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Contests
National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1911 Eotvos Mathematical Competition
1911 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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3^n + 1 is not divisible by 2^n
Prove that
3
n
+
1
3^n + 1
3
n
+
1
is not divisible by
2
n
2^n
2
n
for any integer
n
>
1
n > 1
n
>
1
.
2
1
Hide problems
sum of 4th powers of a point to 4 diameters of rexygar octagon is fixed
Let
Q
Q
Q
be any point on a circle and let
P
1
P
2
P
3
.
.
.
P
8
P_1P_2P_3...P_8
P
1
P
2
P
3
...
P
8
be a regular inscribed octagon. Prove that the sum of the fourth powers of the distances from
Q
Q
Q
to the diameters
P
1
P
5
P_1P_5
P
1
P
5
,
P
2
P
6
P_2P_6
P
2
P
6
,
P
3
P
7
P_3P_7
P
3
P
7
,
P
4
P
8
P_4P_8
P
4
P
8
is independent of the position of
Q
Q
Q
.
1
1
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AC - B^2 < 0 if aC -2bB + cA = 0 and ac - bz > 0
Show that, if the real numbers
a
,
b
,
c
,
A
,
B
,
C
a, b, c, A, B, C
a
,
b
,
c
,
A
,
B
,
C
satisfy
a
C
−
2
b
B
+
c
A
=
0
a
n
d
a
c
−
b
2
>
0
,
aC -2bB + cA = 0 \ \ and \ \ ac - b^2 > 0,
a
C
−
2
b
B
+
c
A
=
0
an
d
a
c
−
b
2
>
0
,
then
A
C
−
B
2
≤
0.
AC - B^2 \le 0.
A
C
−
B
2
≤
0.