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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1897 Eotvos Mathematical Competition
1897 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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Let $ABCD$ be a rectangle and let $M, N$ and $P, Q$ be the points of intersectio
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle and let
M
,
N
M, N
M
,
N
and
P
,
Q
P, Q
P
,
Q
be the points of intersections of some line
e
e
e
with the sides
A
B
,
C
D
AB, CD
A
B
,
C
D
and
A
D
,
B
C
AD, BC
A
D
,
BC
, respectively (or their extensions). Given the points
M
,
N
,
P
,
Q
M, N, P, Q
M
,
N
,
P
,
Q
and the length
p
p
p
of side
A
B
AB
A
B
, construct the rectangle. Under what conditions can this problem be solved, and how many solutions does it have?
2
1
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Show that, if $\alpha$, $\beta$ and $\gamma$ are angles of an arbitrary triangle
Show that, if
α
\alpha
α
,
β
\beta
β
and
γ
\gamma
γ
are angles of an arbitrary triangle,
sin
α
2
sin
β
2
sin
γ
2
<
1
4
.
\text{sin } \frac{\alpha}{2} \text{ sin } \frac{\beta}{2} \text{ sin } \frac{\gamma}{2} < \frac14.
sin
2
α
sin
2
β
sin
2
γ
<
4
1
.
.
1
1
Hide problems
Prove, for angles $\alpha$, $\beta$ and $\gamma$ of a right triangle, the follow
Prove, for angles
α
\alpha
α
,
β
\beta
β
and
γ
\gamma
γ
of a right triangle, the following relation:
sin
α
sin
β
sin
(
α
−
β
)
+
sin
β
sin
γ
sin
(
β
−
γ
)
+
sin
γ
sin
α
sin
(
γ
−
α
)
+
sin
(
α
−
β
)
sin
(
β
−
γ
)
sin
(
γ
−
α
)
=
0.
\text{sin } \alpha \text{ sin } \beta \text{ sin } (\alpha-\beta) \text{ } + \text{ sin } \beta \text{ sin } \gamma \text{ sin } (\beta-\gamma) \text{ }+ \text{ sin } \gamma \text{ sin } \alpha \text{ sin } (\gamma-\alpha) \text{ }+ \text{ sin } (\alpha-\beta) \text{ sin } (\beta-\gamma) \text{ sin } (\gamma-\alpha) = 0.
sin
α
sin
β
sin
(
α
−
β
)
+
sin
β
sin
γ
sin
(
β
−
γ
)
+
sin
γ
sin
α
sin
(
γ
−
α
)
+
sin
(
α
−
β
)
sin
(
β
−
γ
)
sin
(
γ
−
α
)
=
0.