MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand National Olympiad
2022 Thailand Mathematical Olympiad
2022 Thailand Mathematical Olympiad
Part of
Thailand National Olympiad
Subcontests
(10)
10
1
Hide problems
On the multiple and sum of digits
For each positive integers
u
u
u
and
n
n
n
, say that
u
u
u
is a friend of
n
n
n
if and only if there exists a positive integer
N
N
N
that is a multiple of
n
n
n
and the sum of digits of
N
N
N
(in base 10) is equal to
u
u
u
. Determine all positive integers
n
n
n
that only finitely many positive integers are not a friend of
n
n
n
.
9
1
Hide problems
Funny side bash
Let
P
Q
R
S
PQRS
PQRS
be a quadrilateral that has an incircle and
P
Q
≠
Q
R
PQ\neq QR
PQ
=
QR
. Its incircle touches sides
P
Q
,
Q
R
,
R
S
,
PQ,QR,RS,
PQ
,
QR
,
RS
,
and
S
P
SP
SP
at
A
,
B
,
C
,
A,B,C,
A
,
B
,
C
,
and
D
D
D
, respectively. Line
R
P
RP
RP
intersects lines
B
A
BA
B
A
and
B
C
BC
BC
at
T
T
T
and
M
M
M
, respectively. Place point
N
N
N
on line
T
B
TB
TB
such that
N
M
NM
NM
bisects
∠
T
M
B
\angle TMB
∠
TMB
. Lines
C
N
CN
CN
and
T
M
TM
TM
intersect at
K
K
K
, and lines
B
K
BK
B
K
and
C
D
CD
C
D
intersect at
H
H
H
. Prove that
∠
N
M
H
=
9
0
∘
\angle NMH=90^{\circ}
∠
NM
H
=
9
0
∘
.
8
1
Hide problems
Another NT Sequence (Rational)
Determine all possible values of
a
1
a_1
a
1
for which there exists a sequence
a
1
,
a
2
,
…
a_1, a_2, \dots
a
1
,
a
2
,
…
of rational numbers satisfying
a
n
+
1
2
−
a
n
+
1
=
a
n
a_{n+1}^2-a_{n+1}=a_n
a
n
+
1
2
−
a
n
+
1
=
a
n
for all positive integers
n
n
n
.
7
1
Hide problems
a_n=P(a_{n-1})
Let
d
≥
2
d \geq 2
d
≥
2
be a positive integer. Define the sequence
a
1
,
a
2
,
…
a_1,a_2,\dots
a
1
,
a
2
,
…
by
a
1
=
1
and
a
n
+
1
=
a
n
d
+
1
for all
n
≥
1.
a_1=1 \ \text{and} \ a_{n+1}=a_n^d+1 \ \text{for all }n\geq 1.
a
1
=
1
and
a
n
+
1
=
a
n
d
+
1
for all
n
≥
1.
Determine all pairs of positive integers
(
p
,
q
)
(p, q)
(
p
,
q
)
such that
a
p
a_p
a
p
divides
a
q
a_q
a
q
.
6
1
Hide problems
Covid Combo
In an examination, there are
3600
3600
3600
students sitting in a
60
×
60
60 \times 60
60
×
60
grid, where everyone is facing toward the top of the grid. After the exam, it is discovered that there are
901
901
901
students who got infected by COVID-19. Each infected student has a spreading region, which consists of students to the left, to the right, or in the front of them. Student in spreading region of at least two students are considered a close contact. Given that no infected student sat in the spreading region of other infected student, prove that there is at least one close contact.
5
1
Hide problems
Two Variables Functions in a Cartesian plane
Determine all functions
f
:
R
×
R
→
R
f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}
f
:
R
×
R
→
R
that satisfies the equation
f
(
x
+
y
+
z
3
,
a
+
b
+
c
3
)
=
f
(
x
,
a
)
f
(
y
,
b
)
f
(
z
,
c
)
f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)
f
(
3
x
+
y
+
z
,
3
a
+
b
+
c
)
=
f
(
x
,
a
)
f
(
y
,
b
)
f
(
z
,
c
)
for any real numbers
x
,
y
,
z
,
a
,
b
,
c
x,y,z,a,b,c
x
,
y
,
z
,
a
,
b
,
c
such that
a
z
+
b
x
+
c
y
≠
a
y
+
b
z
+
c
x
az+bx+cy\neq ay+bz+cx
a
z
+
b
x
+
cy
=
a
y
+
b
z
+
c
x
.
4
1
Hide problems
P(a)P(b)\neq P(c) (Generalization of China South East MO 2015)
Find all positive integers
n
n
n
such that there exists a monic polynomial
P
(
x
)
P(x)
P
(
x
)
of degree
n
n
n
with integers coefficients satisfying
P
(
a
)
P
(
b
)
≠
P
(
c
)
P(a)P(b)\neq P(c)
P
(
a
)
P
(
b
)
=
P
(
c
)
for all integers
a
,
b
,
c
a,b,c
a
,
b
,
c
.
3
1
Hide problems
Existence of a sector
Let
Ω
\Omega
Ω
be a circle in a plane.
2022
2022
2022
pink points and
2565
2565
2565
blue points are placed inside
Ω
\Omega
Ω
such that no point has two colors and no two points are collinear with the center of
Ω
\Omega
Ω
. Prove that there exists a sector of
Ω
\Omega
Ω
such that the angle at the center is acute and the number of blue points inside the sector is greater than the number of pink points by exactly
100
100
100
. (Note: such sector may contain no pink points.)
2
1
Hide problems
Parity Inequality
Define a function
f
:
N
×
N
→
{
−
1
,
1
}
f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}
f
:
N
×
N
→
{
−
1
,
1
}
such that
f
(
m
,
n
)
=
{
1
if
m
,
n
have the same parity, and
−
1
if
m
,
n
have different parity
f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}
f
(
m
,
n
)
=
{
1
−
1
if
m
,
n
have the same parity, and
if
m
,
n
have different parity
for every positive integers
m
,
n
m,n
m
,
n
. Determine the minimum possible value of
∑
1
≤
i
<
j
≤
2565
i
j
f
(
x
i
,
x
j
)
\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)
1
≤
i
<
j
≤
2565
∑
ij
f
(
x
i
,
x
j
)
across all permutations
x
1
,
x
2
,
x
3
,
…
,
x
2565
x_1,x_2,x_3,\dots,x_{2565}
x
1
,
x
2
,
x
3
,
…
,
x
2565
of
1
,
2
,
…
,
2565
1,2,\dots,2565
1
,
2
,
…
,
2565
.
1
1
Hide problems
Equal segment wanted
Let
B
C
BC
BC
be a chord of a circle
Γ
\Gamma
Γ
and
A
A
A
be a point inside
Γ
\Gamma
Γ
such that
∠
B
A
C
\angle BAC
∠
B
A
C
is acute. Outside
△
A
B
C
\triangle ABC
△
A
BC
, construct two isosceles triangles
△
A
C
P
\triangle ACP
△
A
CP
and
△
A
B
R
\triangle ABR
△
A
BR
such that
∠
A
C
P
\angle ACP
∠
A
CP
and
∠
A
B
R
\angle ABR
∠
A
BR
are right angles. Let lines
B
A
BA
B
A
and
C
A
CA
C
A
meet
Γ
\Gamma
Γ
again at points
E
E
E
and
F
F
F
, respectively. Let lines
E
P
EP
EP
and
F
R
FR
FR
meet
Γ
\Gamma
Γ
again at points
X
X
X
and
Y
Y
Y
, respectively. Prove that
B
X
=
C
Y
BX=CY
BX
=
C
Y
.