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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1895 Eotvos Mathematical Competition
1895 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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Given the circumradius $R$ of a triangle, a side length $c$, and the ratio $a/b$
Given the circumradius
R
R
R
of a triangle, a side length
c
c
c
, and the ratio
a
/
b
a/b
a
/
b
of the other two side lengths, determine all three sides and angles of this triangle.
2
1
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Construct a point $N$ inside a given right triangle $ABC$ such that the angles $
Construct a point
N
N
N
inside a given right triangle
A
B
C
ABC
A
BC
such that the angles
∠
N
B
C
\angle NBC
∠
NBC
,
∠
N
C
A
\angle NCA
∠
NC
A
and
∠
N
A
B
\angle NAB
∠
N
A
B
are equal.
1
1
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Prove that there are exactly $2(2^{n-1}-1)$ ways of dealing $n$ cards to two per
Prove that there are exactly
2
(
2
n
−
1
−
1
)
2(2^{n-1}-1)
2
(
2
n
−
1
−
1
)
ways of dealing
n
n
n
cards to two persons. (The persons may receive unequal numbers of cards.)