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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2000 Turkey MO (2nd round)
2000 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
3
2
Hide problems
Turkey NMO 2000 Problem 3, maximum number of couples (x,y)
Let
f
(
x
,
y
)
f(x,y)
f
(
x
,
y
)
and
g
(
x
,
y
)
g(x,y)
g
(
x
,
y
)
be real valued functions defined for every
x
,
y
∈
{
1
,
2
,
.
.
,
2000
}
x,y \in \{1,2,..,2000\}
x
,
y
∈
{
1
,
2
,
..
,
2000
}
. If there exist
X
,
Y
⊂
{
1
,
2
,
.
.
,
2000
}
X,Y \subset \{1,2,..,2000\}
X
,
Y
⊂
{
1
,
2
,
..
,
2000
}
such that
s
(
X
)
=
s
(
Y
)
=
1000
s(X)=s(Y)=1000
s
(
X
)
=
s
(
Y
)
=
1000
and
x
∉
X
x\notin X
x
∈
/
X
and
y
∉
Y
y\notin Y
y
∈
/
Y
implies that
f
(
x
,
y
)
=
g
(
x
,
y
)
f(x,y)=g(x,y)
f
(
x
,
y
)
=
g
(
x
,
y
)
than, what is the maximum number of
(
x
,
y
)
(x,y)
(
x
,
y
)
couples where
f
(
x
,
y
)
≠
g
(
x
,
y
)
f(x,y)\neq g(x,y)
f
(
x
,
y
)
=
g
(
x
,
y
)
.
Turkey NMO 2000 Problem 6, Find All Functions Such That ...
Find all continuous functions
f
:
[
0
,
1
]
→
[
0
,
1
]
f:[0,1]\to [0,1]
f
:
[
0
,
1
]
→
[
0
,
1
]
for which there exists a positive integer
n
n
n
such that
f
n
(
x
)
=
x
f^{n}(x)=x
f
n
(
x
)
=
x
for
x
∈
[
0
,
1
]
x \in [0,1]
x
∈
[
0
,
1
]
where
f
0
(
x
)
=
x
f^{0} (x)=x
f
0
(
x
)
=
x
and
f
k
+
1
=
f
(
f
k
(
x
)
)
f^{k+1}=f(f^{k}(x))
f
k
+
1
=
f
(
f
k
(
x
))
for every positive integer
k
k
k
.
2
2
Hide problems
Turkey NMO 2000 Problem 2
Let define
P
n
(
x
)
=
x
n
−
1
+
x
n
−
2
+
x
n
−
3
+
⋯
+
x
+
1
P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1
P
n
(
x
)
=
x
n
−
1
+
x
n
−
2
+
x
n
−
3
+
⋯
+
x
+
1
for every positive integer
n
n
n
. Prove that for every positive integer
a
a
a
one can find a positive integer
n
n
n
and polynomials
R
(
x
)
R(x)
R
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
with integer coefficients such that
P
n
(
x
)
=
[
1
+
a
x
+
x
2
R
(
x
)
]
Q
(
x
)
.
P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).
P
n
(
x
)
=
[
1
+
a
x
+
x
2
R
(
x
)]
Q
(
x
)
.
Turkey NMO 2000 Problem 5, Circles Have a Common Point
A positive real number
a
a
a
and two rays wich intersect at point
A
A
A
are given. Show that all the circles which pass through
A
A
A
and intersect these rays at points
B
B
B
and
C
C
C
where
∣
A
B
∣
+
∣
A
C
∣
=
a
|AB|+|AC|=a
∣
A
B
∣
+
∣
A
C
∣
=
a
have a common point other than
A
A
A
.
1
2
Hide problems
Turkey NMO 2000 Problem 1, locus of point P_B
A circle with center
O
O
O
and a point
A
A
A
in this circle are given. Let
P
B
P_{B}
P
B
is the intersection point of
[
A
B
]
[AB]
[
A
B
]
and the internal bisector of
∠
A
O
B
\angle AOB
∠
A
OB
where
B
B
B
is a point on the circle such that
B
B
B
doesn't lie on the line
O
A
OA
O
A
, Find the locus of
P
B
P_{B}
P
B
as
B
B
B
varies.
Turkey NMO 2000 Problem 4, OT's Favourite Question
Let
p
p
p
be a prime number.
T
(
x
)
T(x)
T
(
x
)
is a polynomial with integer coefficients and degree from the set
{
0
,
1
,
.
.
.
,
p
−
1
}
\{0,1,...,p-1\}
{
0
,
1
,
...
,
p
−
1
}
and such that
T
(
n
)
≡
T
(
m
)
(
m
o
d
p
)
T(n) \equiv T(m) (mod p)
T
(
n
)
≡
T
(
m
)
(
m
o
d
p
)
for some integers m and n implies that
m
≡
n
(
m
o
d
p
)
m \equiv n (mod p)
m
≡
n
(
m
o
d
p
)
. Determine the maximum possible value of degree of
T
(
x
)
T(x)
T
(
x
)