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Problems
Contests
National and Regional Contests
Azerbaijan Contests
Lotfi Zadeh Olympiad
2021 Lotfi Zadeh Olympiad
2021 Lotfi Zadeh Olympiad
Part of
Lotfi Zadeh Olympiad
Subcontests
(4)
4
1
Hide problems
# of sequences of 0, 1 satisfying two properties
Find the number of sequences of
0
,
1
0, 1
0
,
1
with length
n
n
n
satisfying both of the following properties:[*] There exists a simple polygon such that its
i
i
i
-th angle is less than
180
180
180
degrees if and only if the
i
i
i
-th element of the sequence is
1
1
1
. [*] There exists a convex polygon such that its
i
i
i
-th angle is less than
90
90
90
degrees if and only if the
i
i
i
-th element of the sequence is
1
1
1
.
3
1
Hide problems
find min [lcm(a,b)+lcm(b,c)+lcm(c,a)]/[gcd(a,b)+gcd(b,c)+gcd(c,a)]
Find the least possible value for the fraction
l
c
m
(
a
,
b
)
+
l
c
m
(
b
,
c
)
+
l
c
m
(
c
,
a
)
g
c
d
(
a
,
b
)
+
g
c
d
(
b
,
c
)
+
g
c
d
(
c
,
a
)
\frac{lcm(a,b)+lcm(b,c)+lcm(c,a)}{gcd(a,b)+gcd(b,c)+gcd(c,a)}
g
c
d
(
a
,
b
)
+
g
c
d
(
b
,
c
)
+
g
c
d
(
c
,
a
)
l
c
m
(
a
,
b
)
+
l
c
m
(
b
,
c
)
+
l
c
m
(
c
,
a
)
over all distinct positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
. By
l
c
m
(
x
,
y
)
lcm(x, y)
l
c
m
(
x
,
y
)
we mean the least common multiple of
x
,
y
x, y
x
,
y
and by
g
c
d
(
x
,
y
)
gcd(x, y)
g
c
d
(
x
,
y
)
we mean the greatest common divisor of
x
,
y
x, y
x
,
y
.
2
1
Hide problems
2 sequences given. Prove a_i=b_i or a_i constant for i>N
Let
a
1
,
a
2
,
⋯
,
a
n
a_1, a_2,\cdots , a_n
a
1
,
a
2
,
⋯
,
a
n
and
b
1
,
b
2
,
⋯
,
b
n
b_1, b_2,\cdots , b_n
b
1
,
b
2
,
⋯
,
b
n
be (not necessarily distinct) positive integers. We continue the sequences as follows: For every
i
>
n
i>n
i
>
n
,
a
i
a_i
a
i
is the smallest positive integer which is not among
b
1
,
b
2
,
⋯
,
b
i
−
1
b_1, b_2,\cdots , b_{i-1}
b
1
,
b
2
,
⋯
,
b
i
−
1
, and
b
i
b_i
b
i
is the smallest positive integer which is not among
a
1
,
a
2
,
⋯
,
a
i
−
1
a_1, a_2,\cdots , a_{i-1}
a
1
,
a
2
,
⋯
,
a
i
−
1
. Prove that there exists
N
N
N
such that for every
i
>
N
i>N
i
>
N
we have
a
i
=
b
i
a_i=b_i
a
i
=
b
i
or for every
i
>
N
i>N
i
>
N
we have
a
i
+
1
=
a
i
a_{i+1}=a_i
a
i
+
1
=
a
i
.
1
1
Hide problems
Cyclic quadrilateral given. Parallelity wanted
In the inscribed quadrilateral
A
B
C
D
ABCD
A
BC
D
,
P
P
P
is the intersection point of diagonals and
M
M
M
is the midpoint of arc
A
B
AB
A
B
. Prove that line
M
P
MP
MP
passes through the midpoint of segment
C
D
CD
C
D
, if and only if lines
A
B
,
C
D
AB, CD
A
B
,
C
D
are parallel.