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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2017 Turkey MO (2nd round)
2017 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
6
1
Hide problems
Turkey nmo 2017 p6
Finite number of
2017
2017
2017
units long sticks are fixed on a plate. Each stick has a bead that can slide up and down on it. Beads can only stand on integer heights
(
1
,
2
,
3
,
.
.
.
,
2017
)
( 1, 2, 3,..., 2017 )
(
1
,
2
,
3
,
...
,
2017
)
. Some of the bead pairs are connected with elastic bands.
T
h
e
The
T
h
e
y
o
u
n
g
young
yo
u
n
g
a
n
t
ant
an
t
can go to every bead, starting from any bead by using the elastic bands.
T
h
e
The
T
h
e
o
l
d
old
o
l
d
a
n
t
ant
an
t
can use an elastic band if the difference in height of the beads which are connected by the band, is smaller than or equal to
1
1
1
. If the heights of the beads which are connected to each other are different, we call it
v
a
l
i
d
valid
v
a
l
i
d
s
i
t
u
a
t
i
o
n
situation
s
i
t
u
a
t
i
o
n
. If there exists at least one
v
a
l
i
d
valid
v
a
l
i
d
s
i
t
u
a
t
i
o
n
situation
s
i
t
u
a
t
i
o
n
, prove that we can create a
v
a
l
i
d
valid
v
a
l
i
d
s
i
t
u
a
t
i
o
n
situation
s
i
t
u
a
t
i
o
n
, by arranging the heights of the beads, in which
t
h
e
the
t
h
e
o
l
d
old
o
l
d
a
n
t
ant
an
t
can go to every bead, starting from any bead.
1
1
Hide problems
Turkey NMO 2017 p1
A wedding is going to be held in a city with
25
25
25
types of meals, to which some of the
2017
2017
2017
citizens will be invited. All of the citizens like some meals and each meal is liked by at least one person. A "
s
u
i
t
a
b
l
e
suitable
s
u
i
t
ab
l
e
l
i
s
t
list
l
i
s
t
" is a set of citizens, such that each meal is liked by at least one person in the set. A "
k
a
m
b
e
r
kamber
kamb
er
g
r
o
u
p
group
g
ro
u
p
" is a set that contains at least one person from each "
s
u
i
t
a
b
l
e
suitable
s
u
i
t
ab
l
e
l
i
s
t
list
l
i
s
t
". Given a "
k
a
m
b
e
r
kamber
kamb
er
g
r
o
u
p
group
g
ro
u
p
", which has no subset (other than itself) that is also a "
k
a
m
b
e
r
kamber
kamb
er
g
r
o
u
p
group
g
ro
u
p
", prove that there exists a meal, which is liked by everyone in the group.
5
1
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Turkey NMO 2017 p5
Let
x
0
,
…
,
x
2017
x_0,\dots,x_{2017}
x
0
,
…
,
x
2017
are positive integers and
x
2017
≥
⋯
≥
x
0
=
1
x_{2017}\geq\dots\geq x_0=1
x
2017
≥
⋯
≥
x
0
=
1
such that
A
=
{
x
1
,
…
,
x
2017
}
A=\{x_1,\dots,x_{2017}\}
A
=
{
x
1
,
…
,
x
2017
}
consists of exactly
25
25
25
different numbers. Prove that
∑
i
=
2
2017
(
x
i
−
x
i
−
2
)
x
i
≥
623
\sum_{i=2}^{2017}(x_i-x_{i-2})x_i\geq 623
∑
i
=
2
2017
(
x
i
−
x
i
−
2
)
x
i
≥
623
, and find the number of sequences that holds the case of equality.
4
1
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Turkey NMO 2017 p4
Let
d
(
n
)
d(n)
d
(
n
)
be number of prime divisors of
n
n
n
. Prove that one can find
k
,
m
k,m
k
,
m
positive integers for any positive integer
n
n
n
such that
k
−
m
=
n
k-m=n
k
−
m
=
n
and
d
(
k
)
−
d
(
m
)
=
1
d(k)-d(m)=1
d
(
k
)
−
d
(
m
)
=
1
3
1
Hide problems
Turkey NMO 2017 p3
Denote the sequence
a
i
,
j
a_{i,j}
a
i
,
j
in positive reals such that
a
i
,
j
a_{i,j}
a
i
,
j
.
a
j
,
i
=
1
a_{j,i}=1
a
j
,
i
=
1
. Let
c
i
=
∑
k
=
1
n
a
k
,
i
c_i=\sum_{k=1}^{n}a_{k,i}
c
i
=
∑
k
=
1
n
a
k
,
i
. Prove that
1
≥
1\ge
1
≥
∑
i
=
1
n
1
c
i
\sum_{i=1}^{n}\dfrac {1}{c_i}
∑
i
=
1
n
c
i
1
2
1
Hide problems
Turkey NMO 2017 p2
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral such that line
A
B
AB
A
B
intersects
C
D
CD
C
D
at
X
X
X
. Denote circles with inradius
r
1
r_1
r
1
and centers
A
,
B
A, B
A
,
B
as
w
a
w_a
w
a
and
w
b
w_b
w
b
with inradius
r
2
r_2
r
2
and centers
C
,
D
C, D
C
,
D
as
w
c
w_c
w
c
and
w
d
w_d
w
d
.
w
a
w_a
w
a
intersects
w
d
w_d
w
d
at
P
,
Q
P, Q
P
,
Q
.
w
b
w_b
w
b
intersects
w
c
w_c
w
c
at
R
,
S
R, S
R
,
S
. Prove that if
X
A
.
X
B
+
r
2
2
=
X
C
.
X
D
+
r
1
2
XA.XB+r_2^2=XC.XD+r_1^2
X
A
.
XB
+
r
2
2
=
XC
.
X
D
+
r
1
2
, then
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
are cyclic.