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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1936 Eotvos Mathematical Competition
1936 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
1
1
Hide problems
sum 1/(2k-1)2k
Prove that for all positive integers
n
n
n
,
1
1
⋅
2
+
1
3
⋅
4
+
.
.
.
+
1
(
2
n
−
1
)
2
n
=
1
n
+
1
1
n
+
2
+
.
.
.
+
1
2
n
\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+ ...+ \frac{1}{(2n - 1)2n}=\frac{1}{n + 1}\frac{1}{n + 2}+ ... +\frac{1}{2n}
1
⋅
2
1
+
3
⋅
4
1
+
...
+
(
2
n
−
1
)
2
n
1
=
n
+
1
1
n
+
2
1
+
...
+
2
n
1
3
1
Hide problems
x +1/2 (x + y - 1)(x + y- 2) = a
Let
a
a
a
be any positive integer. Prove that there exists a unique pair of positive integers
x
x
x
and
y
y
y
such that
x
+
1
2
(
x
+
y
−
1
)
(
x
+
y
−
2
)
=
a
.
x +\frac12 (x + y - 1)(x + y- 2) = a.
x
+
2
1
(
x
+
y
−
1
)
(
x
+
y
−
2
)
=
a
.
2
1
Hide problems
S centroid if [ABS]=[BCS]=[CAS]
S
S
S
is a point inside triangle
A
B
C
ABC
A
BC
such that the areas of the triangles
A
B
S
ABS
A
BS
,
B
C
S
BCS
BCS
and
C
A
S
CAS
C
A
S
are all equal. Prove that
S
S
S
is the centroid of
A
B
C
ABC
A
BC
.