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Moscow Mathematical Olympiad
1956 Moscow Mathematical Olympiad
332
MMO 332 Moscow MO 1956 4x4 parameter system
MMO 332 Moscow MO 1956 4x4 parameter system
Source:
August 19, 2019
algebra
system of equations
parameterization
Problem Statement
Prove that the system of equations
{
x
1
−
x
2
=
a
x
3
−
x
4
=
b
x
1
+
x
2
+
x
3
+
x
4
=
1
\begin{cases} x_1 - x_2 = a \\ x_3 - x_4 = b \\ x_1 + x_2 + x_3 + x_4 = 1\end{cases}
⎩
⎨
⎧
x
1
−
x
2
=
a
x
3
−
x
4
=
b
x
1
+
x
2
+
x
3
+
x
4
=
1
has at least one solution in positive numbers (
x
1
,
x
2
,
x
3
,
x
4
>
0
x_1 ,x_2 ,x_3 ,x_4>0
x
1
,
x
2
,
x
3
,
x
4
>
0
) if and only if
∣
a
∣
+
∣
b
∣
<
1
|a| + |b| < 1
∣
a
∣
+
∣
b
∣
<
1
.
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