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2018 Indonesia MO
5
A bit of a minimum, more of a maximum
A bit of a minimum, more of a maximum
Source: Indonesian National Science Olympiad 2018, Mathematics P5
July 6, 2018
algebra
Problem Statement
Find all triples of reals
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
satisfying:
{
1
3
min
{
x
,
y
}
+
2
3
max
{
x
,
y
}
=
2017
1
3
min
{
y
,
z
}
+
2
3
max
{
y
,
z
}
=
2018
1
3
min
{
z
,
x
}
+
2
3
max
{
z
,
x
}
=
2019
\begin{cases} \frac{1}{3} \min \{x,y\} + \frac{2}{3} \max \{x,y\} = 2017 \\ \frac{1}{3} \min \{y,z\} + \frac{2}{3} \max \{y,z\} = 2018 \\ \frac{1}{3} \min \{z,x\} + \frac{2}{3} \max \{z,x\} = 2019 \\ \end{cases}
⎩
⎨
⎧
3
1
min
{
x
,
y
}
+
3
2
max
{
x
,
y
}
=
2017
3
1
min
{
y
,
z
}
+
3
2
max
{
y
,
z
}
=
2018
3
1
min
{
z
,
x
}
+
3
2
max
{
z
,
x
}
=
2019
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