HMMT 二月 2000 · ORAL 赛 · 第 9 题
HMMT February 2000 — ORAL Round — Problem 9
题目详情
- [60] Let v , v , v , v and v be vectors in three dimensions. Show that for some i, j in 1 2 3 4 5 − → − → 1 , 2 , 3 , 4 , 5, v · v ≥ 0. i j
解析
- This is a special case of the problem with n + 2 vectors in n dimensions. First it is clear that we can take all the vectors to be of length 1. Then we induct on n. The first case is n=1. Here the statement is that given a, b, c real numbers, the at least one of ab , bc , and ac is nonnegative. Without loss of generality, we can assume that a and b are of the same sign, but then ab ≥ 0. Now assume the statement is false for the n- − − → dimensional case. Choose some vector, say v , and project the other vectors onto the n +2 − → ′ − − → − → − → − − → − − → space perpendicular to v to get v = v − ( v · v ) v . This is essentially taking n +2 i i n +2 n +2 i − − → out the space parallel to v and reducing the problem by one dimension. The only n +2 − → − → ′ ′ − → − → thing left to check is that if v · v < 0 for all i,j then v · v < 0 for all i,j. This is just a i j i j − → − → ′ ′ − → − → − → − − → − → − − → − − → − − → − → − − → − → − − → calculation: v · v = v · v − 2( v · v )( v · v )+( v · v )( v · v )( v · v ) and i j j n +2 i n +2 n +2 n +2 j n +2 i n +2 i j − − → − − → − → − → − → − − → − → − − → − → − − → − → − − → since ( v · v ) = 1, this is just v · v − ( v · v )( v · v ). ( v · v ) and ( v · v ) n +2 n +2 i j j n +2 i n +2 j n +2 i n +2 − → − → ′ ′ − → − → are both negative by assumption so their product is positive and v · v < v · v < 0. i j i j 23