HMMT 二月 2000 · ORAL 赛 · 第 9 题
HMMT February 2000 — ORAL Round — Problem 9
题目详情
英文原题
- [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
英文解析
- This is a special case of the problem with n + 2 vectors in n dimensions. First it isclear that we can take all the vectors to be of length 1. Then we induct on n. The firstcase is n=1. Here the statement is that given a, b, c real numbers, the at least one ofab , bc , and ac is nonnegative. Without loss of generality, we can assume that a and bare 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 then +2 − →
′
− − → − → − → − − → − − →
space perpendicular to v to get v = v − ( v · v ) v . This is essentially takingn +2 i i n +2 n +2 − − →iout the space parallel to v and reducing the problem by one dimension. The onlyn +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 ai ji j − → − →
′ ′
− → − → − → − − → − → − − → − − → − − → − → − − → − → − − →
calculation: v · v = v · v − 2( v · v )( v · v )+( v · v )( v · v )( v · v ) andi 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 ji j
23