HMMT 二月 2013 · 冲刺赛 · 第 10 题

英文解析

10. [ 6 ] Wesyu is a farmer, and she’s building a cao (a relative of the cow) pasture. She starts with a
    ° °
    triangle A A A where angle A is 90 , angle A is 60 , and A A is 1. She then extends the pasture.
    0 1 2 0 1 0 1
    First, she extends A A to A such that A A = A A and the new pasture is triangle A A A .1
    2 0 3 3 0 2 0 1 2 3
    Next, she extends A A to A such that A A = A A . She continues, each time extending A A12
    3 1 4 4 1 3 1 n n − 2
    to A such that A A = A A . What is the smallest K such that her pasture never 16
    n +1 n +1 n − 2 n n n − 2
    2 − 2
    exceeds an area of K ?
    √
    Answer: 3 First, note that for any i , after performing the operation on triangle A A A ,
    i i +1 i +2
    the resulting pasture is triangle A A A . Let K be the area of triangle A A A . Fromi +1 i +2 i +3 i i i +1 i +2
    Guts Round
    1 1
    A A = A A and A A = A A + A A , we have A A = (1+ ) A A .
    n +1 n − 2 n n n − 2 n n +1 n n − 2 n − 2 n +1 n n +1 n n n − 2
    2 − 2 2 − 2
    We also know that the area of a triangle is half the product of its base and height, so if we let the base oftriangle A A A be A A , its area is K = h a A . The area of triangle A A A1
    n − 2 n − 1 n n n − 2 n − 2 n n − 2 n − 1 n n +1
    is K = h a A . The h ’s are equal because the distance from A to the base does not change.12
    n − 1 n n +1 n − 1
    n 2
    K A A
    K K K Kn − 1 n n +1 1 2 − 1 3 7 3 7
    1 2 2 1
    We now have = = 1 + = . Therefore, = , = = · = ,
    n n
    K A A 2 − 2 2 − 2 K 2 K K K 6 2 4
    n − 2 n n − 2 0 0 1 0
    n +1
    K K K K
    15 7 15 2 − 1
    3 3 2 n = = · = . We see the pattern = , which can be easily proven by induction.
    K K K 14 4 8 K 2 n
    0 2 0 0
    As n approaches infinity, grows arbitrarily close to 2, so the smallest K such that the pasture nevern k
    0K
    √
    exceeds an area of K is 2 K = 3. 0