MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2002 Turkey MO (2nd round)
2002 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
1
2
Hide problems
Stable Configuration
Let
(
a
1
,
a
2
,
…
,
a
n
)
(a_1, a_2,\ldots , a_n)
(
a
1
,
a
2
,
…
,
a
n
)
be a permutation of
1
,
2
,
…
,
n
,
1, 2, \ldots , n,
1
,
2
,
…
,
n
,
where n \geq 2. For each
k
=
1
,
…
,
n
k = 1, \ldots , n
k
=
1
,
…
,
n
, we know that
a
k
a_k
a
k
apples are placed at the point
k
k
k
on the real axis. Children named
A
,
B
,
C
A,B,C
A
,
B
,
C
are assigned respective points
x
A
,
x
B
,
x
C
∈
{
1
,
…
,
n
}
.
x_A, x_B, x_C \in \{1, \ldots , n\}.
x
A
,
x
B
,
x
C
∈
{
1
,
…
,
n
}
.
For each
k
,
k,
k
,
the children whose points are closest to
k
k
k
divide
a
k
a_k
a
k
apples equally among themselves. We call
(
x
A
,
x
B
,
x
C
)
(x_A, x_B, x_C)
(
x
A
,
x
B
,
x
C
)
a stable configuration if no child’s total share can be increased by assigning a new point to this child and not changing the points of the other two. Determine the values of
n
n
n
for which a stable configuration exists for some distribution
(
a
1
,
…
,
a
n
)
(a_1, \ldots, a_n)
(
a
1
,
…
,
a
n
)
of the apples.
On the equation y^2 ≡ x^3 - x (mod p)
Find all prime numbers
p
p
p
for which the number of ordered pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
with
0
≤
x
,
y
<
p
0\leq x, y < p
0
≤
x
,
y
<
p
satisfying the condition y^2 \equiv x^3 - x \pmod p is exactly
p
.
p.
p
.
2
2
Hide problems
A is the incenter if the centers of circles aren't collinear
Two circles are externally tangent to each other at a point
A
A
A
and internally tangent to a third circle
Γ
\Gamma
Γ
at points
B
B
B
and
C
.
C.
C
.
Let
D
D
D
be the midpoint of the secant of
Γ
\Gamma
Γ
which is tangent to the smaller circles at
A
.
A.
A
.
Show that
A
A
A
is the incenter of the triangle
B
C
D
BCD
BC
D
if the centers of the circles are not collinear.
Equal to distance between the incenter and the circumcenter
Let
A
B
C
ABC
A
BC
be a triangle, and points
D
,
E
D,E
D
,
E
are on
B
A
,
C
A
BA,CA
B
A
,
C
A
respectively such that
D
B
=
B
C
=
C
E
DB=BC=CE
D
B
=
BC
=
CE
. Let
O
,
I
O,I
O
,
I
be the circumcenter, incenter of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that the circumradius of
△
A
D
E
\triangle ADE
△
A
D
E
is equal to
O
I
OI
O
I
.
3
2
Hide problems
Graph Airlines (GA)
Graph Airlines
(
G
A
)
(GA)
(
G
A
)
operates flights between some of the cities of the Republic of Graphia. There are at least three
G
A
GA
G
A
flights from each city, and it is possible to travel from any city in Graphia to any city in Graphia using
G
A
GA
G
A
flights.
G
A
GA
G
A
decides to discontinue some of its flights. Show that this can be done in such a way that it is still possible to travel between any two cities using
G
A
GA
G
A
flights, yet at least
2
/
9
2/9
2/9
of the cities have only one flight.
Show that there exists real number d
Let
n
n
n
be a positive integer and let
T
T
T
denote the collection of points
(
x
1
,
x
2
,
…
,
x
n
)
∈
R
n
(x_1, x_2, \ldots, x_n) \in \mathbb R^n
(
x
1
,
x
2
,
…
,
x
n
)
∈
R
n
for which there exists a permutation
σ
\sigma
σ
of
1
,
2
,
…
,
n
1, 2, \ldots , n
1
,
2
,
…
,
n
such that
x
σ
(
i
)
−
x
σ
(
i
+
1
)
≥
1
x_{\sigma(i)} - x_{\sigma(i+1) } \geq 1
x
σ
(
i
)
−
x
σ
(
i
+
1
)
≥
1
for each
i
=
1
,
2
,
…
,
n
.
i=1, 2, \ldots , n.
i
=
1
,
2
,
…
,
n
.
Prove that there is a real number
d
d
d
satisfying the following condition: For every
(
a
1
,
a
2
,
…
,
a
n
)
∈
R
n
(a_1, a_2, \ldots, a_n) \in \mathbb R^n
(
a
1
,
a
2
,
…
,
a
n
)
∈
R
n
there exist points
(
b
1
,
…
,
b
n
)
(b_1, \ldots, b_n)
(
b
1
,
…
,
b
n
)
and
(
c
1
,
…
,
c
n
)
(c_1,\ldots, c_n)
(
c
1
,
…
,
c
n
)
in
T
T
T
such that, for each
i
=
1
,
.
.
.
,
n
,
i = 1, . . . , n,
i
=
1
,
...
,
n
,
a_i=\frac 12 (b_i+c_i) , |a_i - b_i| \leq d, \text{and} |a_i - c_i| \leq d.