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Greece Contests
Greece Team Selection Test
2005 Greece Team Selection Test
2005 Greece Team Selection Test
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Greece Team Selection Test
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Side legnths are roots, then the altitudes are roots
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
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Polynomial
Let the polynomial
P
(
x
)
=
x
3
+
19
x
2
+
94
x
+
a
P(x)=x^3+19x^2+94x+a
P
(
x
)
=
x
3
+
19
x
2
+
94
x
+
a
where
a
∈
N
a\in\mathbb{N}
a
∈
N
. If
p
p
p
a prime number, prove that no more than three numbers of the numbers
P
(
0
)
,
P
(
1
)
,
…
,
P
(
p
−
1
)
P(0), P(1),\ldots, P(p-1)
P
(
0
)
,
P
(
1
)
,
…
,
P
(
p
−
1
)
are divisible by
p
p
p
.