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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1941 Eotvos Mathematical Competition
1941 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
1
1
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prod (1 + x) ... (1 + x^2{k-1} )
Prove that
(
1
+
x
)
(
1
+
x
2
)
(
1
+
x
4
)
(
1
+
x
8
)
.
.
.
(
1
+
x
2
k
−
1
)
=
1
+
x
+
x
2
+
x
3
+
.
.
.
+
x
2
k
−
1
(1 + x)(1 + x^2)(1 + x^4)(1 + x^8) ... (1 + x^{2^{k-1}} ) = 1 + x + x^2 + x^3 +... + x^{2^{k-1}}
(
1
+
x
)
(
1
+
x
2
)
(
1
+
x
4
)
(
1
+
x
8
)
...
(
1
+
x
2
k
−
1
)
=
1
+
x
+
x
2
+
x
3
+
...
+
x
2
k
−
1
3
1
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equilateral bye midpoints of 2 equal sides to radius, of cyclic hexagon
The hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is inscribed in a circle. The sides
A
B
AB
A
B
,
C
D
CD
C
D
and
E
F
EF
EF
are all equal in length to the radius. Prove that the midpoints of the other three sides determine an equilateral triangle.
2
1
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all four vertices of a parallelogram are lattice points , area >1
Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds
1
1
1
.