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Problems
Contests
International Contests
Francophone Mathematical Olympiad
2021 Francophone Mathematical Olympiad
2021 Francophone Mathematical Olympiad
Part of
Francophone Mathematical Olympiad
Subcontests
(4)
4
2
Hide problems
GCD (f(m),n ) + LCM (m,f(n)) = LCM(f(m),n ) + GCD (m,f(n))
Let
N
⩾
1
\mathbb{N}_{\geqslant 1}
N
⩾
1
be the set of positive integers. Find all functions
f
:
N
⩾
1
→
N
⩾
1
f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}
f
:
N
⩾
1
→
N
⩾
1
such that, for all positive integers
m
m
m
and
n
n
n
:
G
C
D
(
f
(
m
)
,
n
)
+
L
C
M
(
m
,
f
(
n
)
)
=
G
C
D
(
m
,
f
(
n
)
)
+
L
C
M
(
f
(
m
)
,
n
)
.
\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) = \mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).
GCD
(
f
(
m
)
,
n
)
+
LCM
(
m
,
f
(
n
)
)
=
GCD
(
m
,
f
(
n
)
)
+
LCM
(
f
(
m
)
,
n
)
.
Note: if
a
a
a
and
b
b
b
are positive integers,
G
C
D
(
a
,
b
)
\mathrm{GCD}(a,b)
GCD
(
a
,
b
)
is the largest positive integer that divides both
a
a
a
and
b
b
b
, and
L
C
M
(
a
,
b
)
\mathrm{LCM}(a,b)
LCM
(
a
,
b
)
is the smallest positive integer that is a multiple of both
a
a
a
and
b
b
b
.
n = (f(2n)-f(n) )(2 f(n) - f(2n) )
Let
N
≥
1
\mathbb{N}_{\ge 1}
N
≥
1
be the set of positive integers. Find all functions
f
:
N
≥
1
→
N
≥
1
f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}
f
:
N
≥
1
→
N
≥
1
such that, for all positive integers
m
m
m
and
n
n
n
:(a)
n
=
(
f
(
2
n
)
−
f
(
n
)
)
(
2
f
(
n
)
−
f
(
2
n
)
)
n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)
n
=
(
f
(
2
n
)
−
f
(
n
)
)
(
2
f
(
n
)
−
f
(
2
n
)
)
, (b)
f
(
m
)
f
(
n
)
−
f
(
m
n
)
=
(
f
(
2
m
)
−
f
(
m
)
)
(
2
f
(
n
)
−
f
(
2
n
)
)
+
(
f
(
2
n
)
−
f
(
n
)
)
(
2
f
(
m
)
−
f
(
2
m
)
)
f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)
f
(
m
)
f
(
n
)
−
f
(
mn
)
=
(
f
(
2
m
)
−
f
(
m
)
)
(
2
f
(
n
)
−
f
(
2
n
)
)
+
(
f
(
2
n
)
−
f
(
n
)
)
(
2
f
(
m
)
−
f
(
2
m
)
)
, (c)
m
−
n
m-n
m
−
n
divides
f
(
2
m
)
−
f
(
2
n
)
f(2m)-f(2n)
f
(
2
m
)
−
f
(
2
n
)
if
m
m
m
and
n
n
n
are distinct odd prime numbers.
3
2
Hide problems
every point in the plane was colored in red or blue
Every point in the plane was colored in red or blue. Prove that one the two following statements is true:
∙
\bullet
∙
there exist two red points at distance
1
1
1
from each other;
∙
\bullet
∙
there exist four blue points
B
1
B_1
B
1
,
B
2
B_2
B
2
,
B
3
B_3
B
3
,
B
4
B_4
B
4
such that the points
B
i
B_i
B
i
and
B
j
B_j
B
j
are at distance
∣
i
−
j
∣
|i - j|
∣
i
−
j
∣
from each other, for all integers
i
i
i
and
j
j
j
such as
1
≤
i
≤
4
1 \le i \le 4
1
≤
i
≤
4
and
1
≤
j
≤
4
1 \le j \le 4
1
≤
j
≤
4
.
tangent line to incircle of square wanted
Let
A
B
C
D
ABCD
A
BC
D
be a square with incircle
Γ
\Gamma
Γ
. Let
M
M
M
be the midpoint of the segment
[
C
D
]
[CD]
[
C
D
]
. Let
P
≠
B
P \neq B
P
=
B
be a point on the segment
[
A
B
]
[AB]
[
A
B
]
. Let
E
≠
M
E \neq M
E
=
M
be the point on
Γ
\Gamma
Γ
such that
(
D
P
)
(DP)
(
D
P
)
and
(
E
M
)
(EM)
(
EM
)
are parallel. The lines
(
C
P
)
(CP)
(
CP
)
and
(
A
D
)
(AD)
(
A
D
)
meet each other at
F
F
F
. Prove that the line
(
E
F
)
(EF)
(
EF
)
is tangent to
Γ
\Gamma
Γ
2
2
Hide problems
Sophie colors every triangle side in red, green or blue in 12 triangles fig.
Evariste has drawn twelve triangles as follows, so that two consecutive triangles share exactly one edge. https://cdn.artofproblemsolving.com/attachments/6/2/50377e7ad5fb1c40e36725e43c7eeb1e3c2849.png Sophie colors every triangle side in red, green or blue. Among the
3
24
3^{24}
3
24
possible colorings, how many have the property that every triangle has one edge of each color?
Slbert and Beatrice play a game with 2021 on a table
Albert and Beatrice play a game.
2021
2021
2021
stones lie on a table. Starting with Albert, they alternatively remove stones from the table, while obeying the following rule. At the
n
n
n
-th turn, the active player (Albert if
n
n
n
is odd, Beatrice if
n
n
n
is even) can remove from
1
1
1
to
n
n
n
stones. Thus, Albert first removes
1
1
1
stone; then, Beatrice can remove
1
1
1
or
2
2
2
stones, as she wishes; then, Albert can remove from
1
1
1
to
3
3
3
stones, and so on. The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player's moves?
1
2
Hide problems
R < 1/15 < S, R=/2 x 3/4 x 5/6 x ...x 223/ 224
Let
R
R
R
and
S
S
S
be the numbers defined by
R
=
1
2
×
3
4
×
5
6
×
⋯
×
223
224
and
S
=
2
3
×
4
5
×
6
7
×
⋯
×
224
225
.
R = \dfrac{1}{2} \times \dfrac{3}{4} \times \dfrac{5}{6} \times \cdots \times \dfrac{223}{224} \text{ and } S = \dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7} \times \cdots \times \dfrac{224}{225}.
R
=
2
1
×
4
3
×
6
5
×
⋯
×
224
223
and
S
=
3
2
×
5
4
×
7
6
×
⋯
×
225
224
.
Prove that
R
<
1
15
<
S
R < \dfrac{1}{15} < S
R
<
15
1
<
S
.
exists c : b_n=c a_n. a_{n+2}=a_n+a_{n+1}, b_{n+2}=b_n+b_{n+1}, a_n / b_n
Let
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\ldots
a
1
,
a
2
,
a
3
,
…
and
b
1
,
b
2
,
b
3
,
…
b_1,b_2,b_3,\ldots
b
1
,
b
2
,
b
3
,
…
be positive integers such that
a
n
+
2
=
a
n
+
a
n
+
1
a_{n+2} = a_n + a_{n+1}
a
n
+
2
=
a
n
+
a
n
+
1
and
b
n
+
2
=
b
n
+
b
n
+
1
b_{n+2} = b_n + b_{n+1}
b
n
+
2
=
b
n
+
b
n
+
1
for all
n
≥
1
n \ge 1
n
≥
1
. Assume that
a
n
a_n
a
n
divides
b
n
b_n
b
n
for infinitely many values of
n
n
n
. Prove that there exists an integer
c
c
c
such that
b
n
=
c
a
n
b_n = c a_n
b
n
=
c
a
n
for all
n
≥
1
n \ge 1
n
≥
1
.