3 , it must be at most ℓ ( x, y ) + 3 . Thus, ℓ (2000 , 2000) ≤ ℓ (0 , 0) + 3 · 2000 , and ℓ (2000 , 2024) ≤ ℓ (2000 , 2000) + 2 · 24 , so ℓ (2000 , 2024) ≤ 6048 . Construction. Consider lines ℓ of the form x + 2 y = n (so (2000 , 2024) lies on ℓ ). Then any three points of the n 6048 form ( x, y ) , ( x, y + 1) , and ( x + 1 , y ) lie on three consecutive lines ℓ , ℓ , ℓ in some order. Thus, n n +1 n +2 for any k which is a multiple of 3 , if we label every point on line ℓ with max( i mod 3 , i − k ) , any three i consecutive lines ℓ , ℓ , ℓ will either be labelled 0 , 1 , and 2 in some order, or n − k , n − k + 1 , n n +1 n +2 n − k + 2 , both of which consist of three consecutive numbers. Below is an example with k = 6 . 8 8 9 10 11 12 13 14 15 7 6 6 7 8 9 10 13 11 12 5 4 4 5 6 7 8 9 10 11 3 2 3 5 6 7 8 9 2 4 1 → 0 0 1 2 3 4 5 6 7 2 1 0 3 5 1 2 1 2 4 0 2 2 0 1 2 0 1 2 3 1 0 0 0 0 1 2 1 2 1 Any such labelling is valid, and letting k range from 0 to 6048 , we see (2000 , 2024) can take any label of the form 6048 − k , which spans all such multiples of 3 . Hence the possible labels are precisely the multiples of 3 from 0 to 6048 . 2 2