HMMT 十一月 2008 · GEN2 赛 · 第 3 题

HMMT November 2008 — GEN2 Round — Problem 3

[ 5 ] Let DEF be a triangle and H the foot of the altitude from D to EF . If DE = 60, DF = 35, and DH = 21, what is the difference between the minimum and the maximum possible values for the area of DEF ? 3

解析

[ 5 ] Joe B. now clears the board. How many ways can he place 3 white rooks and 3 black rooks on the board so that no two rooks of opposite color can attack each other? Answer: 608 Consider first placing the white rooks. They will occupy either 3 columns and 1 row, 3 columns and 2 rows, 3 columns and 3 rows, 2 rows and 2 columns, 2 columns and 3 rows, or 1 column and 3 rows. First note that placing the black rooks is impossible in the second, third, and fifth cases. The first case can happen in 4 · 4 ways, and each leads to a unique way to place the black ( ) 4 rooks. In the fourth case, we can choose the row with 2 rooks in 4 ways, place the rooks in ways, 2 ( ) 4 choose the column of the other rook in 2 ways, and place it in 3 ways, for a total of 4 · · 2 · 3 = 144 2 ways to place the white rooks in this configuration; the black rooks can then be placed in any of 4 possible locations, and there are 4 ways to do this, leading to 576 possibilities in this case. Finally, the sixth case is analogous to the first, giving 16 possibilities. Summing, we find 16 + 576 + 16 = 608 total placements.