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Regional Mathematical Olympiad
2024 India Regional Mathematical Olympiad
4
RMO 2024 Q4
RMO 2024 Q4
Source: RMO 2024 Q4
November 3, 2024
algebra
inequalities
Problem Statement
Let
a
1
,
a
2
,
a
3
,
a
4
a_1,a_2,a_3,a_4
a
1
,
a
2
,
a
3
,
a
4
be real numbers such that
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
=
1
a_1^2 + a_2^2 + a_3^2 + a_4^2 = 1
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
=
1
. Show that there exist
i
,
j
i,j
i
,
j
with
1
≤
i
<
j
≤
4
1 \leq i < j \leq 4
1
≤
i
<
j
≤
4
, such that
(
a
i
−
a
j
)
2
≤
1
5
(a_i - a_j)^2 \leq \frac{1}{5}
(
a
i
−
a
j
)
2
≤
5
1
.
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