MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2012 Turkey Team Selection Test
2012 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
3
3
Hide problems
Three-variable inequality
For all positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying
a
b
+
b
c
+
c
a
≤
1
,
ab+bc+ca \leq 1,
ab
+
b
c
+
c
a
≤
1
,
prove that
a
+
b
+
c
+
3
≥
8
a
b
c
(
1
a
2
+
1
+
1
b
2
+
1
+
1
c
2
+
1
)
a+b+c+\sqrt{3} \geq 8abc \left(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)
a
+
b
+
c
+
3
≥
8
ab
c
(
a
2
+
1
1
+
b
2
+
1
1
+
c
2
+
1
1
)
A game in a 1xm board
Two players
A
A
A
and
B
B
B
play a game on a
1
×
m
1\times m
1
×
m
board, using
2012
2012
2012
pieces numbered from
1
1
1
to
2012.
2012.
2012.
At each turn,
A
A
A
chooses a piece and
B
B
B
places it to an empty place. After
k
k
k
turns, if all pieces are placed on the board increasingly, then
B
B
B
wins, otherwise
A
A
A
wins. For which values of
(
m
,
k
)
(m,k)
(
m
,
k
)
pairs can
B
B
B
guarantee to win?
Find a subset of Z+ satisfying a property
Let
Z
+
\mathbb{Z^+}
Z
+
and
P
\mathbb{P}
P
denote the set of positive integers and the set of prime numbers, respectively. A set
A
A
A
is called
S
−
proper
S-\text{proper}
S
−
proper
where
A
,
S
⊂
Z
+
A, S \subset \mathbb{Z^+}
A
,
S
⊂
Z
+
if there exists a positive integer
N
N
N
such that for all
a
∈
A
a \in A
a
∈
A
and for all
0
≤
b
<
a
0 \leq b <a
0
≤
b
<
a
there exist
s
1
,
s
2
,
…
,
s
n
∈
S
s_1, s_2, \ldots, s_n \in S
s
1
,
s
2
,
…
,
s
n
∈
S
satisfying
b
≡
s
1
+
s
2
+
⋯
+
s
n
(
m
o
d
a
)
b \equiv s_1+s_2+\cdots+s_n \pmod a
b
≡
s
1
+
s
2
+
⋯
+
s
n
(
mod
a
)
and
1
≤
n
≤
N
.
1 \leq n \leq N.
1
≤
n
≤
N
.
Find a subset
S
S
S
of
Z
+
\mathbb{Z^+}
Z
+
for which
P
\mathbb{P}
P
is
S
−
proper
S-\text{proper}
S
−
proper
but
Z
+
\mathbb{Z^+}
Z
+
is not.
2
3
Hide problems
Sum of perfect squares
A positive integer
n
n
n
is called good if for all positive integers
a
a
a
which can be written as
a
=
n
2
∑
i
=
1
n
x
i
2
a=n^2 \sum_{i=1}^n {x_i}^2
a
=
n
2
∑
i
=
1
n
x
i
2
where
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots ,x_n
x
1
,
x
2
,
…
,
x
n
are integers, it is possible to express
a
a
a
as
a
=
∑
i
=
1
n
y
i
2
a=\sum_{i=1}^n {y_i}^2
a
=
∑
i
=
1
n
y
i
2
where
y
1
,
y
2
,
…
,
y
n
y_1, y_2, \ldots, y_n
y
1
,
y
2
,
…
,
y
n
are integers with none of them is divisible by
n
.
n.
n
.
Find all good numbers.
An isosceles trapezoid
In an acute triangle
A
B
C
,
ABC,
A
BC
,
let
D
D
D
be a point on the side
B
C
.
BC.
BC
.
Let
M
1
,
M
2
,
M
3
,
M
4
,
M
5
M_1, M_2, M_3, M_4, M_5
M
1
,
M
2
,
M
3
,
M
4
,
M
5
be the midpoints of the line segments
A
D
,
A
B
,
A
C
,
B
D
,
C
D
,
AD, AB, AC, BD, CD,
A
D
,
A
B
,
A
C
,
B
D
,
C
D
,
respectively and
O
1
,
O
2
,
O
3
,
O
4
O_1, O_2, O_3, O_4
O
1
,
O
2
,
O
3
,
O
4
be the circumcenters of triangles
A
B
D
,
A
C
D
,
M
1
M
2
M
4
,
M
1
M
3
M
5
,
ABD, ACD, M_1M_2M_4, M_1M_3M_5,
A
B
D
,
A
C
D
,
M
1
M
2
M
4
,
M
1
M
3
M
5
,
respectively. If
S
S
S
and
T
T
T
are midpoints of the line segments
A
O
1
AO_1
A
O
1
and
A
O
2
,
AO_2,
A
O
2
,
respectively, prove that
S
O
3
O
4
T
SO_3O_4T
S
O
3
O
4
T
is an isosceles trapezoid.
Geometric inequality with centroid
In a plane, the six different points
A
,
B
,
C
,
A
′
,
B
′
,
C
′
A, B, C, A', B', C'
A
,
B
,
C
,
A
′
,
B
′
,
C
′
are given such that triangles
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
are congruent, i.e.
A
B
=
A
′
B
′
,
B
C
=
B
′
C
′
,
C
A
=
C
′
A
′
.
AB=A'B', BC=B'C', CA=C'A'.
A
B
=
A
′
B
′
,
BC
=
B
′
C
′
,
C
A
=
C
′
A
′
.
Let
G
G
G
be the centroid of
A
B
C
ABC
A
BC
and
A
1
A_1
A
1
be an intersection point of the circle with diameter
A
A
′
AA'
A
A
′
and the circle with center
A
′
A'
A
′
and passing through
G
.
G.
G
.
Define
B
1
B_1
B
1
and
C
1
C_1
C
1
similarly. Prove that
A
A
1
2
+
B
B
1
2
+
C
C
1
2
≤
A
B
2
+
B
C
2
+
C
A
2
AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2
A
A
1
2
+
B
B
1
2
+
C
C
1
2
≤
A
B
2
+
B
C
2
+
C
A
2
1
3
Hide problems
Find the number of functions
Let
A
=
{
1
,
2
,
…
,
2012
}
,
B
=
{
1
,
2
,
…
,
19
}
A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}
A
=
{
1
,
2
,
…
,
2012
}
,
B
=
{
1
,
2
,
…
,
19
}
and
S
S
S
be the set of all subsets of
A
.
A.
A
.
Find the number of functions
f
:
S
→
B
f : S\to B
f
:
S
→
B
satisfying
f
(
A
1
∩
A
2
)
=
min
{
f
(
A
1
)
,
f
(
A
2
)
}
f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}
f
(
A
1
∩
A
2
)
=
min
{
f
(
A
1
)
,
f
(
A
2
)}
for all
A
1
,
A
2
∈
S
.
A_1, A_2 \in S.
A
1
,
A
2
∈
S
.
A circumradius equation
In a triangle
A
B
C
,
ABC,
A
BC
,
incircle touches the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
,
D, E, F,
D
,
E
,
F
,
respectively. A circle
ω
\omega
ω
passing through
A
A
A
and tangent to line
B
C
BC
BC
at
D
D
D
intersects the line segments
B
F
BF
BF
and
C
E
CE
CE
at
K
K
K
and
L
,
L,
L
,
respectively. The line passing through
E
E
E
and parallel to
D
L
DL
D
L
intersects the line passing through
F
F
F
and parallel to
D
K
DK
DK
at
P
.
P.
P
.
If
R
1
,
R
2
,
R
3
,
R
4
R_1, R_2, R_3, R_4
R
1
,
R
2
,
R
3
,
R
4
denotes the circumradius of the triangles
A
F
D
,
A
E
D
,
F
P
D
,
E
P
D
,
AFD, AED, FPD, EPD,
A
F
D
,
A
E
D
,
FP
D
,
EP
D
,
respectively, prove that
R
1
R
4
=
R
2
R
3
.
R_1R_4=R_2R_3.
R
1
R
4
=
R
2
R
3
.
Sum of powers
Let
S
r
(
n
)
=
1
r
+
2
r
+
⋯
+
n
r
S_r(n)=1^r+2^r+\cdots+n^r
S
r
(
n
)
=
1
r
+
2
r
+
⋯
+
n
r
where
n
n
n
is a positive integer and
r
r
r
is a rational number. If
S
a
(
n
)
=
(
S
b
(
n
)
)
c
S_a(n)=(S_b(n))^c
S
a
(
n
)
=
(
S
b
(
n
)
)
c
for all positive integers
n
n
n
where
a
,
b
a, b
a
,
b
are positive rationals and
c
c
c
is positive integer then we call
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
as nice triple. Find all nice triples.