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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1906 Eotvos Mathematical Competition
1906 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
1
1
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tan (a/2) is rational iff cos a and sin a are rational
Prove that, if
tan
(
a
/
2
)
\tan (a/2)
tan
(
a
/2
)
is rational (or else, if
a
a
a
is an odd multiple of
π
\pi
π
so that
tan
(
a
/
2
)
\tan (a/2)
tan
(
a
/2
)
is not defined), then
cos
a
\cos a
cos
a
and
sin
a
\sin a
sin
a
are rational. And, conversely, if
cos
a
\cos a
cos
a
and
sin
a
\sin a
sin
a
are rational, then
tan
(
a
/
2
)
\tan (a/2)
tan
(
a
/2
)
is rational unless
a
a
a
is an odd multiple of
π
\pi
π
so that
tan
(
a
/
2
)
\tan (a/2)
tan
(
a
/2
)
is not defined.
3
1
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product (a_1 - 1)(a_2 - 2) ... (a_n -n) is an even number
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ...,a_n
a
1
,
a
2
,
...
,
a
n
represent an arbitrary arrangement of the numbers
1
,
2
,
.
.
.
,
n
1, 2, ...,n
1
,
2
,
...
,
n
. Prove that, if
n
n
n
is odd, the product
(
a
1
−
1
)
(
a
2
−
2
)
.
.
.
(
a
n
−
n
)
(a_1 - 1)(a_2 - 2) ... (a_n -n)
(
a
1
−
1
)
(
a
2
−
2
)
...
(
a
n
−
n
)
is an even number.
2
1
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centers of squares erected onsides (outside) of rhombus form a square
Let
K
,
L
,
M
,
N
K, L,M,N
K
,
L
,
M
,
N
designate the centers of the squares erected on the four sides (outside) of a rhombus. Prove that the polygon
K
L
M
N
KLMN
K
L
MN
is a square.