MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand Online MO
2022 Thailand Online MO
2022 Thailand Online MO
Part of
Thailand Online MO
Subcontests
(10)
5
1
Hide problems
OXY and HBC are similar (complex moment)
Let
A
B
C
ABC
A
BC
be an acute triangle with circumcenter
O
O
O
and orthocenter
H
H
H
. Let
M
B
M_B
M
B
and
M
C
M_C
M
C
be the midpoints of
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. Place points
X
X
X
and
Y
Y
Y
on line
B
C
BC
BC
such that
∠
H
M
B
X
=
∠
H
M
C
Y
=
9
0
∘
\angle HM_BX = \angle HM_CY = 90^{\circ}
∠
H
M
B
X
=
∠
H
M
C
Y
=
9
0
∘
. Prove that triangles
O
X
Y
OXY
OX
Y
and
H
B
C
HBC
H
BC
are similar.
4
1
Hide problems
Remove 1011 signs from 2022 signs and the remaning can be any color
There are
2022
2022
2022
signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length
1011
1011
1011
whose each term is either red or blue, Auto can always remove
1011
1011
1011
signs from the line so that the remaining
1011
1011
1011
signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.
3
1
Hide problems
mn+1|f(m)f(n)+1
Let
N
\mathbb{N}
N
be the set of positive integers. Across all function
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
such that
m
n
+
1
divides
f
(
m
)
f
(
n
)
+
1
mn+1\text{ divides } f(m)f(n)+1
mn
+
1
divides
f
(
m
)
f
(
n
)
+
1
for all positive integers
m
m
m
and
n
n
n
, determine all possible values of
f
(
101
)
.
f(101).
f
(
101
)
.
2
1
Hide problems
Prove that intersection of two circumcircle lie on a line
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid such that
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
and
A
B
>
C
D
AB > CD
A
B
>
C
D
. Points
X
X
X
and
Y
Y
Y
are on the side
A
B
AB
A
B
such that
X
Y
=
A
B
−
C
D
XY = AB-CD
X
Y
=
A
B
−
C
D
and
X
X
X
lies between
A
A
A
and
Y
Y
Y
. Prove that one intersection of the circumcircles of triangles
A
Y
D
AYD
A
Y
D
and
B
X
C
BXC
BXC
is on line
C
D
CD
C
D
.
1
1
Hide problems
x^3+y+z=x+y^3+z=x+y+z^3=-xyz
Determine, with proof, all triples of real numbers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
satisfying the equations
x
3
+
y
+
z
=
x
+
y
3
+
z
=
x
+
y
+
z
3
=
−
x
y
z
.
x^3+y+z=x+y^3+z=x+y+z^3=-xyz.
x
3
+
y
+
z
=
x
+
y
3
+
z
=
x
+
y
+
z
3
=
−
x
yz
.
6
1
Hide problems
Chef Kao cut a pizza into 2k pieces
Let
n
n
n
and
k
k
k
be positive integers. Chef Kao cuts a circular pizza through
k
k
k
diameters, dividing the pizza into
2
k
2k
2
k
equal pieces. Then, he dresses the pizza with
n
n
n
toppings. For each topping, he chooses
k
k
k
consecutive pieces of pizza and puts that topping on all of the chosen pieces. Then, for each piece of pizza, Chef Kao counts the number of distinct toppings on it, yielding
2
k
2k
2
k
numbers. Among these numbers, let
m
m
m
and
M
M
M
being the minimum and maximum, respectively. Prove that
m
+
M
=
n
m + M = n
m
+
M
=
n
.
7
1
Hide problems
p|a_ib_j-a_jb_i
Let
p
p
p
be a prime number, and let
a
1
,
a
2
,
…
,
a
p
a_1, a_2, \dots , a_p
a
1
,
a
2
,
…
,
a
p
and
b
1
,
b
2
,
…
,
b
p
b_1, b_2, \dots , b_p
b
1
,
b
2
,
…
,
b
p
be
2
p
2p
2
p
(not necessarily distinct) integers chosen from the set
{
1
,
2
,
…
,
p
−
1
}
\{1, 2, \dots , p - 1\}
{
1
,
2
,
…
,
p
−
1
}
. Prove that there exist integers
i
i
i
and
j
j
j
such that
1
≤
i
<
j
≤
p
1 \le i < j \le p
1
≤
i
<
j
≤
p
and
p
p
p
divides
a
i
b
j
−
a
j
b
i
a_ib_j-a_jb_i
a
i
b
j
−
a
j
b
i
.
8
1
Hide problems
Prove that two circumcircle intersect on a line
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
A
D
=
B
C
AD = BC
A
D
=
BC
,
∠
B
A
C
+
∠
D
C
A
=
18
0
∘
\angle BAC+\angle DCA = 180^{\circ}
∠
B
A
C
+
∠
D
C
A
=
18
0
∘
, and
∠
B
A
C
≠
9
0
∘
.
\angle BAC \neq 90^{\circ}.
∠
B
A
C
=
9
0
∘
.
Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the circumcenters of triangles
A
B
C
ABC
A
BC
and
C
A
D
CAD
C
A
D
, respectively. Prove that one intersection point of the circumcircles of triangles
O
1
B
C
O_1BC
O
1
BC
and
O
2
A
D
O_2AD
O
2
A
D
lies on
A
C
AC
A
C
.
10
1
Hide problems
f(x+y)-f(x)-f(y) and f(xy)=f(x)f(y) are integers (f:Q--->Q)
Let
Q
\mathbb{Q}
Q
be the set of rational numbers. Determine all functions
f
:
Q
→
Q
f : \mathbb{Q}\to\mathbb{Q}
f
:
Q
→
Q
satisfying both of the following conditions.[*]
f
(
a
)
f(a)
f
(
a
)
is not an integer for some rational number
a
a
a
. [*] For any rational numbers
x
x
x
and
y
y
y
, both
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
f(x + y) - f(x) - f(y)
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
and
f
(
x
y
)
−
f
(
x
)
f
(
y
)
f(xy) - f(x)f(y)
f
(
x
y
)
−
f
(
x
)
f
(
y
)
are integers.
9
1
Hide problems
Write ab or 1/a+1/b+1/ab on the board
The number
1
1
1
is written on the blackboard. At any point, Kornny may pick two (not necessary distinct) of the numbers
a
a
a
and
b
b
b
written on the board and write either
a
b
ab
ab
or
1
a
+
1
b
+
1
a
b
\frac{1}{a}+\frac{1}{b}+\frac{1}{ab}
a
1
+
b
1
+
ab
1
on the board as well. Determine all possible numbers that Kornny can write on the board in finitely many steps.