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Problems
Contests
National and Regional Contests
Thailand Contests
Thailand Online MO
2023 Thailand Online MO
2023 Thailand Online MO
Part of
Thailand Online MO
Subcontests
(10)
10
1
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Game ensuring a base-2 representation is in S
Let
n
n
n
be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set
S
S
S
containing
m
m
m
integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing
i
∈
{
1
,
2
,
…
,
n
}
i\in\{1,2,\dots, n\}
i
∈
{
1
,
2
,
…
,
n
}
that has not been chosen and setting the value of
v
i
v_i
v
i
to either
0
0
0
or
1
1
1
. At the end of the game, when all of
v
1
,
v
2
,
…
,
v
n
v_1,v_2,\dots,v_n
v
1
,
v
2
,
…
,
v
n
have been set, the expression
E
=
v
1
⋅
2
0
+
v
2
⋅
2
1
+
⋯
+
v
n
⋅
2
n
−
1
E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}
E
=
v
1
⋅
2
0
+
v
2
⋅
2
1
+
⋯
+
v
n
⋅
2
n
−
1
is calculated. Determine the minimum
m
m
m
such that Alice can always ensure that
E
∈
S
E\in S
E
∈
S
regardless of how Bob plays.
9
1
Hide problems
(n^2+1)a_n=n(a_{n^2}+1)
Find all sequences of positive integers
a
1
,
a
2
,
…
a_1,a_2,\dots
a
1
,
a
2
,
…
such that
(
n
2
+
1
)
a
n
=
n
(
a
n
2
+
1
)
(n^2+1)a_n = n(a_{n^2}+1)
(
n
2
+
1
)
a
n
=
n
(
a
n
2
+
1
)
for all positive integers
n
n
n
.
8
1
Hide problems
Prove that B is the incenter of DEC’
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and
E
E
E
be the foot of altitude from
B
B
B
to
A
C
AC
A
C
. The point
C
′
C'
C
′
is the reflection of
C
C
C
across
A
M
AM
A
M
. The point
D
D
D
not equal to
C
C
C
is placed on line
B
C
BC
BC
such that
A
D
=
A
C
AD=AC
A
D
=
A
C
. Prove that
B
B
B
is the incenter of triangle
D
E
C
′
DEC'
D
E
C
′
.
7
1
Hide problems
minimum of two consecutive terms is less than 1
Let
a
0
,
a
1
,
…
a_0,a_1,\dots
a
0
,
a
1
,
…
be a sequence of positive reals such that
a
n
+
2
≤
2023
a
n
a
n
a
n
+
1
+
2023
a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}
a
n
+
2
≤
a
n
a
n
+
1
+
2023
2023
a
n
for all integers
n
≥
0
n\geq 0
n
≥
0
. Prove that either
a
2023
<
1
a_{2023}<1
a
2023
<
1
or
a
2024
<
1
a_{2024}<1
a
2024
<
1
.
6
1
Hide problems
circumcenter of AXY lies on circumcircle of ABC
Let
A
B
C
ABC
A
BC
be a triangle. Construct point
X
X
X
such that
B
X
=
B
A
BX=BA
BX
=
B
A
and
X
X
X
and
C
C
C
lies on the same side of line
A
B
AB
A
B
. Construct point
Y
Y
Y
such that
C
Y
=
C
A
CY=CA
C
Y
=
C
A
and
Y
Y
Y
and
B
B
B
lies on different sides of line
A
C
AC
A
C
. Suppose that triangle
B
A
X
BAX
B
A
X
and triangle
C
A
Y
CAY
C
A
Y
are similar, prove that the circumcenter of triangle
A
X
Y
AXY
A
X
Y
lies on the circumcircle of triangle
A
B
C
ABC
A
BC
.
5
1
Hide problems
FE involving number-theoretic functions under composition with f
For each positive integer
k
k
k
, let
d
(
k
)
d(k)
d
(
k
)
be the number of positive divisors of
k
k
k
and
σ
(
k
)
\sigma(k)
σ
(
k
)
be the sum of positive divisors of
k
k
k
. Let
N
\mathbb N
N
be the set of all positive integers. Find all functions
f
:
N
→
N
f: \mathbb{N} \to \mathbb N
f
:
N
→
N
such that \begin{align*} f(d(n+1)) &= d(f(n)+1) \text{and} \\ f(\sigma(n+1)) &= \sigma(f(n)+1) \end{align*} for all positive integers
n
n
n
.
4
1
Hide problems
Time to length bash
Let
A
B
C
ABC
A
BC
be a triangle, and let
D
D
D
and
D
1
D_1
D
1
be points on segment
B
C
BC
BC
such that
B
D
=
C
D
1
BD = CD_1
B
D
=
C
D
1
. Construct point
E
E
E
such that
E
C
⊥
B
C
EC\perp BC
EC
⊥
BC
and
E
D
⊥
A
C
ED\perp AC
E
D
⊥
A
C
. Similarly, construct point
F
F
F
such that
F
B
⊥
B
C
FB\perp BC
FB
⊥
BC
and
F
D
⊥
A
B
FD\perp AB
F
D
⊥
A
B
. Prove that
E
F
⊥
A
D
1
EF\perp AD_1
EF
⊥
A
D
1
.
3
1
Hide problems
n!|a^{n!}-1
Let
a
a
a
and
n
n
n
be positive integers such that the greatest common divisor of
a
a
a
and
n
!
n!
n
!
is
1
1
1
. Prove that
n
!
n!
n
!
divides
a
n
!
−
1
a^{n!}-1
a
n
!
−
1
.
2
1
Hide problems
not all roots of x^3P(x)+1 are real
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with real coefficients. Prove that not all roots of
x
3
P
(
x
)
+
1
x^3P(x)+1
x
3
P
(
x
)
+
1
are real.
1
1
Hide problems
Chef Kao arranges 2n cups of ice cream
Let
n
n
n
be a positive integer. Chef Kao has
n
n
n
different flavors of ice cream. He wants to serve one small cup and one large cup for each flavor. He arranges the
2
n
2n
2
n
ice cream cups into two rows of
n
n
n
cups on a tray. He wants the tray to be colorful, so he arranges the ice cream cups with the following conditions: [*]each row contains all ice cream flavors, and [*]each column has different sizes of ice cream cup. Determine the number of ways that Chef Kao can arrange cups of ice cream with the above conditions.