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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1931 Eotvos Mathematical Competition
1931 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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max 1/(1 + AP)+1/(1+BP) whn P lies on line AB, with AB=1
Let
A
A
A
and
B
B
B
be two given points, distance
1
1
1
apart. Determine a point
P
P
P
on the line
A
B
AB
A
B
such that
1
1
+
A
P
+
1
1
+
B
P
\frac{1}{1 + AP}+\frac{1}{1 + BP}
1
+
A
P
1
+
1
+
BP
1
is a maximum.
2
1
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a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2, not all odd`
Let
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
+
a
5
2
=
b
2
a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2
a
1
2
+
a
2
2
+
a
3
2
+
a
4
2
+
a
5
2
=
b
2
, where
a
1
a_1
a
1
,
a
2
a_2
a
2
,
a
3
a_3
a
3
,
a
4
a_4
a
4
,
a
5
a_5
a
5
, and
b
b
b
are integers. Prove that not all of these numbers can be odd.
1
1
Hide problems
2/p can be expressed in exactly one way in the form 1/x+1/y
Let
p
p
p
be a prime greater than
2
2
2
. Prove that
2
p
\frac{2}{p}
p
2
can be expressed in exactly one way in the form
1
x
+
1
y
\frac{1}{x}+\frac{1}{y}
x
1
+
y
1
where
x
x
x
and
y
y
y
are positive integers with
x
>
y
x > y
x
>
y
.