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Problems
Contests
National and Regional Contests
Azerbaijan Contests
Istek Lyceum Math Olympiads
Istek Lyceum Math Olympiad 2016
Istek Lyceum Math Olympiad 2016
Part of
Istek Lyceum Math Olympiads
Subcontests
(4)
2
1
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Geometry
Let
ω
\omega
ω
be the semicircle with diameter
P
Q
PQ
PQ
. A circle
k
k
k
is tangent internally to
ω
\omega
ω
and to segment
P
Q
PQ
PQ
at
C
C
C
. Let
A
B
AB
A
B
be the tangent to
K
K
K
perpendicular to
P
Q
PQ
PQ
, with
A
A
A
on
ω
\omega
ω
and
B
B
B
on the segment
C
Q
CQ
CQ
. Show that
A
C
AC
A
C
bisects angle
∠
P
A
B
\angle PAB
∠
P
A
B
1
1
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Functional Equation
Find all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
for which
f
(
x
+
y
)
=
f
(
x
−
y
)
+
f
(
f
(
1
−
x
y
)
)
f(x+y)=f(x-y)+f(f(1-xy))
f
(
x
+
y
)
=
f
(
x
−
y
)
+
f
(
f
(
1
−
x
y
))
holds for all real numbers
x
x
x
and
y
y
y
3
1
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Number theory
Let
n
n
n
,
m
m
m
and
k
k
k
be positive integers satisfying
(
n
−
1
)
n
(
n
+
1
)
=
m
k
.
(n-1)n(n+1)=m^k.
(
n
−
1
)
n
(
n
+
1
)
=
m
k
.
Prove that
k
=
1.
k=1.
k
=
1.
4
1
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Zeroes in all cells
Zeroes are written in all cells of a
5
×
5
5\times 5
5
×
5
board. We can take an arbitrary cell and increase by 1 the number in this cell and the cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously?