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Problems
Contests
International Contests
Francophone Mathematical Olympiad
2022 Francophone Mathematical Olympiad
2022 Francophone Mathematical Olympiad
Part of
Francophone Mathematical Olympiad
Subcontests
(4)
2
2
Hide problems
color k cells of nxn table, n tokens
We consider an
n
×
n
n \times n
n
×
n
table, with
n
≥
1
n\ge1
n
≥
1
. Aya wishes to color
k
k
k
cells of this table so that that there is a unique way to place
n
n
n
tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of
k
k
k
for which Aya's wish is achievable?
exist at least 18^n passwords of length n
To connect to the OFM site, Alice must choose a password. The latter must be consisting of
n
n
n
characters among the following
27
27
27
characters:
A
,
B
,
C
,
.
.
.
,
Y
,
Z
,
#
A, B, C, . . ., Y , Z, \#
A
,
B
,
C
,
...
,
Y
,
Z
,
#
We say that a password
m
m
m
is redundant if we can color in red and blue a block of consecutive letters of
m
m
m
in such a way that the word formed from the red letters is identical to the word formed from blue letters. For example, the password
H
#
Z
B
Z
J
B
J
Z
H\#ZBZJBJZ
H
#
ZBZ
J
B
J
Z
is redundant, because it contains the [color=#00f]ZB[color=#f00]Z[color=#00f]J[color=#f00]BJ block, where the word
Z
B
J
ZBJ
ZB
J
appears in both blue and red. At otherwise, the
A
B
C
A
C
B
ABCACB
A
BC
A
CB
password is not redundant. Show that, for any integer
n
≥
1
n \ge 1
n
≥
1
, there exist at least
1
8
n
18^n
1
8
n
passwords of length
n
n
n
, that is to say formed of
n
n
n
characters each, which are not redundant.
3
2
Hide problems
P3 froncophone
Let
△
A
B
C
\triangle ABC
△
A
BC
a triangle, and
D
D
D
the intersection of the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
and the perpendicular bisector of
A
C
AC
A
C
. the line parallel to
A
C
AC
A
C
passing by the point
B
B
B
, intersect the line
A
D
AD
A
D
at
X
X
X
. the line parallel to
C
X
CX
CX
passing by the point
B
B
B
, intersect
A
C
AC
A
C
at
Y
Y
Y
.
E
=
(
A
Y
B
)
∩
B
X
E = (AYB) \cap BX
E
=
(
A
Y
B
)
∩
BX
. prove that
C
C
C
,
D
D
D
and
E
E
E
collinear.
BC//EF wanted, 3 circles related
Let
A
B
C
ABC
A
BC
be a triangle and
Γ
\Gamma
Γ
its circumcircle. Denote
Δ
\Delta
Δ
the tangent at
A
A
A
to the circle
Γ
\Gamma
Γ
.
Γ
1
\Gamma_1
Γ
1
is a circle tangent to the lines
Δ
\Delta
Δ
,
(
A
B
)
(AB)
(
A
B
)
and
(
B
C
)
(BC)
(
BC
)
, and
E
E
E
its touchpoint with the line
(
A
B
)
(AB)
(
A
B
)
. Let
Γ
2
\Gamma_2
Γ
2
be a circle tangent to the lines
Δ
\Delta
Δ
,
(
A
C
)
(AC)
(
A
C
)
and
(
B
C
)
(BC)
(
BC
)
, and
F
F
F
its touchpoint with the line
(
A
C
)
(AC)
(
A
C
)
. We suppose that
E
E
E
and
F
F
F
belong respectively to the segments
[
A
B
]
[AB]
[
A
B
]
and
[
A
C
]
[AC]
[
A
C
]
, and that the two circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
lie outside triangle
A
B
C
ABC
A
BC
. Show that the lines
(
B
C
)
(BC)
(
BC
)
and
(
E
F
)
(EF)
(
EF
)
are parallel.
4
2
Hide problems
equation with 4 variables
find the smallest integer
n
≥
1
n\geq1
n
≥
1
such that the equation :
a
2
+
b
2
+
c
2
−
n
d
2
=
0
a^2+b^2+c^2-nd^2=0
a
2
+
b
2
+
c
2
−
n
d
2
=
0
has
(
0
,
0
,
0
,
0
)
(0,0,0,0)
(
0
,
0
,
0
,
0
)
as unique solution .
problem involving the digits of a^b+1
find all positive integer
a
≥
2
a\geq 2
a
≥
2
and
b
≥
2
b\geq2
b
≥
2
such that
a
a
a
is even and all the digits of
a
b
+
1
a^b+1
a
b
+
1
are equals.
1
2
Hide problems
number theory involing floor of sqrt(n)
find all the integer
n
≥
1
n\geq1
n
≥
1
such that
⌊
n
⌋
∣
n
\lfloor\sqrt{n}\rfloor \mid n
⌊
n
⌋
∣
n
functional equation from Z to Z
find all functions
f
:
Z
→
Z
f:\mathbb{Z} \to \mathbb{Z}
f
:
Z
→
Z
such that
f
(
m
+
n
)
+
f
(
m
)
f
(
n
)
=
n
2
(
f
(
m
)
+
1
)
+
m
2
(
f
(
n
)
+
1
)
+
m
n
(
2
−
m
n
)
f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)
f
(
m
+
n
)
+
f
(
m
)
f
(
n
)
=
n
2
(
f
(
m
)
+
1
)
+
m
2
(
f
(
n
)
+
1
)
+
mn
(
2
−
mn
)
holds for all
m
,
n
∈
Z
m,n \in \mathbb{Z}
m
,
n
∈
Z