MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand TST Selection Test
2024 Thailand October Camp
2024 Thailand October Camp
Part of
Thailand TST Selection Test
Subcontests
(5)
1
1
Hide problems
Why is this test so weird
In a test,
201
201
201
students are trying to solve
6
6
6
problems.We know that for each of
5
5
5
first problems, there are at least
140
140
140
students, who can solve it. Moreover, there is exactly
60
60
60
students, who can solve
6
t
h
6^{th}
6
t
h
problem. Show that there exist
2
2
2
students, such that two of them combined are able to solve all
6
6
6
question. (For example, number
1
1
1
do
1
,
2
,
3
,
4
1,2,3,4
1
,
2
,
3
,
4
and number
2
2
2
do
3
,
5
,
6
3,5,6
3
,
5
,
6
)
4
2
Hide problems
a_1 is anything but 2
The sequence
(
a
n
)
n
∈
N
(a_n)_{n\in\mathbb{N}}
(
a
n
)
n
∈
N
is defined by
a
1
=
3
a_1=3
a
1
=
3
and
a
n
=
a
1
a
2
⋯
a
n
−
1
−
1
a_n=a_1a_2\cdots a_{n-1}-1
a
n
=
a
1
a
2
⋯
a
n
−
1
−
1
Show that there exist infinitely many prime number that divide at least one number in this sequences
Lovely geo
Let
A
B
C
ABC
A
BC
be an acute triangle with altitudes
A
D
,
B
E
AD,BE
A
D
,
BE
and
C
F
CF
CF
. Denote
ω
1
,
ω
2
\omega_1,\omega_2
ω
1
,
ω
2
the circumcircles of
△
A
E
B
,
△
A
F
C
\triangle AEB, \triangle AFC
△
A
EB
,
△
A
FC
, respectively. Suppose the line through
A
A
A
parallel to
E
F
EF
EF
intersects
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
at
P
P
P
and
Q
Q
Q
, respectively. Show that the circumcenter of
△
P
Q
D
\triangle PQD
△
PQ
D
lies on
A
D
AD
A
D
5
1
Hide problems
Points on plane
Find the maximal number of points, such that there exist a configuration of
2023
2023
2023
lines on the plane, with each lines pass at least
2
2
2
points.
6
1
Hide problems
Sth simple but not simple
A polynomial
A
(
x
)
A(x)
A
(
x
)
is said to be simple if
A
(
x
)
A(x)
A
(
x
)
is divisible by
x
x
x
but not divisible by
x
2
x^2
x
2
. Suppose that a polynomial
P
(
x
)
P(x)
P
(
x
)
has a simple polynomial
Q
(
x
)
Q(x)
Q
(
x
)
such that
P
(
Q
(
x
)
)
−
Q
(
2
x
)
P(Q(x))-Q(2x)
P
(
Q
(
x
))
−
Q
(
2
x
)
is divisible by
x
2
x^2
x
2
. Prove that there exists a simple polynomial
R
(
x
)
R(x)
R
(
x
)
such that
P
(
R
(
x
)
)
−
R
(
2
x
)
P(R(x))-R(2x)
P
(
R
(
x
))
−
R
(
2
x
)
is divisible by
x
2023
x^{2023}
x
2023
.
3
2
Hide problems
Product of primitive roots is not primitive root
Recall that for an arbitrary prime
p
p
p
, we define a primitive root modulo
p
p
p
as an integer
r
r
r
for which the least positive integer
v
v
v
such that
r
v
≡
1
(
m
o
d
p
)
r^{v}\equiv 1\pmod{p}
r
v
≡
1
(
mod
p
)
is
p
−
1
p-1
p
−
1
.\\ Prove or disprove the following statement: For every prime
p
>
2023
p>2023
p
>
2023
, there exists positive integers
1
⩽
a
<
b
<
c
<
p
1\leqslant a<b<c<p
1
⩽
a
<
b
<
c
<
p
\\ such that
a
,
b
a,b
a
,
b
and
c
c
c
are primitive roots modulo
p
p
p
but
a
b
c
abc
ab
c
is not a primitive root modulo
p
p
p
.
square of hell :)
Let triangle
A
B
C
ABC
A
BC
be an acute-angled triangle. Square
A
E
F
B
AEFB
A
EFB
and
A
D
G
C
ADGC
A
D
GC
lie outside triangle
A
B
C
ABC
A
BC
.
B
D
BD
B
D
intersects
C
E
CE
CE
at point
H
H
H
, and
B
G
BG
BG
intersects
C
F
CF
CF
at point
I
I
I
. The circumcircle of triangle
B
F
I
BFI
BF
I
intersects the circumcircle of triangle
C
G
I
CGI
CG
I
again at point
K
K
K
. Prove that line segment
H
K
HK
HK
bisects
B
C
BC
BC
.