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Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan Junior National Olympiad
2020 Azerbaijan National Olympiad
2020 Azerbaijan National Olympiad
Part of
Azerbaijan Junior National Olympiad
Subcontests
(4)
4
1
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Problem 4
There is a non-equilateral triangle
A
B
C
ABC
A
BC
.Let
A
B
C
ABC
A
BC
's Incentri
I
I
I
.Point
D
D
D
is on the
B
C
BC
BC
side.The circle drawn outside the triangle
I
B
D
IBD
I
B
D
and
I
C
D
ICD
I
C
D
intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
E
E
E
and
F
.
F.
F
.
The circle drawn outside the triangle
D
E
F
DEF
D
EF
intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
at
N
N
N
and
M
M
M
.Prove that
E
M
∥
F
N
EM\parallel FN
EM
∥
FN
.
3
1
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Problem 3
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive numbers.
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
Prove that:
∑
a
2
+
6
2
a
2
+
2
b
2
+
2
c
2
+
2
a
−
1
≤
3
\sum \frac{a^2+6}{2a^2+2b^2+2c^2+2a-1}\leq 3
∑
2
a
2
+
2
b
2
+
2
c
2
+
2
a
−
1
a
2
+
6
≤
3
2
1
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Problem 2
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive integer. Solve the equation:
2
a
!
+
2
b
!
=
c
3
2^{a!}+2^{b!}=c^3
2
a
!
+
2
b
!
=
c
3
1
1
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Problem 1
13
13
13
fractions are corrected by using each of the numbers
1
,
2
,
.
.
.
,
26
1,2,...,26
1
,
2
,
...
,
26
once.Example:
12
5
,
18
26
.
.
.
.
\frac{12}{5},\frac{18}{26}....
5
12
,
26
18
....
What is the maximum number of fractions which are integers?