MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand TST Selection Test
2022 Thailand TSTST
2022 Thailand TSTST
Part of
Thailand TST Selection Test
Subcontests
(3)
2
2
Hide problems
Cringe IE
Let
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
satisfy
a
≥
b
≥
c
a\geq b\geq c
a
≥
b
≥
c
. Prove that
4
a
2
(
b
+
c
)
+
4
b
2
(
c
+
a
)
+
4
c
2
(
a
+
b
)
≤
(
∑
c
y
c
a
2
+
1
b
2
)
(
∑
c
y
c
b
3
a
2
(
a
3
+
2
b
3
)
)
.
\frac{4}{a^2(b+c)}+\frac{4}{b^2(c+a)}+\frac{4}{c^2(a+b)} \leq \left(\sum_{cyc} \frac{a^2+1} {b^2} \right)\left(\sum_{cyc} \frac{b^3}{a^2(a^3+2b^3)}\right).
a
2
(
b
+
c
)
4
+
b
2
(
c
+
a
)
4
+
c
2
(
a
+
b
)
4
≤
(
cyc
∑
b
2
a
2
+
1
)
(
cyc
∑
a
2
(
a
3
+
2
b
3
)
b
3
)
.
some guy literally cry over this problem
An acute triangle
A
B
C
ABC
A
BC
has
A
B
AB
A
B
as one of its longest sides. The incircle of
A
B
C
ABC
A
BC
has center
I
I
I
and radius
r
r
r
. Line
C
I
CI
C
I
meets the circumcircle of
A
B
C
ABC
A
BC
at
D
D
D
. Let
E
E
E
be a point on the minor arc
B
C
BC
BC
of the circumcircle of
A
B
C
ABC
A
BC
with
∠
A
B
E
>
∠
B
A
D
\angle ABE > \angle BAD
∠
A
BE
>
∠
B
A
D
and
E
∉
{
B
,
C
}
E\notin \{B,C\}
E
∈
/
{
B
,
C
}
. Line
A
B
AB
A
B
meets
D
E
DE
D
E
at
F
F
F
and line
A
D
AD
A
D
meets
B
E
BE
BE
at
G
G
G
. Let
P
P
P
be a point inside triangle
A
G
E
AGE
A
GE
with
∠
A
P
E
=
∠
A
F
E
\angle APE=\angle AFE
∠
A
PE
=
∠
A
FE
and
P
≠
F
P\neq F
P
=
F
. Let
X
X
X
be a point on side
A
E
AE
A
E
with
X
P
∥
E
G
XP\parallel EG
XP
∥
EG
and let
S
S
S
be a point on side
E
G
EG
EG
with
P
S
∥
A
E
PS\parallel AE
PS
∥
A
E
. Suppose
X
S
XS
XS
and
G
P
GP
GP
meet on the circumcircle of
A
G
E
AGE
A
GE
. Determine the possible positions of
E
E
E
as well as the minimum value of
B
E
r
\frac{BE}{r}
r
BE
.
3
3
Hide problems
XY bisects BC
An acute scalene triangle
A
B
C
ABC
A
BC
with circumcircle
Ω
\Omega
Ω
is given. The altitude from
B
B
B
intersects side
A
C
AC
A
C
at
B
1
B_1
B
1
and circle
Ω
\Omega
Ω
at
B
2
B_2
B
2
. The circle with diameter
B
1
B
2
B_1B_2
B
1
B
2
intersects circle
Ω
\Omega
Ω
again at
B
3
B_3
B
3
. Similarly, the altitude from
C
C
C
intersects side
A
B
AB
A
B
at
C
1
C_1
C
1
and circle
Ω
\Omega
Ω
at
C
2
C_2
C
2
. The circle with diameter
C
1
C
2
C_1C_2
C
1
C
2
intersects circle
Ω
\Omega
Ω
again at
C
3
C_3
C
3
. Let
X
X
X
be the intersection of lines
B
1
B
3
B_1B_3
B
1
B
3
and
C
1
C
3
C_1C_3
C
1
C
3
, and let
Y
Y
Y
be the intersection of lines
B
3
C
B_3C
B
3
C
and
C
3
B
C_3B
C
3
B
. Prove that line
X
Y
XY
X
Y
bisects side
B
C
BC
BC
.
Weird function
Let
S
S
S
be the set of the positive integers greater than
1
1
1
, and let
n
n
n
be from
S
S
S
. Does there exist a function
f
f
f
from
S
S
S
to itself such that for all pairwise distinct positive integers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
from
S
S
S
, we have
f
(
a
1
)
f
(
a
2
)
.
.
.
f
(
a
n
)
=
f
(
a
1
n
a
2
n
.
.
.
a
n
n
)
f(a_1)f(a_2)...f(a_n)=f(a_1^na_2^n...a_n^n)
f
(
a
1
)
f
(
a
2
)
...
f
(
a
n
)
=
f
(
a
1
n
a
2
n
...
a
n
n
)
?
n sums are positive
An odd positive integer
n
n
n
is called pretty if there exists at least one permutation
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,..., a_n
a
1
,
a
2
,
...
,
a
n
, of
1
,
2
,
.
.
.
,
n
1,2,...,n
1
,
2
,
...
,
n
, such that all
n
n
n
sums
a
1
−
a
2
+
a
3
−
.
.
.
+
a
n
a_1-a_2+a_3-...+a_n
a
1
−
a
2
+
a
3
−
...
+
a
n
,
a
2
−
a
3
+
a
4
−
.
.
.
+
a
1
a_2-a_3+a_4-...+a_1
a
2
−
a
3
+
a
4
−
...
+
a
1
,...,
a
n
−
a
1
+
a
2
−
.
.
.
+
a
n
−
1
a_n-a_1+a_2-...+a_{n-1}
a
n
−
a
1
+
a
2
−
...
+
a
n
−
1
are positive. Find all pretty integers.
1
3
Hide problems
p^2022|n^2022+2022
An odd prime
p
p
p
is called a prime of the year
2022
2022
2022
if there is a positive integer
n
n
n
such that
p
2022
p^{2022}
p
2022
divides
n
2022
+
2022
n^{2022}+2022
n
2022
+
2022
. Show that there are infinitely many primes of the year
2022
2022
2022
.
Polynomial equation
Find all polynomials
f
,
g
,
h
f, g, h
f
,
g
,
h
with real coefficients, such that
f
(
x
)
2
+
(
x
+
1
)
g
(
x
)
2
=
(
x
3
+
x
)
h
(
x
)
2
f(x)^2+(x+1)g(x)^2=(x^3+x)h(x)^2
f
(
x
)
2
+
(
x
+
1
)
g
(
x
)
2
=
(
x
3
+
x
)
h
(
x
)
2
ok combi
Let
n
≥
3
n\geq 3
n
≥
3
be an integer. Each vertex of a regular
n
n
n
-gon is labelled with a real number not exceeding
1
1
1
. For real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
on any three consecutive vertices which are arranged clockwise in such an order, we have
c
=
∣
a
−
b
∣
c=|a-b|
c
=
∣
a
−
b
∣
. Determine the maximum value of the sum of all numbers in terms of
n
n
n
.