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National and Regional Contests
France Contests
France Team Selection Test
2007 France Team Selection Test
2007 France Team Selection Test
Part of
France Team Selection Test
Subcontests
(3)
3
1
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France TST 2007
Let
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be four distinct points on a circle such that the lines
(
A
C
)
(AC)
(
A
C
)
and
(
B
D
)
(BD)
(
B
D
)
intersect at
E
E
E
, the lines
(
A
D
)
(AD)
(
A
D
)
and
(
B
C
)
(BC)
(
BC
)
intersect at
F
F
F
and such that
(
A
B
)
(AB)
(
A
B
)
and
(
C
D
)
(CD)
(
C
D
)
are not parallel. Prove that
C
,
D
,
E
,
F
C,D,E,F
C
,
D
,
E
,
F
are on the same circle if, and only if,
(
E
F
)
⊥
(
A
B
)
(EF)\bot(AB)
(
EF
)
⊥
(
A
B
)
.
2
2
Hide problems
France TST 2007
Find all functions
f
:
Z
→
Z
f: \mathbb{Z}\rightarrow\mathbb{Z}
f
:
Z
→
Z
such that for all
x
,
y
∈
Z
x,y \in \mathbb{Z}
x
,
y
∈
Z
:
f
(
x
−
y
+
f
(
y
)
)
=
f
(
x
)
+
f
(
y
)
.
f(x-y+f(y))=f(x)+f(y).
f
(
x
−
y
+
f
(
y
))
=
f
(
x
)
+
f
(
y
)
.
France TST 2007
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive reals such taht
a
+
b
+
c
+
d
=
1
a+b+c+d=1
a
+
b
+
c
+
d
=
1
. Prove that:
6
(
a
3
+
b
3
+
c
3
+
d
3
)
≥
a
2
+
b
2
+
c
2
+
d
2
+
1
8
.
6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.
6
(
a
3
+
b
3
+
c
3
+
d
3
)
≥
a
2
+
b
2
+
c
2
+
d
2
+
8
1
.
1
2
Hide problems
France TST 2007
For a positive integer
a
a
a
,
a
′
a'
a
′
is the integer obtained by the following method: the decimal writing of
a
′
a'
a
′
is the inverse of the decimal writing of
a
a
a
(the decimal writing of
a
′
a'
a
′
can begin by zeros, but not the one of
a
a
a
); for instance if
a
=
2370
a=2370
a
=
2370
,
a
′
=
0732
a'=0732
a
′
=
0732
, that is
732
732
732
. Let
a
1
a_{1}
a
1
be a positive integer, and
(
a
n
)
n
≥
1
(a_{n})_{n \geq 1}
(
a
n
)
n
≥
1
the sequence defined by
a
1
a_{1}
a
1
and the following formula for
n
≥
1
n \geq 1
n
≥
1
:
a
n
+
1
=
a
n
+
a
n
′
.
a_{n+1}=a_{n}+a'_{n}.
a
n
+
1
=
a
n
+
a
n
′
.
Can
a
7
a_{7}
a
7
be prime?
France TST 2007
Do there exist
5
5
5
points in the space, such that for all
n
∈
{
1
,
2
,
…
,
10
}
n\in\{1,2,\ldots,10\}
n
∈
{
1
,
2
,
…
,
10
}
there exist two of them at distance between them
n
n
n
?