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Moscow Mathematical Olympiad
1997 Moscow Mathematical Olympiad
4
1997 MMO Grade 10 #4
1997 MMO Grade 10 #4
Source:
October 3, 2016
Grade 10
1997
Problem Statement
Given real numbers
a
1
≤
a
2
≤
a
3
a_1\leq{a_2}\leq{a_3}
a
1
≤
a
2
≤
a
3
and
b
1
≤
b
2
≤
b
3
b_1\leq{b_2}\leq{b_3}
b
1
≤
b
2
≤
b
3
such that
a
1
+
a
2
+
a
3
=
b
1
+
b
2
+
b
3
,
a_1+a_2+a_3=b_1+b_2+b_3,
a
1
+
a
2
+
a
3
=
b
1
+
b
2
+
b
3
,
a
1
a
2
+
a
2
a
3
+
a
1
a
3
=
b
1
b
2
+
b
2
b
3
+
b
1
b
3
.
a_1a_2+a_2a_3+a_1a_3=b_1b_2+b_2b_3+b_1b_3.
a
1
a
2
+
a
2
a
3
+
a
1
a
3
=
b
1
b
2
+
b
2
b
3
+
b
1
b
3
.
Prove that if
a
1
≤
b
1
,
a_1\leq{b_1},
a
1
≤
b
1
,
then
a
3
≤
b
3
a_3\leq{b_3}
a
3
≤
b
3
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