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Korea Winter Program Practice Test
2020 Korean MO winter camp
#2
Minimizing sum of squares
Minimizing sum of squares
Source: 2020 Korean MO winter camp Test 1 P2
September 7, 2020
algebra
Problem Statement
X
X
X
is a set of
2020
2020
2020
distinct real numbers. Prove that there exist
a
,
b
∈
R
a,b\in \mathbb{R}
a
,
b
∈
R
and
A
⊂
X
A\subset X
A
⊂
X
such that
∑
x
∈
A
(
x
−
a
)
2
+
∑
x
∈
X
\
A
(
x
−
b
)
2
≤
1009
1010
∑
x
∈
X
x
2
\sum_{x\in A}(x-a)^2 +\sum_{x\in X\backslash A}(x-b)^2\le \frac{1009}{1010}\sum_{x\in X}x^2
x
∈
A
∑
(
x
−
a
)
2
+
x
∈
X
\
A
∑
(
x
−
b
)
2
≤
1010
1009
x
∈
X
∑
x
2
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