MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2010 Turkey Team Selection Test
2010 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
3
2
Hide problems
Turkey TST 2010 Q3
A teacher wants to divide the
2010
2010
2010
questions she asked in the exams during the school year into three folders of
670
670
670
questions and give each folder to a student who solved all
670
670
670
questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.
Turkey TST 2010 Q6
Let
Λ
\Lambda
Λ
be the set of points in the plane whose coordinates are integers and let
F
F
F
be the collection of all functions from
Λ
\Lambda
Λ
to
{
1
,
−
1
}
.
\{1,-1\}.
{
1
,
−
1
}
.
We call a function
f
f
f
in
F
F
F
perfect if every function
g
g
g
in
F
F
F
that differs from
f
f
f
at finitely many points satisfies the condition
∑
0
<
d
(
P
,
Q
)
<
2010
f
(
P
)
f
(
Q
)
−
g
(
P
)
g
(
Q
)
d
(
P
,
Q
)
≥
0
\sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0
0
<
d
(
P
,
Q
)
<
2010
∑
d
(
P
,
Q
)
f
(
P
)
f
(
Q
)
−
g
(
P
)
g
(
Q
)
≥
0
where
d
(
P
,
Q
)
d(P,Q)
d
(
P
,
Q
)
denotes the distance between
P
P
P
and
Q
.
Q.
Q
.
Show that there exist infinitely many perfect functions that are not translates of each other.
2
2
Hide problems
Turkey TST 2010 Q2
Show that
∑
c
y
c
(
a
2
+
b
2
)
(
a
2
−
a
b
+
b
2
)
2
4
≤
2
3
(
a
2
+
b
2
+
c
2
)
(
1
a
+
b
+
1
b
+
c
+
1
c
+
a
)
\sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)
cyc
∑
4
2
(
a
2
+
b
2
)
(
a
2
−
ab
+
b
2
)
≤
3
2
(
a
2
+
b
2
+
c
2
)
(
a
+
b
1
+
b
+
c
1
+
c
+
a
1
)
for all positive real numbers
a
,
b
,
c
.
a, \: b, \: c.
a
,
b
,
c
.
Turkey TST 2010 Q5
For an interior point
D
D
D
of a triangle
A
B
C
,
ABC,
A
BC
,
let
Γ
D
\Gamma_D
Γ
D
denote the circle passing through the points
A
,
E
,
D
,
F
A, \: E, \: D, \: F
A
,
E
,
D
,
F
if these points are concyclic where
B
D
∩
A
C
=
{
E
}
BD \cap AC=\{E\}
B
D
∩
A
C
=
{
E
}
and
C
D
∩
A
B
=
{
F
}
.
CD \cap AB=\{F\}.
C
D
∩
A
B
=
{
F
}
.
Show that all circles
Γ
D
\Gamma_D
Γ
D
pass through a second common point different from
A
A
A
as
D
D
D
varies.
1
2
Hide problems
Turkey TST 2010 Q1
D
,
E
,
F
D, \: E , \: F
D
,
E
,
F
are points on the sides
A
B
,
B
C
,
C
A
,
AB, \: BC, \: CA,
A
B
,
BC
,
C
A
,
respectively, of a triangle
A
B
C
ABC
A
BC
such that
A
D
=
A
F
,
B
D
=
B
E
,
AD=AF, \: BD=BE,
A
D
=
A
F
,
B
D
=
BE
,
and
D
E
=
D
F
.
DE=DF.
D
E
=
D
F
.
Let
I
I
I
be the incenter of the triangle
A
B
C
,
ABC,
A
BC
,
and let
K
K
K
be the point of intersection of the line
B
I
BI
B
I
and the tangent line through
A
A
A
to the circumcircle of the triangle
A
B
I
.
ABI.
A
B
I
.
Show that
A
K
=
E
K
AK=EK
A
K
=
E
K
if
A
K
=
A
D
.
AK=AD.
A
K
=
A
D
.
Turkey TST 2010 Q4
Let
0
≤
k
<
n
0 \leq k < n
0
≤
k
<
n
be integers and
A
=
{
a
:
a
≡
k
(
m
o
d
n
)
}
.
A=\{a \: : \: a \equiv k \pmod n \}.
A
=
{
a
:
a
≡
k
(
mod
n
)}
.
Find the smallest value of
n
n
n
for which the expression
a
m
+
3
m
a
2
−
3
a
+
1
\frac{a^m+3^m}{a^2-3a+1}
a
2
−
3
a
+
1
a
m
+
3
m
does not take any integer values for
(
a
,
m
)
∈
A
×
Z
+
.
(a,m) \in A \times \mathbb{Z^+}.
(
a
,
m
)
∈
A
×
Z
+
.