I'm dealing with the test of the International Mathematics Competition for University Students, 2011, and I've had a lot of difficulties, so I hope someone could help me to discuss the questions. I've posted the question 2 at here.
The question 5 says:
Let be $n$ a positive integer and $V$ a vector space with dimension $2n-1$ over the field of two elements. Prove that for any arbitrary vectors $v_1,v_2,\dots,v_{4n-1}$, there is a sequence $1\leq i_1\lt i_2\lt \dots \lt i_{2n}\leq 4n-1$ of indexs such that $v_{i_1}+v_{i_2}+\dots+v_{i_{2n}}=0$.
Well, my little attempts was these:
Consider all the sets linearly independent (LI) of those vectors. Once the sets have finite amount, we have some that has the maximum amount of vectors. Supose that are $\{v_1,v_2,\dots, v_k\}$.
As the dimension of $V$ is $2n-1$, we have $k\leq 2n-1$.
Take $v_1+v_2+\dots+v_{4k-1}$. We have that this sum can be writed as a linear combination of $\{v_1,v_2,\dots, v_k\}$.
But we are on binary system and the linear combinations are simple sums. In fact, the constants are only $0$ and $1$. So, we can write $v_1+v_2+\dots+v_{4k-1}$ as a sum of vectors from $\{v_1,v_2,\dots, v_k\}$, with $d$ terms ($d\leq k\leq 2n-1$).
From that equality, we cancel these $d$ vectors in both sides and get a sum of $4n-1-d\geq 2n$ vectors that is null. Is it correct?
Well, we can take the same way and get other smaller sums $=0$. But I cannot prove that there's a null sum of EXACTLY $2n$ vectors.
Thanks very much for help.
Important Edit (October, 04)
I've found a document with these solutions and I'm studying them. These are the documents: http://www.imc-math.org.uk/imc2011/imc2011-day2-solutions.pdf and http://www.imc-math.org.uk/imc2011/imc2011-day1-solutions.pdf.
Edited October, 11
I need a help at the answer of the document above in that part:
So, I couldn't understand very well because $\tilde{v_i}$ or sets, not vector as I'm familiarized. But I think I can get this; however, I couldn't understand as I pass from here to
"Then, $v_1+v_2,v_3+v_4,\cdots, v_{2d-1}+v_{2d}$ is a basis of $V$".
Could someone explain better or indicate some material? Thank you very much.