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Problems
Contests
International Contests
Francophone Mathematical Olympiad
2020 Francophone Mathematical Olympiad
2020 Francophone Mathematical Olympiad
Part of
Francophone Mathematical Olympiad
Subcontests
(4)
2
2
Hide problems
new colony, n cities, 2020 languages, 2,020 galactic credits cheapest ticket
Emperor Zorg wishes to found a colony on a new planet. Each of the
n
n
n
cities that he will establish there will have to speak exactly one of the Empire's
2020
2020
2020
official languages. Some towns in the colony will be connected by a direct air link, each link can be taken in both directions. The emperor fixed the cost of the ticket for each connection to
1
1
1
galactic credit. He wishes that, given any two cities speaking the same language, it is always possible to travel from one to the other via these air links, and that the cheapest trip between these two cities costs exactly
2020
2020
2020
galactic credits. For what values of
n
n
n
can Emperor Zorg fulfill his dream?
A combinatorics problem with palindromic sequences
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
be a finite sequence of non negative integers, its subsequences are the sequences of the form
a
i
,
a
i
+
1
,
…
,
a
j
a_i,a_{i+1},\ldots,a_j
a
i
,
a
i
+
1
,
…
,
a
j
with
1
≤
i
≤
j
≤
n
1\le i\le j \le n
1
≤
i
≤
j
≤
n
. Two subsequences are said to be equal if they have the same length and have the same terms, that is, two subsequences
a
i
,
a
i
+
1
,
…
,
a
j
a_i,a_{i+1},\ldots,a_j
a
i
,
a
i
+
1
,
…
,
a
j
and
a
u
,
a
u
+
1
,
…
a
v
a_u,a_{u+1},\ldots a_v
a
u
,
a
u
+
1
,
…
a
v
are equal iff
j
−
i
=
u
−
v
j-i=u-v
j
−
i
=
u
−
v
and
a
i
+
k
=
a
u
+
k
a_{i+k}=a_{u+k}
a
i
+
k
=
a
u
+
k
forall integers
k
k
k
such that
0
≤
k
≤
j
−
1
0\le k\le j-1
0
≤
k
≤
j
−
1
. Finally, we say that a subsequence
a
i
,
a
i
+
1
,
…
,
a
j
a_i,a_{i+1},\ldots,a_j
a
i
,
a
i
+
1
,
…
,
a
j
is palindromic if
a
i
+
k
=
a
j
−
k
a_{i+k}=a_{j-k}
a
i
+
k
=
a
j
−
k
forall integers
k
k
k
such that
0
≤
k
≤
j
−
i
0\le k \le j-i
0
≤
k
≤
j
−
i
What is the greatest number of different palindromic subsequences that can a palindromic sequence of length
n
n
n
contain?
4
2
Hide problems
2^x + 9 \cdot 7^y = z^3 for integers x,y,z>=0
Find all the integers
x
,
y
x, y
x
,
y
and
z
z
z
greater than or equal to
0
0
0
such that
2
x
+
9
⋅
7
y
=
z
3
2^x + 9 \cdot 7^y = z^3
2
x
+
9
⋅
7
y
=
z
3
Prove that the set for which the number of prime divisors is bounded is finite
Let
(
a
i
)
i
∈
N
(a_i)_{i\in \mathbb{N}}
(
a
i
)
i
∈
N
a sequence of positive integers, such that for any finite, non-empty subset
S
S
S
of
N
\mathbb{N}
N
, the integer
Π
k
∈
S
a
k
−
1
\Pi_{k\in S} a_k -1
Π
k
∈
S
a
k
−
1
is prime. Prove that the number of
a
i
a_i
a
i
's with
i
∈
N
i\in \mathbb{N}
i
∈
N
such that
a
i
a_i
a
i
has less than
m
m
m
distincts prime factors is finite.
3
2
Hide problems
min k such among k reals, exist a,b in R of which |a -b|<1/n or |a-b| >n
Let
n
n
n
be an integer greater than or equal to
1
1
1
. Find, as a function of
n
n
n
, the smallest integer
k
≥
2
k\ge 2
k
≥
2
such that, among any
k
k
k
real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than
1
/
n
1 / n
1/
n
, either strictly greater than
n
n
n
.
Find n for which k<=a_n<k+1
Let
(
a
i
)
i
∈
N
(a_i)_{i\in \mathbb{N}}
(
a
i
)
i
∈
N
be a sequence with
a
1
=
3
2
a_1=\frac{3}2
a
1
=
2
3
such that
a
n
+
1
=
1
+
n
a
n
a_{n+1}=1+\frac{n}{a_n}
a
n
+
1
=
1
+
a
n
n
Find
n
n
n
such that
2020
≤
a
n
<
2021
2020\le a_n <2021
2020
≤
a
n
<
2021
1
2
Hide problems
congruent triangles wanted, incircle, 2 excircles and circumcircle related
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
<
A
C
AB <AC
A
B
<
A
C
,
ω
\omega
ω
its inscribed circle and
Γ
\Gamma
Γ
its circumscribed circle. Let also
ω
b
\omega_b
ω
b
be the excircle relative to vertex
B
B
B
, then
B
′
B'
B
′
is the point of tangency between
ω
b
\omega_b
ω
b
and
(
A
C
)
(AC)
(
A
C
)
. Similarly, let the circle
ω
c
\omega_c
ω
c
be the excircle exinscribed relative to vertex
C
C
C
, then
C
′
C'
C
′
is the point of tangency between
ω
c
\omega_c
ω
c
and
(
A
B
)
(AB)
(
A
B
)
. Finally, let
I
I
I
be the center of
ω
\omega
ω
and
X
X
X
the point of
Γ
\Gamma
Γ
such that
∠
X
A
I
\angle XAI
∠
X
A
I
is a right angle. Prove that the triangles
X
B
C
′
XBC'
XB
C
′
and
X
C
B
′
XCB'
XC
B
′
are congruent.
another geometry problem with sharky-devil point
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
C
>
A
B
AC>AB
A
C
>
A
B
, Let
D
E
F
DEF
D
EF
be the intouch triangle with
D
∈
(
B
C
)
D \in (BC)
D
∈
(
BC
)
,
E
∈
(
A
C
)
E \in (AC)
E
∈
(
A
C
)
,
F
∈
(
A
B
)
F \in (AB)
F
∈
(
A
B
)
,, let
G
G
G
be the intersecttion of the perpendicular from
D
D
D
to
E
F
EF
EF
with
A
B
AB
A
B
, and
X
=
(
A
B
C
)
∩
(
A
E
F
)
X=(ABC)\cap (AEF)
X
=
(
A
BC
)
∩
(
A
EF
)
. Prove that
B
,
D
,
G
B,D,G
B
,
D
,
G
and
X
X
X
are concylic