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National and Regional Contests
China Contests
(China) National High School Mathematics League
2022 China Second Round
3
China Second Round Olympiad 2022 Test 2 Q 3
China Second Round Olympiad 2022 Test 2 Q 3
Source:
September 11, 2022
algebra
Inequality
China
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
100
a_1,a_2,\cdots ,a_{100}
a
1
,
a
2
,
⋯
,
a
100
be non-negative integers such that
(
1
)
(1)
(
1
)
There are positive integers
k
≤
100
k\leq 100
k
≤
100
such that
a
1
≤
a
2
≤
⋯
≤
a
k
a_1\leq a_2\leq \cdots\leq a_{k}
a
1
≤
a
2
≤
⋯
≤
a
k
and
a
i
=
0
a_i=0
a
i
=
0
(
i
>
k
)
;
(i>k);
(
i
>
k
)
;
(
2
)
(2)
(
2
)
a
1
+
a
2
+
a
3
+
⋯
+
a
100
=
100
;
a_1+a_2+a_3+\cdots +a_{100}=100;
a
1
+
a
2
+
a
3
+
⋯
+
a
100
=
100
;
(
3
)
(3)
(
3
)
a
1
+
2
a
2
+
3
a
3
+
⋯
+
100
a
100
=
2022.
a_1+2a_2+3a_3+\cdots +100a_{100}=2022.
a
1
+
2
a
2
+
3
a
3
+
⋯
+
100
a
100
=
2022.
Find the minimum of
a
1
+
2
2
a
2
+
3
2
a
3
+
⋯
+
10
0
2
a
100
.
a_1+2^2a_2+3^2a_3+\cdots +100^2a_{100}.
a
1
+
2
2
a
2
+
3
2
a
3
+
⋯
+
10
0
2
a
100
.
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