## SEEMOUS 2011 Problem 3

Given vectors , show that

, with the usual notations.

*SEEMOUS 2010 Problem 3*

**Solution: **First, reduce the problem to . How we can do that? Consider an orthonormal basis for which the first three vectors span the vector space spanned by . An orthonormal transformation preserves the norm and the inner product, therefore, it would be enough to prove the inequality in the new basis, a basis in which have only the first three components non-zero. Therefore, if we solve the problem for we are done. But in we can reduce the problem to a geometric one. First, note that changing the sign of some of the vectors does not change anything in the inequality. If we denote . Arrange the signs such that and drawing a trihedron angle with supports of its sides determined by the projection of on the plane would be inside the angle .

It is easy to prove that if satisfy then . The inequality we want to prove is . Note that if one of is then we are done.

Project on the plane in point , and denote . Project on in . By expressing cosine in a right triangle, we get . Similar to this we get . Use the given identity for and get . Let’s prove that

$latex 1+2\cos\alpha_1\cos\beta_1\cos\gamma-\cos^2\gamma$ . Of course, if we are done. Else, the inequality is equivalent to . Finally, this last inequality is equivalent to , which is true.

This solves the problem for . For the inequality is trivial, and for we do something similar to the arguments above, but we are no longer in space, and in the plane, the problem is even easier.

Eu tot geometric ma gandisem. Si tot ca tine am presupus ca . Trebuie sa aratam ca . Avem deci deci putem presupune ca . Insa sunt unghirile triedrului (care poate fi si degenerat) deci avem . Rezulta ca , care, impreuna cu implica .

E adevarat ca la problema asta s-au dat mai multe solutii distincte? Asa am auzit.

O solutie foarte rapida ar fi sa scriem determinantul Gramm pentru 3 vectori oarecare a,b,c avand dimensiunea n. Stim ca are determinantul mai mare sau egal decat 0 si se obtine imediat concluzia.

(Enghish: A fast solution would be to write the Gramm determinant for the three vectors.)

Ce stiti despre studentul care a fost descalificat pentru ca a dat solutiile din lista scurta?

Nu am auzit despre acest incident. Daca intr-adevar au fost scurgeri de informatii de la profesori catre studenti, acest fapt este foarte trist.