MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2011 Turkey Team Selection Test
2011 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
1
2
Hide problems
Incenters and an equation
Let
D
D
D
be a point different from the vertices on the side
B
C
BC
BC
of a triangle
A
B
C
.
ABC.
A
BC
.
Let
I
,
I
1
I, \: I_1
I
,
I
1
and
I
2
I_2
I
2
be the incenters of the triangles
A
B
C
,
A
B
D
ABC, \: ABD
A
BC
,
A
B
D
and
A
D
C
,
ADC,
A
D
C
,
respectively. Let
E
E
E
be the second intersection point of the circumcircles of the triangles
A
I
1
I
AI_1I
A
I
1
I
and
A
D
I
2
,
ADI_2,
A
D
I
2
,
and
F
F
F
be the second intersection point of the circumcircles of the triangles
A
I
I
2
AII_2
A
I
I
2
and
A
I
1
D
.
AI_1D.
A
I
1
D
.
Prove that if
A
I
1
=
A
I
2
,
AI_1=AI_2,
A
I
1
=
A
I
2
,
then
E
I
F
I
⋅
E
D
F
D
=
E
I
1
2
F
I
1
2
.
\frac{EI}{FI} \cdot \frac{ED}{FD}=\frac{{EI_1}^2}{{FI_1}^2}.
F
I
E
I
⋅
F
D
E
D
=
F
I
1
2
E
I
1
2
.
Geometric inequality
Let
K
K
K
be a point in the interior of an acute triangle
A
B
C
ABC
A
BC
and
A
R
B
P
C
Q
ARBPCQ
A
RBPCQ
be a convex hexagon whose vertices lie on the circumcircle
Γ
\Gamma
Γ
of the triangle
A
B
C
.
ABC.
A
BC
.
Let
A
1
A_1
A
1
be the second point where the circle passing through
K
K
K
and tangent to
Γ
\Gamma
Γ
at
A
A
A
intersects the line
A
P
.
AP.
A
P
.
The points
B
1
B_1
B
1
and
C
1
C_1
C
1
are defined similarly. Prove that
min
{
P
A
1
A
A
1
,
Q
B
1
B
B
1
,
R
C
1
C
C
1
}
≤
1.
\min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.
min
{
A
A
1
P
A
1
,
B
B
1
Q
B
1
,
C
C
1
R
C
1
}
≤
1.
3
3
Hide problems
Show that there exists a function
Let
A
A
A
and
B
B
B
be sets with
201
1
2
2011^2
201
1
2
and
2010
2010
2010
elements, respectively. Show that there is a function
f
:
A
×
A
→
B
f:A \times A \to B
f
:
A
×
A
→
B
satisfying the condition
f
(
x
,
y
)
=
f
(
y
,
x
)
f(x,y)=f(y,x)
f
(
x
,
y
)
=
f
(
y
,
x
)
for all
(
x
,
y
)
∈
A
×
A
(x,y) \in A \times A
(
x
,
y
)
∈
A
×
A
such that for every function
g
:
A
→
B
g:A \to B
g
:
A
→
B
there exists
(
a
1
,
a
2
)
∈
A
×
A
(a_1,a_2) \in A \times A
(
a
1
,
a
2
)
∈
A
×
A
with
g
(
a
1
)
=
f
(
a
1
,
a
2
)
=
g
(
a
2
)
g(a_1)=f(a_1,a_2)=g(a_2)
g
(
a
1
)
=
f
(
a
1
,
a
2
)
=
g
(
a
2
)
and
a
1
≠
a
2
.
a_1 \neq a_2.
a
1
=
a
2
.
Binary representations and sum of the digits
Let
t
(
n
)
t(n)
t
(
n
)
be the sum of the digits in the binary representation of a positive integer
n
,
n,
n
,
and let
k
≥
2
k \geq 2
k
≥
2
be an integer.a. Show that there exists a sequence
(
a
i
)
i
=
1
∞
(a_i)_{i=1}^{\infty}
(
a
i
)
i
=
1
∞
of integers such that
a
m
≥
3
a_m \geq 3
a
m
≥
3
is an odd integer and
t
(
a
1
a
2
⋯
a
m
)
=
k
t(a_1a_2 \cdots a_m)=k
t
(
a
1
a
2
⋯
a
m
)
=
k
for all
m
≥
1.
m \geq 1.
m
≥
1.
b. Show that there is an integer
N
N
N
such that
t
(
3
⋅
5
⋯
(
2
m
+
1
)
)
>
k
t(3 \cdot 5 \cdots (2m+1))>k
t
(
3
⋅
5
⋯
(
2
m
+
1
))
>
k
for all integers
m
≥
N
.
m \geq N.
m
≥
N
.
Determine the number of functions
Let
p
p
p
be a prime,
n
n
n
be a positive integer, and let
Z
p
n
\mathbb{Z}_{p^n}
Z
p
n
denote the set of congruence classes modulo
p
n
.
p^n.
p
n
.
Determine the number of functions
f
:
Z
p
n
→
Z
p
n
f: \mathbb{Z}_{p^n} \to \mathbb{Z}_{p^n}
f
:
Z
p
n
→
Z
p
n
satisfying the condition
f
(
a
)
+
f
(
b
)
≡
f
(
a
+
b
+
p
a
b
)
(
m
o
d
p
n
)
f(a)+f(b) \equiv f(a+b+pab) \pmod{p^n}
f
(
a
)
+
f
(
b
)
≡
f
(
a
+
b
+
p
ab
)
(
mod
p
n
)
for all
a
,
b
∈
Z
p
n
.
a,b \in \mathbb{Z}_{p^n}.
a
,
b
∈
Z
p
n
.
2
3
Hide problems
Prove that two lines are perpendicular
Let
I
I
I
be the incenter and
A
D
AD
A
D
be a diameter of the circumcircle of a triangle
A
B
C
.
ABC.
A
BC
.
If the point
E
E
E
on the ray
B
A
BA
B
A
and the point
F
F
F
on the ray
C
A
CA
C
A
satisfy the condition
B
E
=
C
F
=
A
B
+
B
C
+
C
A
2
BE=CF=\frac{AB+BC+CA}{2}
BE
=
CF
=
2
A
B
+
BC
+
C
A
show that the lines
E
F
EF
EF
and
D
I
DI
D
I
are perpendicular.
Inequality with a condition
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers satisfying
a
2
+
b
2
+
c
2
≥
3.
a^2+b^2+c^2 \geq 3.
a
2
+
b
2
+
c
2
≥
3.
Prove that
(
a
+
1
)
(
b
+
2
)
(
b
+
1
)
(
b
+
5
)
+
(
b
+
1
)
(
c
+
2
)
(
c
+
1
)
(
c
+
5
)
+
(
c
+
1
)
(
a
+
2
)
(
a
+
1
)
(
a
+
5
)
≥
3
2
\frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}
(
b
+
1
)
(
b
+
5
)
(
a
+
1
)
(
b
+
2
)
+
(
c
+
1
)
(
c
+
5
)
(
b
+
1
)
(
c
+
2
)
+
(
a
+
1
)
(
a
+
5
)
(
c
+
1
)
(
a
+
2
)
≥
2
3
Graphistan
Graphistan has
2011
2011
2011
cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer
k
k
k
such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than
k
.
k.
k
.