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2021 Mediterranean Mathematics Olympiad
4
\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5} <= 20
\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5} <= 20
Source: 2021 Mediterranean Mathematical Olympiad P4 MMC
September 11, 2021
inequalities
algebra
Problem Statement
Let
x
1
,
x
2
,
x
3
,
x
4
,
x
5
x_1,x_2,x_3,x_4,x_5
x
1
,
x
2
,
x
3
,
x
4
,
x
5
ve non-negative real numbers, so that
x
1
≤
4
x_1\le4
x
1
≤
4
and
x
1
+
x
2
≤
13
x_1+x_2\le13
x
1
+
x
2
≤
13
and
x
1
+
x
2
+
x
3
≤
29
x_1+x_2+x_3\le29
x
1
+
x
2
+
x
3
≤
29
and
x
1
+
x
2
+
x
3
+
x
4
≤
54
x_1+x_2+x_3+x_4\le54
x
1
+
x
2
+
x
3
+
x
4
≤
54
and
x
1
+
x
2
+
x
3
+
x
4
+
x
5
≤
90
x_1+x_2+x_3+x_4+x_5\le90
x
1
+
x
2
+
x
3
+
x
4
+
x
5
≤
90
. Prove that
x
1
+
x
2
+
x
3
+
x
4
+
x
5
≤
20
\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5}\le20
x
1
+
x
2
+
x
3
+
x
4
+
x
5
≤
20
.
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