MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
JBMO TST - Azerbaijan
2022 Azerbaijan JBMO TST
2022 Azerbaijan JBMO TST
Part of
JBMO TST - Azerbaijan
Subcontests
(3)
C4
1
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3n x 3n table with interesting coloring
n
n
n
is a natural number. Given
3
n
⋅
3
n
3n \cdot 3n
3
n
⋅
3
n
table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen
2
⋅
2
2 \cdot 2
2
⋅
2
square's white cells are colored orange, orange are colored black and black are colored white. Find all
n
n
n
such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)
G3
1
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Triangles, orthocenter and reflection, proving AF intersects middle point BC
In acute, scalene Triangle
A
B
C
ABC
A
BC
,
H
H
H
is orthocenter,
B
D
BD
B
D
and
C
E
CE
CE
are heights.
X
,
Y
X,Y
X
,
Y
are reflection of
A
A
A
from
D
D
D
,
E
E
E
respectively such that the points
X
,
Y
X,Y
X
,
Y
are on segments
D
C
DC
D
C
and
E
B
EB
EB
. The intersection of circles
H
X
Y
HXY
H
X
Y
and
A
D
E
ADE
A
D
E
is
F
.
F.
F
.
(
F
≠
H
F \neq H
F
=
H
). Prove that
A
F
AF
A
F
intersects middle point of
B
C
BC
BC
. (
M
M
M
in the diagram is Midpoint of
B
C
BC
BC
)
A2
1
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An inequality where cyc \frac{1}{a} \ge \frac{3}{abc}
For positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
,
1
a
+
1
b
+
1
c
≥
3
a
b
c
\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}
a
1
+
b
1
+
c
1
≥
ab
c
3
is true. Prove that:
a
2
+
b
2
a
2
+
b
2
+
1
+
b
2
+
c
2
b
2
+
c
2
+
1
+
c
2
+
a
2
c
2
+
a
2
+
1
≥
2
\frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2
a
2
+
b
2
+
1
a
2
+
b
2
+
b
2
+
c
2
+
1
b
2
+
c
2
+
c
2
+
a
2
+
1
c
2
+
a
2
≥
2