MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand Online MO
2021 Thailand Online MO
2021 Thailand Online MO
Part of
Thailand Online MO
Subcontests
(10)
P10
1
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Fill the table with T,M,O
Each cell of the board with
2021
2021
2021
rows and
2022
2022
2022
columns contains exactly one of the three letters
T
T
T
,
M
M
M
, and
O
O
O
in a way that satisfies each of the following conditions:[*] In total, each letter appears exactly
2021
×
674
2021\times 674
2021
×
674
of times on the board. [*] There are no two squares that share a common side and contain the same letter. [*] Any
2
×
2
2\times 2
2
×
2
square contains all three letters
T
T
T
,
M
M
M
, and
O
O
O
.Prove that each letter
T
T
T
,
M
M
M
, and
O
O
O
appears exactly
674
674
674
times on every row.
P9
1
Hide problems
τ(τ(an)) = τ(τ(bn)) for all n implies a=b
For each positive integer
k
k
k
, denote by
τ
(
k
)
\tau(k)
τ
(
k
)
the number of all positive divisors of
k
k
k
, including
1
1
1
and
k
k
k
. Let
a
a
a
and
b
b
b
be positive integers such that
τ
(
τ
(
a
n
)
)
=
τ
(
τ
(
b
n
)
)
\tau(\tau(an)) = \tau(\tau(bn))
τ
(
τ
(
an
))
=
τ
(
τ
(
bn
))
for all positive integers
n
n
n
. Prove that
a
=
b
a=b
a
=
b
.
P8
1
Hide problems
Functional equation with gcd
Let
N
\mathbb N
N
be the set of positive integers. Determine all functions
f
:
N
×
N
→
N
f:\mathbb N\times\mathbb N\to\mathbb N
f
:
N
×
N
→
N
that satisfy both of the following conditions:[*]
f
(
gcd
(
a
,
b
)
,
c
)
=
gcd
(
a
,
f
(
c
,
b
)
)
f(\gcd (a,b),c) = \gcd (a,f(c,b))
f
(
g
cd
(
a
,
b
)
,
c
)
=
g
cd
(
a
,
f
(
c
,
b
))
for all
a
,
b
,
c
∈
N
a,b,c \in \mathbb{N}
a
,
b
,
c
∈
N
. [*]
f
(
a
,
a
)
≥
a
f(a,a) \geq a
f
(
a
,
a
)
≥
a
for all
a
∈
N
a \in \mathbb{N}
a
∈
N
.
P7
1
Hide problems
Two congruent triangles form a cyclic quad
Let
A
B
C
ABC
A
BC
be an acute triangle. Construct a point
X
X
X
on the different side of
C
C
C
with respect to the line
A
B
AB
A
B
and construct a point
Y
Y
Y
on the different side of
B
B
B
with respect to the line
A
C
AC
A
C
such that
B
X
=
A
C
BX=AC
BX
=
A
C
,
C
Y
=
A
B
CY=AB
C
Y
=
A
B
, and
A
X
=
A
Y
AX=AY
A
X
=
A
Y
. Let
A
′
A'
A
′
be the reflection of
A
A
A
across the perpendicular bisector of
B
C
BC
BC
. Suppose that
X
X
X
and
Y
Y
Y
lie on different sides of the line
A
A
′
AA'
A
A
′
, prove that points
A
A
A
,
A
′
A'
A
′
,
X
X
X
, and
Y
Y
Y
lie on a circle.
P6
1
Hide problems
Sequence of rational numbers cannot be too close
Let
m
<
n
m<n
m
<
n
be two positive integers and
x
m
<
x
m
+
1
<
⋯
<
x
n
x_m<x_{m+1}<\cdots<x_n
x
m
<
x
m
+
1
<
⋯
<
x
n
be a sequence of rational numbers. Suppose that
k
x
k
kx_k
k
x
k
is an integer for all integers
k
k
k
which
m
≤
k
≤
n
m\leq k\leq n
m
≤
k
≤
n
. Prove that
x
n
−
x
m
≥
1
m
−
1
n
.
x_n-x_m\geq \frac{1}{m}-\frac{1}{n}.
x
n
−
x
m
≥
m
1
−
n
1
.
P4
1
Hide problems
A-angle bisector and B-foot of altitude
Let
A
B
C
ABC
A
BC
be an acute triangle such that
∠
B
>
∠
C
\angle B > \angle C
∠
B
>
∠
C
. Let
D
D
D
and
E
E
E
be the points on the segments
B
C
BC
BC
and
C
A
CA
C
A
, respectively, such that
A
D
AD
A
D
bisects
∠
A
\angle A
∠
A
and
B
E
⊥
A
C
BE\perp AC
BE
⊥
A
C
. Finally, let
M
M
M
be the midpoint of the side
B
C
BC
BC
. Suppose that the circumcircle of
△
C
D
E
\triangle CDE
△
C
D
E
intersects
A
D
AD
A
D
again at a point
X
X
X
different from
D
D
D
. Prove that
∠
X
M
E
=
9
0
∘
−
∠
B
A
C
\angle XME = 90^{\circ} - \angle BAC
∠
XME
=
9
0
∘
−
∠
B
A
C
.
P1
1
Hide problems
Painting a fence with 100 colors
There is a fence that consists of
n
n
n
planks arranged in a line. Each plank is painted with one of the available
100
100
100
colors. Suppose that for any two distinct colors
i
i
i
and
j
j
j
, there is a plank with color
i
i
i
located to the left of a (not necessarily adjacent) plank with color
j
j
j
. Determine the minimum possible value of
n
n
n
.
P2
1
Hide problems
if gcd$(x,n)=1$, then gcd$(x+101,n)=1$.
Determine all integers
n
>
1
n>1
n
>
1
that satisfy the following condition: for any positive integer
x
x
x
, if gcd
(
x
,
n
)
=
1
(x,n)=1
(
x
,
n
)
=
1
, then gcd
(
x
+
101
,
n
)
=
1
(x+101,n)=1
(
x
+
101
,
n
)
=
1
.
P5
1
Hide problems
$P(a)+P(b)+P(c)\ge 2021$
Prove that there exists a polynomial
P
(
x
)
P(x)
P
(
x
)
with real coefficients and degree greater than 1 such that both of the following conditions are true
⋅
\cdot
⋅
P
(
a
)
+
P
(
b
)
+
P
(
c
)
≥
2021
P(a)+P(b)+P(c)\ge 2021
P
(
a
)
+
P
(
b
)
+
P
(
c
)
≥
2021
holds for all nonnegative real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
⋅
\cdot
⋅
There are infinitely many triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of nonnegative real numbers such that
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
and
P
(
a
)
+
P
(
b
)
+
P
(
c
)
=
2021
P(a)+P(b)+P(c)= 2021
P
(
a
)
+
P
(
b
)
+
P
(
c
)
=
2021
P3
1
Hide problems
$a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1$
Let
a
1
,
a
2
,
⋯
a_1,a_2,\cdots
a
1
,
a
2
,
⋯
be an infinity sequence of positive integers such that
a
1
=
2021
a_1=2021
a
1
=
2021
and
a
n
+
1
=
(
a
1
+
a
2
+
⋯
+
a
n
)
2
−
1
a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1
a
n
+
1
=
(
a
1
+
a
2
+
⋯
+
a
n
)
2
−
1
for all positive integers
n
n
n
. Prove that for any integer
n
≥
2
n\ge 2
n
≥
2
,
a
n
a_n
a
n
is the product of at least
2
n
2n
2
n
(not necessarily distinct) primes.