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Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan BMO TST
2017 Azerbaijan BMO TST
2017 Azerbaijan BMO TST
Part of
Azerbaijan BMO TST
Subcontests
(2)
3
2
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Nice problem
Find all funtions
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
such that:
f
(
x
y
−
1
)
+
f
(
x
)
f
(
y
)
=
2
x
y
−
1
f(xy-1)+f(x)f(y)=2xy-1
f
(
x
y
−
1
)
+
f
(
x
)
f
(
y
)
=
2
x
y
−
1
for all
x
,
y
∈
R
x,y\in \mathbb{R}
x
,
y
∈
R
.
Romania TST 2016 Day 1 P1
Two circles,
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
, centered at
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, meet at points
A
A
A
and
B
B
B
. A line through
B
B
B
meet
ω
1
\omega_1
ω
1
again at
C
C
C
, and
ω
2
\omega_2
ω
2
again at
D
D
D
. The tangents to
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
at
C
C
C
and
D
D
D
, respectively, meet at
E
E
E
, and the line
A
E
AE
A
E
meets the circle
ω
\omega
ω
through
A
,
O
1
,
O
2
A, O_1,O_2
A
,
O
1
,
O
2
again at
F
F
F
. Prove that the length of the segment
E
F
EF
EF
is equal to the diameter of
ω
\omega
ω
.
1
1
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Angle equal
Let
△
A
B
C
\triangle ABC
△
A
BC
be a acute triangle. Let
H
H
H
the foot of the C-altitude in
A
B
AB
A
B
such that
A
H
=
3
B
H
AH=3BH
A
H
=
3
B
H
, let
M
M
M
and
N
N
N
the midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
and let
P
P
P
be a point such that
N
P
=
N
C
NP=NC
NP
=
NC
and
C
P
=
C
B
CP=CB
CP
=
CB
and
B
B
B
,
P
P
P
are located on different sides of the line
A
C
AC
A
C
. Prove that
∡
A
P
M
=
∡
P
B
A
\measuredangle APM=\measuredangle PBA
∡
A
PM
=
∡
PB
A
.