MathDB
Problems
Contests
National and Regional Contests
France Contests
France Team Selection Test
2006 France Team Selection Test
2006 France Team Selection Test
Part of
France Team Selection Test
Subcontests
(3)
2
1
Hide problems
Inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be three positive real numbers such that
a
b
c
=
1
abc=1
ab
c
=
1
. Show that:
a
(
a
+
1
)
(
b
+
1
)
+
b
(
b
+
1
)
(
c
+
1
)
+
c
(
c
+
1
)
(
a
+
1
)
≥
3
4
.
\displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}.
(
a
+
1
)
(
b
+
1
)
a
+
(
b
+
1
)
(
c
+
1
)
b
+
(
c
+
1
)
(
a
+
1
)
c
≥
4
3
.
When is there equality?
1
2
Hide problems
A square
Let
A
B
C
D
ABCD
A
BC
D
be a square and let
Γ
\Gamma
Γ
be the circumcircle of
A
B
C
D
ABCD
A
BC
D
.
M
M
M
is a point of
Γ
\Gamma
Γ
belonging to the arc
C
D
CD
C
D
which doesn't contain
A
A
A
.
P
P
P
and
R
R
R
are respectively the intersection points of
(
A
M
)
(AM)
(
A
M
)
with
[
B
D
]
[BD]
[
B
D
]
and
[
C
D
]
[CD]
[
C
D
]
,
Q
Q
Q
and
S
S
S
are respectively the intersection points of
(
B
M
)
(BM)
(
BM
)
with
[
A
C
]
[AC]
[
A
C
]
and
[
D
C
]
[DC]
[
D
C
]
. Prove that
(
P
S
)
(PS)
(
PS
)
and
(
Q
R
)
(QR)
(
QR
)
are perpendicular.
2n numbers in an array
In a
2
×
n
2\times n
2
×
n
array we have positive reals s.t. the sum of the numbers in each of the
n
n
n
columns is
1
1
1
. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most
n
+
1
4
\frac{n+1}4
4
n
+
1
.
3
1
Hide problems
Partition
Let
M
=
{
1
,
2
,
…
,
3
⋅
n
}
M=\{1,2,\ldots,3 \cdot n\}
M
=
{
1
,
2
,
…
,
3
⋅
n
}
. Partition
M
M
M
into three sets
A
,
B
,
C
A,B,C
A
,
B
,
C
which
c
a
r
d
card
c
a
r
d
A
A
A
=
=
=
c
a
r
d
card
c
a
r
d
B
B
B
=
=
=
c
a
r
d
card
c
a
r
d
C
C
C
=
=
=
n
.
n .
n
.
Prove that there exists
a
a
a
in
A
,
b
A,b
A
,
b
in
B
,
c
B, c
B
,
c
in
C
C
C
such that or
a
=
b
+
c
,
a=b+c,
a
=
b
+
c
,
or
b
=
c
+
a
,
b=c+a,
b
=
c
+
a
,
or
c
=
a
+
b
c=a+b
c
=
a
+
b
Edited by orl.