Difference between revisions of "1976 IMO Problems/Problem 5"
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== Problem == | == Problem == | ||
| − | {{ | + | We consider the following system |
| + | with <math>q = 2p</math>: | ||
| + | |||
| + | <math>\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}</math> | ||
| + | |||
| + | in which every coefficient is an element from the set <math>\{ - 1,0,1\}</math><math>.</math> Prove that there exists a solution <math>x_{1}, \ldots,x_{q}</math> for the system with the properties: | ||
| + | |||
| + | '''a.)''' all <math>x_{j}, j = 1,\ldots,q</math> are integers<math>;</math> | ||
| + | |||
| + | '''b.)''' there exists at least one j for which <math>x_{j} \neq 0;</math> | ||
| + | |||
| + | '''c.)''' <math>|x_{j}| \leq q</math> for any <math>j = 1, \ldots ,q.</math> | ||
== Solution == | == Solution == | ||
Revision as of 10:45, 26 February 2008
Problem
We consider the following system
with 
:
in which every coefficient is an element from the set 
Prove that there exists a solution 
for the system with the properties:
a.) all 
are integers
b.) there exists at least one j for which 
c.) 
for any 
Solution
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See also
| 1976 IMO (Problems) • Resources | ||
| Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
