1976 IMO Problems/Problem 5

Problem

We consider the following system with $q = 2p$ :

$\begin{matrix} a_{11}x_{1} + \ldots + a_{1q}x_{q} = 0, \\ a_{21}x_{1} + \ldots + a_{2q}x_{q} = 0, \\ \ldots , \\ a_{p1}x_{1} + \ldots + a_{pq}x_{q} = 0, \\ \end{matrix}$

in which every coefficient is an element from the set $\{ - 1,0,1\}$ $.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:

a.) all $x_{j}, j = 1,\ldots,q$ are integers $;$

b.) there exists at least one j for which $x_{j} \neq 0;$

c.) $|x_{j}| \leq q$ for any $j = 1, \ldots ,q.$

Solution

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1976 IMO (Problems) • Resources
Preceded by Problem 4	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 6
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1976 IMO Problems/Problem 5

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