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2019 HMIC
3
Four points at large distance
Four points at large distance
Source: HMIC 2019 Problem 3
April 28, 2019
algebra
HMIC
Problem Statement
Do there exist four points
P
i
=
(
x
i
,
y
i
)
∈
R
2
(
1
≤
i
≤
4
)
P_i = (x_i, y_i) \in \mathbb{R}^2\ (1\leq i \leq 4)
P
i
=
(
x
i
,
y
i
)
∈
R
2
(
1
≤
i
≤
4
)
on the plane such that:[*] for all
i
=
1
,
2
,
3
,
4
i = 1,2,3,4
i
=
1
,
2
,
3
,
4
, the inequality
x
i
4
+
y
i
4
≤
x
i
3
+
y
i
3
x_i^4 + y_i^4 \le x_i^3+ y_i^3
x
i
4
+
y
i
4
≤
x
i
3
+
y
i
3
holds, and [*] for all
i
≠
j
i \neq j
i
=
j
, the distance between
P
i
P_i
P
i
and
P
j
P_j
P
j
is greater than
1
1
1
?Pakawut Jiradilok
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