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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1935 Eotvos Mathematical Competition
1935 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
1
1
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sum a_i /b_i >= n where b_i is permutation of a_i
Let
n
n
n
be a positive integer. Prove that
a
1
b
1
+
a
2
b
2
+
.
.
.
+
a
n
b
n
≥
n
\frac{a_1}{b_1}+ \frac{a_2}{b_2}+ ...+\frac{a_n}{b_n} \ge n
b
1
a
1
+
b
2
a
2
+
...
+
b
n
a
n
≥
n
where
(
b
1
,
b
2
,
.
.
.
,
b
n
)
(b_1, b_2, ..., b_n)
(
b
1
,
b
2
,
...
,
b
n
)
is any permutation of the positive real numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
.
3
1
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real numbers on each vertex of a triangular prism
A real number is assigned to each vertex of a triangular prism so that the number on any vertex is the arithmetic mean of the numbers on the three adjacent vertices. Prove that all six numbers are equal.
2
1
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finite point set cannot have more than one center of symmetry
Prove that a finite point set cannot have more than one center of symmetry.