MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand TST Selection Test
2019 Thailand TSTST
2019 Thailand TSTST
Part of
Thailand TST Selection Test
Subcontests
(3)
1
3
Hide problems
2561 points on a circle
Let
2561
2561
2561
given points on a circle be colored either red or green. In each step, all points are recolored simultaneously in the following way: if both direct neighbors of a point
P
P
P
have the same color as
P
P
P
, then the color of
P
P
P
remains unchanged, otherwise
P
P
P
obtains the other color. Starting with the initial coloring
F
1
F_1
F
1
, we obtain the colorings
F
2
,
F
3
,
…
F_2, F_3,\dots
F
2
,
F
3
,
…
after several recoloring steps. Determine the smallest number
n
n
n
such that, for any initial coloring
F
1
F_1
F
1
, we must have
F
n
=
F
n
+
2
F_n = F_{n+2}
F
n
=
F
n
+
2
.
Sequence + Inequality Problem
Let
{
x
i
}
i
=
1
∞
\{x_i\}^{\infty}_{i=1}
{
x
i
}
i
=
1
∞
and
{
y
i
}
i
=
1
∞
\{y_i\}^{\infty}_{i=1}
{
y
i
}
i
=
1
∞
be sequences of real numbers such that
x
1
=
y
1
=
3
x_1=y_1=\sqrt{3}
x
1
=
y
1
=
3
, x_{n+1}=x_n+\sqrt{1+x_n^2} \text{and} y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}} for all
n
≥
1
n\geq 1
n
≥
1
. Prove that
2
<
x
n
y
n
<
3
2<x_ny_n<3
2
<
x
n
y
n
<
3
for all
n
>
1
n>1
n
>
1
.
(p-3)^p+p^2 is perfect square
Find all primes
p
p
p
such that
(
p
−
3
)
p
+
p
2
(p-3)^p+p^2
(
p
−
3
)
p
+
p
2
is a perfect square.
3
3
Hide problems
concurrent wanted, perimeters equal to CA+AB,AB +BC,BC +CA
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
X
,
B
Y
AX, BY
A
X
,
B
Y
and
C
Z
CZ
CZ
as its altitudes.
∙
\bullet
∙
Line
ℓ
A
\ell_A
ℓ
A
, which is parallel to
Y
Z
YZ
Y
Z
, intersects
C
A
CA
C
A
at
A
1
A_1
A
1
between
C
C
C
and
A
A
A
, and intersects
A
B
AB
A
B
at
A
2
A_2
A
2
between
A
A
A
and
B
B
B
.
∙
\bullet
∙
Line
ℓ
B
\ell_B
ℓ
B
, which is parallel to
Z
X
ZX
ZX
, intersects
A
B
AB
A
B
at
B
1
B_1
B
1
between
A
A
A
and
B
B
B
, and intersects
B
C
BC
BC
at
B
2
B_2
B
2
between
B
B
B
and
C
C
C
.
∙
\bullet
∙
Line
ℓ
C
\ell_C
ℓ
C
, which is parallel to
X
Y
XY
X
Y
, intersects
B
C
BC
BC
at
C
1
C_1
C
1
between
B
B
B
and
C
C
C
, and intersects
C
A
CA
C
A
at
C
2
C_2
C
2
between
C
C
C
and
A
A
A
. Suppose that the perimeters of the triangles
△
A
A
1
A
2
\vartriangle AA_1A_2
△
A
A
1
A
2
,
△
B
B
1
B
2
\vartriangle BB_1B_2
△
B
B
1
B
2
and
△
C
C
1
C
2
\vartriangle CC_1C_2
△
C
C
1
C
2
are equal to
C
A
+
A
B
,
A
B
+
B
C
CA+AB,AB +BC
C
A
+
A
B
,
A
B
+
BC
and
B
C
+
C
A
BC +CA
BC
+
C
A
, respectively. Prove that
ℓ
A
,
ℓ
B
\ell_A, \ell_B
ℓ
A
,
ℓ
B
and
ℓ
C
\ell_C
ℓ
C
are concurrent.
Extremely Similar to ISL 2014 A4
Find all function
f
:
Z
→
Z
f:\mathbb{Z}\to\mathbb{Z}
f
:
Z
→
Z
satisfying
(i)
\text{(i)}
(i)
f
(
f
(
m
)
+
n
)
+
2
m
=
f
(
n
)
+
f
(
3
m
)
f(f(m)+n)+2m=f(n)+f(3m)
f
(
f
(
m
)
+
n
)
+
2
m
=
f
(
n
)
+
f
(
3
m
)
for every
m
,
n
∈
Z
m,n\in\mathbb{Z}
m
,
n
∈
Z
,
(ii)
\text{(ii)}
(ii)
there exists a
d
∈
Z
d\in\mathbb{Z}
d
∈
Z
such that
f
(
d
)
−
f
(
0
)
=
2
f(d)-f(0)=2
f
(
d
)
−
f
(
0
)
=
2
, and
(iii)
\text{(iii)}
(iii)
f
(
1
)
−
f
(
0
)
f(1)-f(0)
f
(
1
)
−
f
(
0
)
is even.
On the number of terms with odd coefficients
Let
n
≥
2
n\geq 2
n
≥
2
be an integer. Determine the number of terms in the polynomial
∏
1
≤
i
<
j
≤
n
(
x
i
+
x
j
)
\prod_{1\leq i< j\leq n}(x_i+x_j)
1
≤
i
<
j
≤
n
∏
(
x
i
+
x
j
)
whose coefficients are odd integers.
2
3
Hide problems
Two inequality mixed together
Let
a
,
b
,
c
∈
(
0
,
4
3
)
a,b,c\in(0,\frac{4}{3})
a
,
b
,
c
∈
(
0
,
3
4
)
and
a
+
b
+
c
=
3
a + b + c = 3
a
+
b
+
c
=
3
. Prove that
4
a
b
c
(
a
+
b
)
(
a
+
c
)
+
(
a
+
b
)
2
+
(
a
+
c
)
2
(
a
+
b
)
+
(
a
+
c
)
≤
∑
c
y
c
1
a
2
(
3
b
+
3
c
−
5
)
.
\frac{4abc}{(a+b)(a+c)}+\frac{(a+b)^2+(a+c)^2}{(a+b)+(a+c)}\leq\sum_{cyc}\frac{1}{a^2(3b+3c-5)}.
(
a
+
b
)
(
a
+
c
)
4
ab
c
+
(
a
+
b
)
+
(
a
+
c
)
(
a
+
b
)
2
+
(
a
+
c
)
2
≤
cyc
∑
a
2
(
3
b
+
3
c
−
5
)
1
.
collinear wanted, toucpoints of incircle related
Let
Ω
\Omega
Ω
be the inscribed circle of a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
D
,
E
D, E
D
,
E
and
F
F
F
be the tangency points of
Ω
\Omega
Ω
and the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
, respectively, and let
A
D
,
B
E
AD, BE
A
D
,
BE
and
C
F
CF
CF
intersect
Ω
\Omega
Ω
at
K
,
L
K, L
K
,
L
and
M
M
M
, respectively, such that
D
,
E
,
F
,
K
,
L
D, E, F, K, L
D
,
E
,
F
,
K
,
L
and
M
M
M
are all distinct. The tangent line of
Ω
\Omega
Ω
at
K
K
K
intersects
E
F
EF
EF
at
X
X
X
, the tangent line of
Ω
\Omega
Ω
at
L
L
L
intersects
D
E
DE
D
E
at
Y
Y
Y
, and the tangent line of
Ω
\Omega
Ω
at M intersects
D
F
DF
D
F
at
Z
Z
Z
. Prove that
X
,
Y
X,Y
X
,
Y
and
Z
Z
Z
are collinear.
2^x+31^y=z^2
Find all nonnegative integers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfying the equation
2
x
+
3
1
y
=
z
2
.
2^x+31^y=z^2.
2
x
+
3
1
y
=
z
2
.