MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2019 Turkey Team SeIection Test
2019 Turkey Team SeIection Test
Part of
Turkey Team Selection Test
Subcontests
(9)
6
1
Hide problems
Labeling marbles that are connected by ropes
k
k
k
is a positive integer,
R
n
=
−
k
,
−
(
k
−
1
)
,
.
.
.
,
−
1
,
1
,
.
.
.
,
k
−
1
,
k
R_{n}={-k, -(k-1),..., -1, 1,..., k-1, k}
R
n
=
−
k
,
−
(
k
−
1
)
,
...
,
−
1
,
1
,
...
,
k
−
1
,
k
for
n
=
2
k
n=2k
n
=
2
k
R
n
=
−
k
,
−
(
k
−
1
)
,
.
.
.
,
−
1
,
0
,
1
,
.
.
.
,
k
−
1
,
k
R_{n}={-k, -(k-1),..., -1, 0, 1,..., k-1, k}
R
n
=
−
k
,
−
(
k
−
1
)
,
...
,
−
1
,
0
,
1
,
...
,
k
−
1
,
k
for
n
=
2
k
+
1
n=2k+1
n
=
2
k
+
1
. A mechanism consists of some marbles and white/red ropes that connects some marble pairs. If each one of the marbles are written on some numbers from
R
n
R_{n}
R
n
with the property that any two connected marbles have different numbers on them, we call it nice labeling. If each one of the marbles are written on some numbers from
R
n
R_{n}
R
n
with the properties that any two connected marbles with a white rope have different numbers on them and any two connected marbles with a red rope have two numbers with sum not equal to
0
0
0
, we call it precise labeling.
n
≥
3
n\geq{3}
n
≥
3
, if every mechanism that is labeled nicely with
R
n
R_{n}
R
n
, could be labeled precisely with
R
m
R_{m}
R
m
, what is the minimal value of
m
m
m
?
5
1
Hide problems
Polynomial with real roots
P
(
x
)
P(x)
P
(
x
)
is a nonconstant polynomial with real coefficients and its all roots are real numbers. If there exist a
Q
(
x
)
Q(x)
Q
(
x
)
polynomial with real coefficients that holds the equality for all
x
x
x
real numbers
(
P
(
x
)
)
2
=
P
(
Q
(
x
)
)
(P(x))^{2}=P(Q(x))
(
P
(
x
)
)
2
=
P
(
Q
(
x
))
, then prove that all the roots of
P
(
x
)
P(x)
P
(
x
)
are same.
4
1
Hide problems
Deleting digits and creating subdivisors
For an integer
n
n
n
with
b
b
b
digits, let a subdivisor of
n
n
n
be a positive number which divides a number obtained by removing the
r
r
r
leftmost digits and the
l
l
l
rightmost digits of
n
n
n
for nonnegative integers
r
,
l
r,l
r
,
l
with
r
+
l
<
b
r+l<b
r
+
l
<
b
(For example, the subdivisors of
143
143
143
are
1
1
1
,
2
2
2
,
3
3
3
,
4
4
4
,
7
7
7
,
11
11
11
,
13
13
13
,
14
14
14
,
43
43
43
, and
143
143
143
). For an integer
d
d
d
, let
A
d
A_d
A
d
be the set of numbers that don't have
d
d
d
as a subdivisor. Find all
d
d
d
, such that
A
d
A_d
A
d
is finite.
3
1
Hide problems
Bisector and altitude relations
In a triangle
A
B
C
ABC
A
BC
,
A
B
>
A
C
AB>AC
A
B
>
A
C
. The foot of the altitude from
A
A
A
to
B
C
BC
BC
is
D
D
D
, the intersection of bisector of
B
B
B
and
A
D
AD
A
D
is
K
K
K
, the foot of the altitude from
B
B
B
to
C
K
CK
C
K
is
M
M
M
and let
B
M
BM
BM
and
A
K
AK
A
K
intersect at point
N
N
N
. The line through
N
N
N
parallel to
D
M
DM
D
M
intersects
A
C
AC
A
C
at
T
T
T
. Prove that
B
M
BM
BM
is the bisector of angle
T
B
C
^
\widehat{TBC}
TBC
.
2
1
Hide problems
Integer sequence and prime divisors
(
a
n
)
n
=
1
∞
(a_{n})_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
is an integer sequence,
a
1
=
1
a_{1}=1
a
1
=
1
,
a
2
=
2
a_{2}=2
a
2
=
2
and for
n
≥
1
n\geq{1}
n
≥
1
,
a
n
+
2
=
a
n
+
1
2
+
(
n
+
2
)
a
n
+
1
−
a
n
2
−
n
a
n
a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}
a
n
+
2
=
a
n
+
1
2
+
(
n
+
2
)
a
n
+
1
−
a
n
2
−
n
a
n
.
a
)
a)
a
)
Prove that the set of primes that divides at least one term of the sequence can not be finite.
b
)
b)
b
)
Find 3 different prime numbers that do not divide any terms of this sequence.
1
1
Hide problems
Picking stones from boxes in appropriate ways
In each one of the given
2019
2019
2019
boxes, there are
2019
2019
2019
stones numbered as
1
,
2
,
.
.
.
,
2019
1,2,...,2019
1
,
2
,
...
,
2019
with total mass of
1
1
1
kilogram. In all situations satisfying these conditions, if one can pick stones from different boxes with different numbers, with total mass of at least 1 kilogram, in
k
k
k
different ways, what is the maximal of
k
k
k
?
7
1
Hide problems
TST Geometry P7
In a triangle
A
B
C
ABC
A
BC
with
∠
A
C
B
=
9
0
∘
\angle ACB = 90^{\circ}
∠
A
CB
=
9
0
∘
D
D
D
is the foot of the altitude of
C
C
C
. Let
E
E
E
and
F
F
F
be the reflections of
D
D
D
with respect to
A
C
AC
A
C
and
B
C
BC
BC
. Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the circumcenters of
△
E
C
B
\triangle {ECB}
△
ECB
and
△
F
C
A
\triangle {FCA}
△
FC
A
. Show that:
2
O
1
O
2
=
A
B
2O_1O_2=AB
2
O
1
O
2
=
A
B
8
1
Hide problems
pn|(p-1)^n+1
Let
p
>
2
p>2
p
>
2
be a prime number,
m
>
1
m>1
m
>
1
and
n
n
n
be positive integers such that
m
p
n
−
1
m
n
−
1
\frac {m^{pn}-1}{m^n-1}
m
n
−
1
m
p
n
−
1
is a prime number. Show that:
p
n
∣
(
p
−
1
)
n
+
1
pn\mid (p-1)^n+1
p
n
∣
(
p
−
1
)
n
+
1
9
1
Hide problems
Non-symmetric Inequality
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be real numbers such that
y
≥
2
z
≥
4
x
y\geq 2z \geq 4x
y
≥
2
z
≥
4
x
and
2
(
x
3
+
y
3
+
z
3
)
+
15
(
x
y
2
+
y
z
2
+
z
x
2
)
≥
16
(
x
2
y
+
y
2
z
+
z
2
x
)
+
2
x
y
z
.
2(x^3+y^3+z^3)+15(xy^2+yz^2+zx^2)\geq 16(x^2y+y^2z+z^2x)+2xyz.
2
(
x
3
+
y
3
+
z
3
)
+
15
(
x
y
2
+
y
z
2
+
z
x
2
)
≥
16
(
x
2
y
+
y
2
z
+
z
2
x
)
+
2
x
yz
.
Prove that:
4
x
+
y
≥
4
z
4x+y\geq 4z
4
x
+
y
≥
4
z