HMMT 二月 2011 · 冲刺赛 · 第 23 题

英文解析

23. [ 14 ] Let S be the set of points ( x, y, z ) in R such that x , y , and z are positive integers less than orequal to 100. Let f be a bijective map between S and the { 1 , 2 , . . . , 1000000 } that satisfies the followingproperty: if x ≤ x , y ≤ y , and z ≤ z , then f ( x , y , z ) ≤ f ( x , y , z ). Define
    1 2 1 2 1 2 1 1 1 2 2 2
    100 100
    ∑ ∑
    A = f ( i, j, k ) ,
    j =1 k =1 i
    100 100
    ∑ ∑
    B = f ( j, i, k ) ,
    j =1 k =1 i
    100 100
    ∑ ∑
    and C = f ( j, k, i ) .
    j =1 k =1 i
    Determine the minimum value of A − A + B − B + C − C .
    i +1 i j +1 j k +1 k
    Answer: 30604
    We examine the 6 planes, their intersections and the lines between 2 points in one of the three pairsof parallel planes. The expression is equivalent to summing differences in values along all these lines.
    We examine the planes intersections. There is one cube, 3 · 98 squares and 3 · 98 · 98 lines. Theminimum value of the difference along a line is 1. For a square, to minimize the differences we takefour consecutive numbers, and the minimum value is 6. To find the minimum value along a cube,
    we take 8 consecutive numbers. Since we are taking differences, we can add or subtract any constantto the numbers, so we assume the numbers are 1-8. Examining the cube, we see there’s 1 spotwhere the number is multiplied by -3, 3 spots where the number is multiplied by -1, 1 spot where
    Guts Roundthe number is multiplied by 3, and 3 spots where the number is multiplied by 1. 1 and 8 must goin the corners, and 2,3,4,5 must go in spots multiplied by -1, -1, 1,1, respectively. To minimize the differences we put 5 in the final spot multiplied by -1, and 4 in the spot multiplied by 1 opposite