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Contests
International Contests
Francophone Mathematical Olympiad
2024 Francophone Mathematical Olympiad
2024 Francophone Mathematical Olympiad
Part of
Francophone Mathematical Olympiad
Subcontests
(4)
4
2
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Find n integers such that a_i divides a_j^2+1 for any i,j
Find all integers
n
≥
2
n \ge 2
n
≥
2
for which there exists
n
n
n
integers
a
1
,
a
2
,
…
,
a
n
≥
2
a_1,a_2,\dots,a_n \ge 2
a
1
,
a
2
,
…
,
a
n
≥
2
such that for all indices
i
≠
j
i \ne j
i
=
j
, we have
a
i
∣
a
j
2
+
1
a_i \mid a_j^2+1
a
i
∣
a
j
2
+
1
.
Can you pair the divisors of n such that all quotients are close to p?
Let
p
p
p
be a fixed prime number. Find all integers
n
≥
1
n \ge 1
n
≥
1
with the following property: One can partition the positive divisors of
n
n
n
in pairs
(
d
,
d
′
)
(d,d')
(
d
,
d
′
)
satisfying
d
<
d
′
d<d'
d
<
d
′
and
p
∣
⌊
d
′
d
⌋
p \mid \left\lfloor \frac{d'}{d}\right\rfloor
p
∣
⌊
d
d
′
⌋
.
3
2
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Geometry with many parallels and perpendiculars
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
and let
O
O
O
be its circumcenter. Let
D
D
D
be a point on the segment
A
C
AC
A
C
such that
A
B
=
A
D
AB=AD
A
B
=
A
D
. Let
E
E
E
be the intersection of the line
A
B
AB
A
B
with the perpendicular line to
A
O
AO
A
O
through
D
D
D
. Let
F
F
F
be the intersection of the perpendicular line to
O
C
OC
OC
through
C
C
C
with the line parallel to
A
C
AC
A
C
and passing through
E
E
E
. Finally, let the lines
C
E
CE
CE
and
D
F
DF
D
F
intersect in
G
G
G
. Show that
A
G
AG
A
G
and
B
F
BF
BF
are parallel.
Parallel lines in two-circle configuration
Let
A
B
C
ABC
A
BC
be an acute triangle,
ω
\omega
ω
its circumcircle and
O
O
O
its circumcenter. The altitude from
A
A
A
intersects
ω
\omega
ω
in a point
D
≠
A
D \ne A
D
=
A
and the segment
A
C
AC
A
C
intersects the circumcircle of
O
C
D
OCD
OC
D
in a point
E
≠
C
E \ne C
E
=
C
. Finally, let
M
M
M
be the midpoint of
B
E
BE
BE
. Show that
D
E
DE
D
E
is parallel to
O
M
OM
OM
.
2
2
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Alice and Bob move tile around a n-gon, who will win?
Given
n
≥
2
n \ge 2
n
≥
2
points on a circle, Alice and Bob play the following game. Initially, a tile is placed on one of the points and no segment is drawn. The players alternate in turns, with Alice to start. In a turn, a player moves the tile from its current position
P
P
P
to one of the
n
−
1
n-1
n
−
1
other points
Q
Q
Q
and draws the segment
P
Q
PQ
PQ
. This move is not allowed if the segment
P
Q
PQ
PQ
is already drawn. If a player cannot make a move, the game is over and the opponent wins. Determine, for each
n
n
n
, which of the two players has a winning strategy.
3D combinatorial geometry!
Given a positive integer
n
≥
2
n \ge 2
n
≥
2
, let
P
\mathcal{P}
P
and
Q
\mathcal{Q}
Q
be two sets, each consisting of
n
n
n
points in three-dimensional space. Suppose that these
2
n
2n
2
n
points are distinct. Show that it is possible to label the points of
P
\mathcal{P}
P
as
P
1
,
P
2
,
…
,
P
n
P_1,P_2,\dots,P_n
P
1
,
P
2
,
…
,
P
n
and the points of
Q
\mathcal{Q}
Q
as
Q
1
,
Q
2
,
…
,
Q
n
Q_1,Q_2,\dots,Q_n
Q
1
,
Q
2
,
…
,
Q
n
such that for any indices
i
i
i
and
j
j
j
, the balls of diameters
P
i
Q
i
P_iQ_i
P
i
Q
i
and
P
j
Q
j
P_jQ_j
P
j
Q
j
have at least one common point.
1
2
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System with 2024 variables, but how many are equal?
Find the largest integer
k
k
k
with the following property: Whenever real numbers
x
1
,
x
2
,
…
,
x
2024
x_1,x_2,\dots,x_{2024}
x
1
,
x
2
,
…
,
x
2024
satisfy
x
1
2
=
(
x
1
+
x
2
)
2
=
⋯
=
(
x
1
+
x
2
+
⋯
+
x
2024
)
2
,
x_1^2=(x_1+x_2)^2=\dots=(x_1+x_2+\dots+x_{2024})^2,
x
1
2
=
(
x
1
+
x
2
)
2
=
⋯
=
(
x
1
+
x
2
+
⋯
+
x
2024
)
2
,
at least
k
k
k
of them are equal.
Pionocchio and Geppetto play a liar's game with polynomial values
Let
d
d
d
and
m
m
m
be two fixed positive integers. Pinocchio and Geppetto know the values of
d
d
d
and
m
m
m
and play the following game: In the beginning, Pinocchio chooses a polynomial
P
P
P
of degree at most
d
d
d
with integer coefficients. Then Geppetto asks him questions of the following form "What is the value of
P
(
n
)
P(n)
P
(
n
)
?'' for
n
∈
Z
n \in \mathbb{Z}
n
∈
Z
. Pinocchio usually says the truth, but he can lie up to
m
m
m
times. What is, as a function of
d
d
d
and
m
m
m
, the minimal number of questions that Geppetto needs to ask to be sure to determine
P
P
P
, no matter how Pinocchio chooses to reply?