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National and Regional Contests
France Contests
France Team Selection Test
2005 France Team Selection Test
2005 France Team Selection Test
Part of
France Team Selection Test
Subcontests
(6)
6
1
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n integer roots
Let
P
P
P
be a polynom of degree
n
≥
5
n \geq 5
n
≥
5
with integer coefficients given by P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 with
a
i
∈
Z
a_i \in \mathbb{Z}
a
i
∈
Z
,
a
n
≠
0
a_n \neq 0
a
n
=
0
. Suppose that
P
P
P
has
n
n
n
different integer roots (elements of
Z
\mathbb{Z}
Z
) :
0
,
α
2
,
…
,
α
n
0,\alpha_2,\ldots,\alpha_n
0
,
α
2
,
…
,
α
n
. Find all integers
k
∈
Z
k \in \mathbb{Z}
k
∈
Z
such that
P
(
P
(
k
)
)
=
0
P(P(k))=0
P
(
P
(
k
))
=
0
.
5
1
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<PAC=2<CPA
Let
A
B
C
ABC
A
BC
be a triangle such that
B
C
=
A
C
+
1
2
A
B
BC=AC+\frac{1}{2}AB
BC
=
A
C
+
2
1
A
B
. Let
P
P
P
be a point of
A
B
AB
A
B
such that
A
P
=
3
P
B
AP=3PB
A
P
=
3
PB
. Show that
P
A
C
^
=
2
C
P
A
^
.
\widehat{PAC} = 2 \widehat{CPA}.
P
A
C
=
2
CP
A
.
4
1
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All positive integers
Let
X
X
X
be a non empty subset of
N
=
{
1
,
2
,
…
}
\mathbb{N} = \{1,2,\ldots \}
N
=
{
1
,
2
,
…
}
. Suppose that for all
x
∈
X
x \in X
x
∈
X
,
4
x
∈
X
4x \in X
4
x
∈
X
and
⌊
x
⌋
∈
X
\lfloor \sqrt{x} \rfloor \in X
⌊
x
⌋
∈
X
. Prove that
X
=
N
X=\mathbb{N}
X
=
N
.
3
1
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International meeting
In an international meeting of
n
≥
3
n \geq 3
n
≥
3
participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken more that by the half of the participants. What is the least value of
n
n
n
?
2
1
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Two right-angled triangles
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let
S
S
S
(respectively
S
′
S'
S
′
) be the area of the first triangle (respectively of the second triangle). Prove that
S
S
′
≥
3
+
2
2
\frac{S}{S'}\geq 3+2\sqrt{2}
S
′
S
≥
3
+
2
2
.
1
1
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Perfect square
Let
x
x
x
,
y
y
y
be two positive integers such that
3
x
2
+
x
=
4
y
2
+
y
\displaystyle 3x^2+x=4y^2+y
3
x
2
+
x
=
4
y
2
+
y
. Prove that
x
−
y
x-y
x
−
y
is a perfect square.