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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1923 Eotvos Mathematical Competition
1923 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
2
1
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sum B(n+1, k)= 2^n S_n
If
s
n
=
1
+
q
+
q
2
+
.
.
.
+
q
n
s_n = 1 + q + q^2 +... + q^n
s
n
=
1
+
q
+
q
2
+
...
+
q
n
and
S
n
=
1
+
1
+
q
2
+
(
1
+
q
2
)
2
+
.
.
.
+
(
1
+
q
2
)
n
,
S_n = 1 +\frac{1 + q}{2}+ \left( \frac{1 + q}{2}\right)^2 +... + \left( \frac{1 + q}{2}\right)^n,
S
n
=
1
+
2
1
+
q
+
(
2
1
+
q
)
2
+
...
+
(
2
1
+
q
)
n
,
prove that
(
n
+
1
1
)
+
(
n
+
1
2
)
s
1
+
(
n
+
1
3
)
s
2
+
.
.
.
+
(
n
+
1
n
+
1
)
s
n
=
2
n
S
n
{n + 1 \choose 1}+{n + 1 \choose 2} s_1 + {n + 1 \choose 3} s_2 + ... + {n + 1 \choose n + 1} s_n = 2^nS_n
(
1
n
+
1
)
+
(
2
n
+
1
)
s
1
+
(
3
n
+
1
)
s
2
+
...
+
(
n
+
1
n
+
1
)
s
n
=
2
n
S
n
3
1
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ninfinite arithmetic progression of natural numbers cannot have all primes
Prove that, if the terms of an infinite arithmetic progression of natural numbers are not all equal, they cannot all be primes.
1
1
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3 equal concurrent circles
Three circles through the point
O
O
O
and of radius
r
r
r
intersect pairwise in the additional points
A
A
A
,
B
B
B
,
C
C
C
. Prove that the circle through the points
A
A
A
,
B
B
B
, and
C
C
C
also has radius
r
r
r
.