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Problems
Contests
National and Regional Contests
Azerbaijan Contests
JBMO TST - Azerbaijan
2017 Azerbaijan JBMO TST
2017 Azerbaijan JBMO TST
Part of
JBMO TST - Azerbaijan
Subcontests
(4)
4
2
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The central square is illuminated with night lamps
The central square of the City of Mathematicians is an
n
×
n
n\times n
n
×
n
rectangular shape, each paved with
1
×
1
1\times 1
1
×
1
tiles. In order to illuminate the square, night lamps are placed at the corners of the tiles (including the edges of the rectangle) in such a way that each night lamp illuminates all the tiles in its corner. Determine the minimum number of night lamps such that even if one of those night lamps does not work, it is possible to illuminate the entire central square with them.
Gnome country banknotes
The leader of the Gnome country wants to print banknotes in
12
12
12
different denominations (each with an integer number) in such a way that it is possible to pay an arbitrary amount from
1
1
1
to
6543
6543
6543
with these banknotes without a balance, using a maximum of
8
8
8
banknotes. (Several bills with the same denomination can be used during payment.) Can the leader of the land of Gnomes do it?
2
2
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Geometry (similar JBMO 2004)
Let
A
B
C
ABC
A
BC
be isosceles triangle (
A
B
=
B
C
AB=BC
A
B
=
BC
) and
K
K
K
and
M
M
M
be the midpoints of
A
B
AB
A
B
and
A
C
,
AC,
A
C
,
respectively.Let the circumcircle of
△
B
K
C
\triangle BKC
△
B
K
C
meets the line
B
M
BM
BM
at
N
N
N
other than
B
.
B.
B
.
Let the line passing through
N
N
N
and parallel to
A
C
AC
A
C
intersects the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
A
1
A_1
A
1
and
C
1
.
C_1.
C
1
.
Prove that
△
A
1
B
C
1
\triangle A_1BC_1
△
A
1
B
C
1
is equilateral.
Find all the values!
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be 3 different real numbers not equal to
0
0
0
that satisfiying
x
2
−
x
y
=
y
2
−
y
z
=
z
2
−
z
x
x^2-xy=y^2-yz=z^2-zx
x
2
−
x
y
=
y
2
−
yz
=
z
2
−
z
x
. Find all the values of
x
z
+
y
x
+
z
y
\frac{x}{z}+\frac{y}{x}+\frac{z}{y}
z
x
+
x
y
+
y
z
and
(
x
+
y
+
z
)
3
+
9
x
y
z
(x+y+z)^3+9xyz
(
x
+
y
+
z
)
3
+
9
x
yz
.
1
2
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Inequality
Let
x
,
y
,
z
,
t
x,y,z,t
x
,
y
,
z
,
t
be positive numbers.Prove that
x
y
z
t
(
x
+
y
)
(
z
+
t
)
≤
(
x
+
z
)
2
(
y
+
t
)
2
4
(
x
+
y
+
z
+
t
)
2
.
\frac{xyzt}{(x+y)(z+t)}\leq\frac{(x+z)^2(y+t)^2}{4(x+y+z+t)^2}.
(
x
+
y
)
(
z
+
t
)
x
yz
t
≤
4
(
x
+
y
+
z
+
t
)
2
(
x
+
z
)
2
(
y
+
t
)
2
.
Problem 2
a,b,c>0 and
a
b
c
≥
1
abc\ge 1
ab
c
≥
1
.Prove that:
1
a
3
+
2
b
3
+
6
+
1
b
3
+
2
c
3
+
6
+
1
c
3
+
2
a
3
+
6
≤
1
3
\dfrac{1}{a^3+2b^3+6}+\dfrac{1}{b^3+2c^3+6}+\dfrac{1}{c^3+2a^3+6} \le \dfrac{1}{3}
a
3
+
2
b
3
+
6
1
+
b
3
+
2
c
3
+
6
1
+
c
3
+
2
a
3
+
6
1
≤
3
1
3
2
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Nice Number Theory
Let
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
be positive and different divisors of
n
n
n
where
n
∈
Z
+
n \in Z^{+}
n
∈
Z
+
. If
n
=
a
4
+
b
4
+
c
4
+
d
4
+
e
4
n=a^4+b^4+c^4+d^4+e^4
n
=
a
4
+
b
4
+
c
4
+
d
4
+
e
4
let's call
n
n
n
"marvelous" number.
a
)
a)
a
)
Prove that all "marvelous" numbers are divisible by
5
5
5
.
b
)
b)
b
)
Can count of "marvelous" numbers be infinity?
Romania JBMO TST 2016 4
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
and
D
,
E
,
F
D,E,F
D
,
E
,
F
be the contact points of the incircle
(
I
)
(I)
(
I
)
with
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
. Let
M
,
N
M,N
M
,
N
be on
E
F
EF
EF
such that
M
B
⊥
B
C
MB \perp BC
MB
⊥
BC
and
N
C
⊥
B
C
NC \perp BC
NC
⊥
BC
.
M
D
MD
M
D
and
N
D
ND
N
D
intersect the
(
I
)
(I)
(
I
)
in
D
D
D
and
Q
Q
Q
. Prove that
D
P
=
D
Q
DP=DQ
D
P
=
D
Q
.