HMMT 二月 2002 · 团队赛 · 第 7 题
HMMT February 2002 — Team Round — Problem 7
题目详情
英文原题
- [45] Let T = n / 9 for positive integers L . Determine all L for which T is a square
L Ln =1
number.
Luck of the dice. Problems 8–12 concern a two-player game played on a board consisting of fourteenspaces in a row. The leftmost space is labeled START , and the rightmost space is labeled END .
Each of the twelve other squares, which we number 1 through 12 from left to right, may be blankor may be labeled with an arrow pointing to the right. The term blank square will refer to one ofthese twelve squares that is not labeled with an arrow. The set of blank squares on the board will be called a board configuration ; the board below uses the configuration { 1 , 2 , 3 , 4 , 7 , 8 , 10 , 11 , 12 } .
START ⇒ ⇒ ⇒ END
1 2 3 4 5 6 7 8 9 10 11 12
For i ∈ { 1 , 2 } , player i has a die that produces each integer from 1 to s with probability 1 /s . Herei is and s are positive integers fixed before the game begins. The game rules are as follows:
1 2
解析
英文解析
- [45] Let T = n / 9 for positive integers L . Determine all L for which T is a square
L Ln =1
number.
Solution. Since T is square if and only if 9 T is square, we may consider 9 T instead of T .
L L L L
It is well known that n is congruent to 0, 1, or 8 modulo 9 according as n is congruent to 0, 1,3
3 3 2 2 3 3
or 2 modulo 3. (Proof: (3 m + k ) = 27 m + 3(9 m ) k + 3(3 m ) k + k ≡ k (mod 9).) Therefore
⌊ ⌋2
3 3
n − 9 n / 9 is 0, 1, or 8 according as n is congruent to 0, 1, or 2 modulo 3. We find therefore that
⌊ ⌋
∑3
9 T = 9 n
9L
1 ≤ n ≤ L
∑
= n − # { 1 ≤ n ≤ L : n ≡ 1 (mod 3) } − 8# { 1 ≤ n ≤ L : n ≡ 2 (mod 3) }3
1 ≤ n ≤ L
( ) ⌊ ⌋ ⌊ ⌋
1 L + 2 L + 12 = L ( L + 1) − − 8 .
2 3 3
( ) ( )
2 2
Clearly 9 T < L ( L + 1) / 2 for L ≥ 1. We shall prove that 9 T > L ( L + 1) / 2 − 1 for L ≥ 4,
L Lwhence 9 T is not square for L ≥ 4. Because
( ) ( )L
2 2
L ( L + 1) / 2 − 1 = L ( L + 1) / 2 − L ( L + 1) + 1 ,
we need only show that
⌊ ⌋ ⌊ ⌋
L + 2 L + 1 + 8 ≤ L + L − 2 .2
3 3
But the left-hand side of this is bounded above by 3 L +10 / 3, and the inequality 3 L +10 / 3 ≤ L + L − 22
2 2
means exactly L − 2 L − 16 / 3 ≥ 0 or ( L − 1) ≥ 19 / 3, which is true for L ≥ 4, as desired.
Hence T is not square for L ≥ 4. By direct computation we find T = T = 0 and T = 3, so T is
L 1 2 3 Lsquare only for L ∈ { 1 , 2 } .
Luck of the dice. Problems 8–12 concern a two-player game played on a board consisting of fourteenspaces in a row. The leftmost space is labeled START , and the rightmost space is labeled END .
Each of the twelve other squares, which we number 1 through 12 from left to right, may be blankor may be labeled with an arrow pointing to the right. The term blank square will refer to one ofthese twelve squares that is not labeled with an arrow. The set of blank squares on the board will be called a board configuration ; the board below uses the configuration { 1 , 2 , 3 , 4 , 7 , 8 , 10 , 11 , 12 } .
START ⇒ ⇒ ⇒ END
1 2 3 4 5 6 7 8 9 10 11 12
For i ∈ { 1 , 2 } , player i has a die that produces each integer from 1 to s with probability 1 /s . Herei is and s are positive integers fixed before the game begins. The game rules are as follows:
1 2