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Moscow Mathematical Olympiad
1954 Moscow Mathematical Olympiad
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1954 Moscow Mathematical Olympiad
Problems
(1)
MMO 281 Moscow MO 1954 sum x_i^2 >10000, sum x_1<300
Source:
8/13/2019
*. Positive numbers
x
1
,
x
2
,
.
.
.
,
x
100
x_1, x_2, ..., x_{100}
x
1
,
x
2
,
...
,
x
100
satisfy the system
{
x
1
2
+
x
2
2
+
.
.
.
+
x
100
2
>
10000
x
1
+
x
2
+
.
.
.
+
x
100
<
300
\begin{cases} x^2_1+ x^2_2+ ... + x^2_{100} > 10 000 \\ x_1 + x_2 + ...+ x_{100} < 300 \end{cases}
{
x
1
2
+
x
2
2
+
...
+
x
100
2
>
10000
x
1
+
x
2
+
...
+
x
100
<
300
Prove that among these numbers there are three whose sum is greater than
100
100
100
.
Sum
system of inequalities
inequalities
algebra