MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2014 Turkey Team Selection Test
2014 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
1
2
Hide problems
Number of Permutations
Find the number of
(
a
1
,
a
2
,
.
.
.
,
a
2014
)
(a_1,a_2, ... ,a_{2014})
(
a
1
,
a
2
,
...
,
a
2014
)
permutations of the
(
1
,
2
,
.
.
.
,
2014
)
(1,2, . . . ,2014)
(
1
,
2
,
...
,
2014
)
such that, for all
1
≤
i
<
j
≤
2014
1\leq i<j\leq2014
1
≤
i
<
j
≤
2014
,
i
+
a
i
≤
j
+
a
j
i+a_i \leq j+a_j
i
+
a
i
≤
j
+
a
j
.
Angle Equality
Let
P
P
P
be a point inside the acute triangle
A
B
C
ABC
A
BC
with
m
(
P
A
C
^
)
=
m
(
P
C
B
^
)
m(\widehat{PAC})=m(\widehat{PCB})
m
(
P
A
C
)
=
m
(
PCB
)
.
D
D
D
is the midpoint of the segment
P
C
PC
PC
.
A
P
AP
A
P
and
B
C
BC
BC
intersect at
E
E
E
, and
B
P
BP
BP
and
D
E
DE
D
E
intersect at
Q
Q
Q
. Prove that
sin
B
C
Q
^
=
sin
B
A
P
^
\sin\widehat{BCQ}=\sin\widehat{BAP}
sin
BCQ
=
sin
B
A
P
.
3
3
Hide problems
Ttriangles with the same area
Let
r
,
R
r,R
r
,
R
and
r
a
r_a
r
a
be the radii of the incircle, circumcircle and A-excircle of the triangle
A
B
C
ABC
A
BC
with
A
C
>
A
B
AC>AB
A
C
>
A
B
, respectively.
I
,
O
I,O
I
,
O
and
J
A
J_A
J
A
are the centers of these circles, respectively. Let incircle touches the
B
C
BC
BC
at
D
D
D
, for a point
E
∈
(
B
D
)
E \in (BD)
E
∈
(
B
D
)
the condition
A
(
I
E
J
A
)
=
2
A
(
I
E
O
)
A(IEJ_A)=2A(IEO)
A
(
I
E
J
A
)
=
2
A
(
I
EO
)
holds. Prove that
E
D
=
A
C
−
A
B
⟺
R
=
2
r
+
r
a
.
ED=AC-AB \iff R=2r+r_a.
E
D
=
A
C
−
A
B
⟺
R
=
2
r
+
r
a
.
Hard ineq
Prove that for all all non-negative real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
with
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
a
+
b
+
a
+
c
+
b
+
c
≥
5
a
b
c
+
2.
\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c} \geq 5abc+2.
a
+
b
+
a
+
c
+
b
+
c
≥
5
ab
c
+
2.
Worms that are allowed to move one way
At the bottom-left corner of a
2014
×
2014
2014\times 2014
2014
×
2014
chessboard, there are some green worms and at the top-left corner of the same chessboard, there are some brown worms. Green worms can move only to right and up, and brown worms can move only to right and down. After a while, the worms make some moves and all of the unit squares of the chessboard become occupied at least once throughout this process. Find the minimum total number of the worms.
2
3
Hide problems
Hard Functional Equation
Find all
f
f
f
functions from real numbers to itself such that for all real numbers
x
,
y
x,y
x
,
y
the equation
f
(
f
(
y
)
+
x
2
+
1
)
+
2
x
=
y
+
(
f
(
x
+
1
)
)
2
f(f(y)+x^2+1)+2x=y+(f(x+1))^2
f
(
f
(
y
)
+
x
2
+
1
)
+
2
x
=
y
+
(
f
(
x
+
1
)
)
2
holds.
Concurency
A circle
ω
\omega
ω
cuts the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
of the triangle
A
B
C
ABC
A
BC
at
A
1
A_1
A
1
and
A
2
A_2
A
2
;
B
1
B_1
B
1
and
B
2
B_2
B
2
;
C
1
C_1
C
1
and
C
2
C_2
C
2
, respectively. Let
P
P
P
be the center of
ω
\omega
ω
.
A
′
A'
A
′
is the circumcenter of the triangle
A
1
A
2
P
A_1A_2P
A
1
A
2
P
,
B
′
B'
B
′
is the circumcenter of the triangle
B
1
B
2
P
B_1B_2P
B
1
B
2
P
,
C
′
C'
C
′
is the circumcenter of the triangle
C
1
C
2
P
C_1C_2P
C
1
C
2
P
. Prove that
A
A
′
,
B
B
′
AA', BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
concur.
Natural Number Sequence
a
1
=
−
5
a_1=-5
a
1
=
−
5
,
a
2
=
−
6
a_2=-6
a
2
=
−
6
and for all
n
≥
2
n \geq 2
n
≥
2
the
(
a
n
)
∞
n
=
1
{(a_n)^\infty}_{n=1}
(
a
n
)
∞
n
=
1
sequence defined as,
a
n
+
1
=
a
n
+
(
a
1
+
1
)
(
2
a
2
+
1
)
(
3
a
3
+
1
)
⋯
(
(
n
−
1
)
a
n
−
1
+
1
)
(
(
n
2
+
n
)
a
n
+
2
n
+
1
)
)
.
a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)).
a
n
+
1
=
a
n
+
(
a
1
+
1
)
(
2
a
2
+
1
)
(
3
a
3
+
1
)
⋯
((
n
−
1
)
a
n
−
1
+
1
)
((
n
2
+
n
)
a
n
+
2
n
+
1
))
.
If a prime
p
p
p
divides
n
a
n
+
1
na_n+1
n
a
n
+
1
for a natural number n, prove that there is a integer
m
m
m
such that
m
2
≡
5
(
m
o
d
p
)
m^2\equiv5(modp)
m
2
≡
5
(
m
o
d
p
)