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Problems
Contests
National and Regional Contests
Thailand Contests
Thailand National Olympiad
2019 Thailand Mathematical Olympiad
2019 Thailand Mathematical Olympiad
Part of
Thailand National Olympiad
Subcontests
(10)
10
1
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Infinite n such that (2n+1)!+1 composite
Prove that there are infinitely many positive odd integer
n
n
n
such that
n
!
+
1
n!+1
n
!
+
1
is composite number.
9
1
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Dissecting a convex n-gon to special triangles
A chaisri figure is a triangle which the three vertices are vertices of a regular
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-gon. Two different chaisri figure may be formed by different regular
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-gon. A thubkaew figure is a convex polygon which can be dissected into multiple chaisri figure where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon.Determine the maximum number of vertices that a thubkaew figure may have.
7
1
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Condition to maximize an expression
Let
A
=
{
−
2562
,
−
2561
,
.
.
.
,
2561
,
2562
}
A=\{-2562,-2561,...,2561,2562\}
A
=
{
−
2562
,
−
2561
,
...
,
2561
,
2562
}
. Prove that for any bijection (1-1, onto function)
f
:
A
→
A
f:A\to A
f
:
A
→
A
,
∑
k
=
1
2562
∣
f
(
k
)
−
f
(
−
k
)
∣
is maximized if and only if
f
(
k
)
f
(
−
k
)
<
0
for any
k
=
1
,
2
,
.
.
.
,
2562.
\sum_{k=1}^{2562}\left\lvert f(k)-f(-k)\right\rvert\text{ is maximized if and only if } f(k)f(-k)<0\text{ for any } k=1,2,...,2562.
k
=
1
∑
2562
∣
f
(
k
)
−
f
(
−
k
)
∣
is maximized if and only if
f
(
k
)
f
(
−
k
)
<
0
for any
k
=
1
,
2
,
...
,
2562.
6
1
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Functional inequality
Determine all function
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
x
f
(
y
)
+
y
f
(
x
)
⩽
x
y
xf(y)+yf(x)\leqslant xy
x
f
(
y
)
+
y
f
(
x
)
⩽
x
y
for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
.
5
1
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Asymmetric Inequality with abc=1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals such that
a
b
c
=
1
abc=1
ab
c
=
1
. Prove the inequality
4
a
−
1
(
2
b
+
1
)
2
+
4
b
−
1
(
2
c
+
1
)
2
+
4
c
−
1
(
2
a
+
1
)
2
⩾
1.
\frac{4a-1}{(2b+1)^2} + \frac{4b-1}{(2c+1)^2} + \frac{4c-1}{(2a+1)^2}\geqslant 1.
(
2
b
+
1
)
2
4
a
−
1
+
(
2
c
+
1
)
2
4
b
−
1
+
(
2
a
+
1
)
2
4
c
−
1
⩾
1.
4
1
Hide problems
Odd number of ways to jump
A rabbit initially stands at the position
0
0
0
, and repeatedly jumps on the real line. In each jump, the rabbit can jump to any position corresponds to an integer but it cannot stand still. Let
N
(
a
)
N(a)
N
(
a
)
be the number of ways to jump with a total distance of
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and stop at the position
a
a
a
. Determine all integers
a
a
a
such that
N
(
a
)
N(a)
N
(
a
)
is odd.
3
1
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FE on positive reals with a surprise
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R}^+\to\mathbb{R}^+
f
:
R
+
→
R
+
such that
f
(
x
+
y
f
(
x
)
+
y
2
)
=
f
(
x
)
+
2
y
f(x+yf(x)+y^2) = f(x)+2y
f
(
x
+
y
f
(
x
)
+
y
2
)
=
f
(
x
)
+
2
y
for every
x
,
y
∈
R
+
x,y\in\mathbb{R}^+
x
,
y
∈
R
+
.
2
1
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Quotient never be an integer
Let
a
,
b
a,b
a
,
b
be two different positive integers. Suppose that
a
,
b
a,b
a
,
b
are relatively prime. Prove that
2
a
(
a
2
+
b
2
)
a
2
−
b
2
\dfrac{2a(a^2+b^2)}{a^2-b^2}
a
2
−
b
2
2
a
(
a
2
+
b
2
)
is not an integer.
8
1
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collinearity wanted, related to incenter and circumcircle
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
≠
A
C
AB\ne AC
A
B
=
A
C
and
ω
\omega
ω
be the circumcircle of this triangle. Let
I
I
I
be the center of the inscribed circle of
A
B
C
ABC
A
BC
which touches
B
C
BC
BC
at
D
D
D
. Let the circle with diameter
A
I
AI
A
I
meets
ω
\omega
ω
again at
K
K
K
. If the line
A
I
AI
A
I
intersects
ω
\omega
ω
again at
M
M
M
, show that
K
,
D
,
M
K, D, M
K
,
D
,
M
are collinear.
1
1
Hide problems
circumcenter and concyclic wanted, inside a convex pentagon with 2 right angles
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon with
∠
A
E
B
=
∠
B
D
C
=
9
0
o
\angle AEB=\angle BDC=90^o
∠
A
EB
=
∠
B
D
C
=
9
0
o
and line
A
C
AC
A
C
bisects
∠
B
A
E
\angle BAE
∠
B
A
E
and
∠
D
C
B
\angle DCB
∠
D
CB
internally. The circumcircle of
A
B
E
ABE
A
BE
intersects line
A
C
AC
A
C
again at
P
P
P
. (a) Show that
P
P
P
is the circumcenter of
B
D
E
BDE
B
D
E
. (b) Show that
A
,
C
,
D
,
E
A, C, D, E
A
,
C
,
D
,
E
are concyclic.