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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1932 Eotvos Mathematical Competition
1932 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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sin 2a> sin 2b> sin 2c if a<b<c in acute ABC
Let
α
\alpha
α
,
β
\beta
β
and
γ
\gamma
γ
be the interior angles of an acute triangle. Prove that if
α
<
β
<
γ
\alpha < \beta < \gamma
α
<
β
<
γ
then \sin 2\alpha >\ sin 2 \beta > \sin 2\gamma.
2
1
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< FAP < < PAT if ABC is acute triangle.
In triangle
A
B
C
ABC
A
BC
,
A
B
≠
A
C
AB \ne AC
A
B
=
A
C
. Let
A
F
AF
A
F
,
A
P
AP
A
P
and
A
T
AT
A
T
be the median, angle bisector and altitude from vertex
A
A
A
, with
F
,
P
F, P
F
,
P
and
T
T
T
on
B
G
BG
BG
or its extension. (a) Prove that
P
P
P
always lies between
F
F
F
and
T
T
T
. (b) Prove that
∠
F
A
P
<
∠
P
A
T
\angle FAP < \angle PAT
∠
F
A
P
<
∠
P
A
T
if
A
B
C
ABC
A
BC
is an acute triangle.
1
1
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(a+1)^b-1 is divisible by a^{n+1} when b is divisible by a^n
Let
a
,
b
a, b
a
,
b
and
n
n
n
be positive integers such that
b
b
b
is divisible by
a
n
a^n
a
n
. Prove that
(
a
+
1
)
b
−
1
(a+1)^b-1
(
a
+
1
)
b
−
1
is divisible by
a
n
+
1
a^{n+1}
a
n
+
1
.