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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1901 Eotvos Mathematical Competition
1901 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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Let $a$ and $b$ be two natural numbers whose greatest common divisor is $d$. Pro
Let
a
a
a
and
b
b
b
be two natural numbers whose greatest common divisor is
d
d
d
. Prove that exactly
d
d
d
of the numbers
a
,
2
a
,
3
a
,
.
.
.
,
(
b
−
1
)
a
,
b
a
a, 2a, 3a, ..., (b-1)a, ba
a
,
2
a
,
3
a
,
...
,
(
b
−
1
)
a
,
ba
is divisible by
b
b
b
.
2
1
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If $$u=\text{cot} 22^{\circ}30’ \text{ },\text{ } v= \frac{1}{\text{sin}
If
u
=
cot
2
2
∘
30
’
,
v
=
1
sin
2
2
∘
30
’
u=\text{cot} 22^{\circ}30’ \text{ },\text{ } v= \frac{1}{\text{sin} 22^{\circ}30’}
u
=
cot
2
2
∘
30’
,
v
=
sin
2
2
∘
30’
1
prove that
u
u
u
satisfies a quadratic and
v
v
v
a quartic (4th degree) equation with integral coefficients and with leading coefficients
1
1
1
.
1
1
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Prove that, for any positive integer $n$, $$1^n+2^n+3^n+4^n$$ is divisible by $5
Prove that, for any positive integer
n
n
n
,
1
n
+
2
n
+
3
n
+
4
n
1^n+2^n+3^n+4^n
1
n
+
2
n
+
3
n
+
4
n
is divisible by
5
5
5
if and only if
n
n
n
is not divisible by
4
4
4
.