MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2021 Turkey Team Selection Test
2021 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(8)
9
1
Hide problems
Number of elements in a special set
For which positive integer couples
(
k
,
n
)
(k,n)
(
k
,
n
)
, the equality
∣
{
a
∈
Z
+
:
1
≤
a
≤
(
n
k
)
!
,
g
c
d
(
(
a
k
)
,
n
)
=
1
}
∣
=
(
n
k
)
!
6
\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}
{
a
∈
Z
+
:
1
≤
a
≤
(
nk
)!
,
g
c
d
(
(
k
a
)
,
n
)
=
1
}
=
6
(
nk
)!
holds?
5
1
Hide problems
A concurrency problem
In a non isoceles triangle
A
B
C
ABC
A
BC
, let the perpendicular bisector of
[
B
C
]
[BC]
[
BC
]
intersect
(
A
B
C
)
(ABC)
(
A
BC
)
at
M
M
M
and
N
N
N
respectively. Let the midpoints of
[
A
M
]
[AM]
[
A
M
]
and
[
A
N
]
[AN]
[
A
N
]
be
K
K
K
and
L
L
L
respectively. Let
(
A
B
K
)
(ABK)
(
A
B
K
)
and
(
A
B
L
)
(ABL)
(
A
B
L
)
intersect
A
C
AC
A
C
again at
D
D
D
and
E
E
E
respectively, let
(
A
C
K
)
(ACK)
(
A
C
K
)
and
(
A
C
L
)
(ACL)
(
A
C
L
)
intersect
A
B
AB
A
B
again at
F
F
F
and
G
G
G
respectively. Prove that the lines
D
F
DF
D
F
,
E
G
EG
EG
and
M
N
MN
MN
are concurrent.
2
1
Hide problems
Some friendship conditions, proving an equality
In a school with some students, for any three student, there exists at least one student who are friends with all these three students.(Friendships are mutual) For any friends
A
A
A
and
B
B
B
, any two of their common friends are also friends with each other. It's not possible to partition these students into two groups, such that every student in each group are friends with all the students in the other gruop. Prove that any two people who aren't friends with each other, has the same number of common friends.(Each person is a friend of him/herself.)
7
1
Hide problems
radius of circumcircle of T_AT_BT_C is twice the radius of circumcircle of ABC
Given a triangle
A
B
C
ABC
A
BC
with the circumcircle
ω
\omega
ω
and incenter
I
I
I
. Let the line pass through the point
I
I
I
and the intersection of exterior angle bisector of
A
A
A
and
ω
\omega
ω
meets the circumcircle of
I
B
C
IBC
I
BC
at
T
A
T_A
T
A
for the second time. Define
T
B
T_B
T
B
and
T
C
T_C
T
C
similarly. Prove that the radius of the circumcircle of the triangle
T
A
T
B
T
C
T_AT_BT_C
T
A
T
B
T
C
is twice the radius of
ω
\omega
ω
.
6
1
Hide problems
Find positive integers n, i<=x_i<=2i is given
For which positive integers
n
n
n
, one can find real numbers
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots ,x_n
x
1
,
x
2
,
⋯
,
x
n
such that
x
1
2
+
x
2
2
+
⋯
+
x
n
2
(
x
1
+
2
x
2
+
⋯
+
n
x
n
)
2
=
27
4
n
(
n
+
1
)
(
2
n
+
1
)
\dfrac{x_1^2+x_2^2+\cdots+x_n^2}{\left(x_1+2x_2+\cdots+nx_n\right)^2}=\dfrac{27}{4n(n+1)(2n+1)}
(
x
1
+
2
x
2
+
⋯
+
n
x
n
)
2
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
4
n
(
n
+
1
)
(
2
n
+
1
)
27
and
i
≤
x
i
≤
2
i
i\leq x_i\leq 2i
i
≤
x
i
≤
2
i
for all
i
=
1
,
2
,
⋯
,
n
i=1,2,\cdots ,n
i
=
1
,
2
,
⋯
,
n
?
3
1
Hide problems
An angle equality wanted
A point
D
D
D
is taken on the arc
B
C
BC
BC
of the circumcircle of triangle
A
B
C
ABC
A
BC
which does not contain
A
A
A
. A point
E
E
E
is taken at the intersection of the interior region of the triangles
A
B
C
ABC
A
BC
and
A
D
C
ADC
A
D
C
such that
m
(
A
B
E
^
)
=
m
(
B
C
E
^
)
m(\widehat{ABE})=m(\widehat{BCE})
m
(
A
BE
)
=
m
(
BCE
)
. Let the circumcircle of the triangle
A
D
E
ADE
A
D
E
meets the line
A
B
AB
A
B
for the second time at
K
K
K
. Let
L
L
L
be the intersection of the lines
E
K
EK
E
K
and
B
C
BC
BC
,
M
M
M
be the intersection of the lines
E
C
EC
EC
and
A
D
AD
A
D
,
N
N
N
be the intersection of the lines
B
M
BM
BM
and
D
L
DL
D
L
. Prove that
m
(
N
E
L
^
)
=
m
(
N
D
E
^
)
m(\widehat{NEL})=m(\widehat{NDE})
m
(
NE
L
)
=
m
(
N
D
E
)
8
1
Hide problems
Is this function periodic?
Let
c
c
c
be a real number. For all
x
x
x
and
y
y
y
real numbers we have,
f
(
x
−
f
(
y
)
)
=
f
(
x
−
y
)
+
c
(
f
(
x
)
−
f
(
y
)
)
f(x-f(y))=f(x-y)+c(f(x)-f(y))
f
(
x
−
f
(
y
))
=
f
(
x
−
y
)
+
c
(
f
(
x
)
−
f
(
y
))
and
f
(
x
)
f(x)
f
(
x
)
is not constant.
a
)
a)
a
)
Find all possible values of
c
c
c
.
b
)
b)
b
)
Can
f
f
f
be periodic?
1
1
Hide problems
3^n+47 cannot divide 20x5^n-2
Let
n
n
n
be a positive integer. Prove that
20
⋅
5
n
−
2
3
n
+
47
\frac{20 \cdot 5^n-2}{3^n+47}
3
n
+
47
20
⋅
5
n
−
2
is not an integer.