MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand TST Selection Test
2017 Thailand TSTST
2017 Thailand TSTST
Part of
Thailand TST Selection Test
Subcontests
(6)
2
2
Hide problems
sum i of (sum j of (f(i)-g(j))^(2559))
Let
f
,
g
f, g
f
,
g
be bijections on
{
1
,
2
,
3
,
…
,
2016
}
\{1, 2, 3, \dots, 2016\}
{
1
,
2
,
3
,
…
,
2016
}
. Determine the value of
∑
i
=
1
2016
∑
j
=
1
2016
[
f
(
i
)
−
g
(
j
)
]
2559
.
\sum_{i=1}^{2016}\sum_{j=1}^{2016}[f(i)-g(j)]^{2559}.
i
=
1
∑
2016
j
=
1
∑
2016
[
f
(
i
)
−
g
(
j
)
]
2559
.
\frac{2016^m-m^{2016}}{m+2016} is integer
(i)
\text{(i)}
(i)
Does there exist a positive integer
m
>
201
6
2016
m > 2016^{2016}
m
>
201
6
2016
such that
201
6
m
−
m
2016
m
+
2016
\frac{2016^m-m^{2016}}{m+2016}
m
+
2016
201
6
m
−
m
2016
is a positive integer?
(ii)
\text{(ii)}
(ii)
Does there exist a positive integer
m
>
201
7
2017
m > 2017^{2017}
m
>
201
7
2017
such that
201
7
m
−
m
2017
m
+
2017
\frac{2017^m-m^{2017}}{m+2017}
m
+
2017
201
7
m
−
m
2017
is a positive integer?(Serbia MO 2016 P1)
6
2
Hide problems
Game Theory
A
A
A
and
B
B
B
plays a game, with
A
A
A
choosing a positive integer
n
∈
{
1
,
2
,
…
,
1001
}
=
S
n \in \{1, 2, \dots, 1001\} = S
n
∈
{
1
,
2
,
…
,
1001
}
=
S
.
B
B
B
must guess the value of
n
n
n
by choosing several subsets of
S
S
S
, then
A
A
A
will tell
B
B
B
how many subsets
n
n
n
is in.
B
B
B
will do this three times selecting
k
1
,
k
2
k_1, k_2
k
1
,
k
2
then
k
3
k_3
k
3
subsets of
S
S
S
each.What is the least value of
k
1
+
k
2
+
k
3
k_1 + k_2 + k_3
k
1
+
k
2
+
k
3
such that
B
B
B
has a strategy to correctly guess the value of
n
n
n
no matter what
A
A
A
chooses?
f(xy)+f(yz)+f(zx)=f(xy+yz+zx), x+y+z=0
Find all polynomials
f
f
f
with real coefficients such that for all reals
x
,
y
,
z
x, y, z
x
,
y
,
z
such that
x
+
y
+
z
=
0
x+y+z =0
x
+
y
+
z
=
0
, the following relation holds:
f
(
x
y
)
+
f
(
y
z
)
+
f
(
z
x
)
=
f
(
x
y
+
y
z
+
z
x
)
.
f(xy) + f(yz) + f(zx) = f(xy + yz + zx).
f
(
x
y
)
+
f
(
yz
)
+
f
(
z
x
)
=
f
(
x
y
+
yz
+
z
x
)
.
4
3
Hide problems
3mk=3(m+3)^n+1
Suppose that
m
,
n
,
k
m, n, k
m
,
n
,
k
are positive integers satisfying
3
m
k
=
(
m
+
3
)
n
+
1.
3mk=(m+3)^n+1.
3
mk
=
(
m
+
3
)
n
+
1.
Prove that
k
k
k
is odd.
#22 functional equation
Find all function
f
:
N
∗
→
N
∗
f:\mathbb{N}^*\rightarrow \mathbb{N}^*
f
:
N
∗
→
N
∗
that satisfy:
(
f
(
1
)
)
3
+
(
f
(
2
)
)
3
+
.
.
.
+
(
f
(
n
)
)
3
=
(
f
(
1
)
+
f
(
2
)
+
.
.
.
+
f
(
n
)
)
2
(f(1))^3+(f(2))^3+...+(f(n))^3=(f(1)+f(2)+...+f(n))^2
(
f
(
1
)
)
3
+
(
f
(
2
)
)
3
+
...
+
(
f
(
n
)
)
3
=
(
f
(
1
)
+
f
(
2
)
+
...
+
f
(
n
)
)
2
Color black/white in 8*8 table
The cells of a
8
×
8
8 \times 8
8
×
8
table are colored either black or white so that each row has a different number of black squares, and each column has a different number of black squares. What is the maximum number of pairs of adjacent cells of different colors?
3
3
Hide problems
CR = 2XY wanted, AQ = AC, AR = CP, circumcircle realted
In
△
A
B
C
\vartriangle ABC
△
A
BC
with
A
B
>
A
C
AB > AC
A
B
>
A
C
, the tangent to the circumcircle at
A
A
A
intersects line
B
C
BC
BC
at
P
P
P
. Let
Q
Q
Q
be the point on
A
B
AB
A
B
such that
A
Q
=
A
C
AQ = AC
A
Q
=
A
C
, and
A
A
A
lies between
B
B
B
and
Q
Q
Q
. Let
R
R
R
be the point on ray
A
P
AP
A
P
such that
A
R
=
C
P
AR = CP
A
R
=
CP
. Let
X
,
Y
X, Y
X
,
Y
be the midpoints of
A
P
,
C
Q
AP, CQ
A
P
,
CQ
respectively. Prove that
C
R
=
2
X
Y
CR = 2XY
CR
=
2
X
Y
.
f(x-f(x))=f(x)-f(f(x))
Let
f
f
f
be a function on a set
X
X
X
. Prove that
f
(
X
−
f
(
X
)
)
=
f
(
X
)
−
f
(
f
(
X
)
)
,
f(X-f(X))=f(X)-f(f(X)),
f
(
X
−
f
(
X
))
=
f
(
X
)
−
f
(
f
(
X
))
,
where for a set
S
S
S
, the notation
f
(
S
)
f(S)
f
(
S
)
means
{
f
(
a
)
∣
a
∈
S
}
\{f(a) | a \in S\}
{
f
(
a
)
∣
a
∈
S
}
.
Weird Inequality
Let
a
,
b
,
c
∈
R
+
a, b, c \in\mathbb{R}^+
a
,
b
,
c
∈
R
+
. Prove that
∑
c
y
c
a
b
(
1
2
a
+
c
+
1
2
b
+
c
)
<
∑
c
y
c
a
3
+
b
3
c
2
+
a
b
.
\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\sum_{cyc}\frac{a^3+b^3}{c^2+ab}.
cyc
∑
ab
(
2
a
+
c
1
+
2
b
+
c
1
)
<
cyc
∑
c
2
+
ab
a
3
+
b
3
.
1
2
Hide problems
concurrent wanted, midpoints, incenters related
In
△
A
B
C
,
D
,
E
,
F
\vartriangle ABC, D, E, F
△
A
BC
,
D
,
E
,
F
are the midpoints of
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
respectively. Denote by
O
A
,
O
B
,
O
C
O_A, O_B, O_C
O
A
,
O
B
,
O
C
the incenters of
△
A
D
F
,
△
B
E
D
,
△
C
F
E
\vartriangle ADF, \vartriangle BED, \vartriangle CFE
△
A
D
F
,
△
BE
D
,
△
CFE
respectively. Prove that
O
A
E
,
O
B
F
,
O
C
D
O_AE, O_BF, O_CD
O
A
E
,
O
B
F
,
O
C
D
are concurrent.
3 Problems Algebra, Combinaroric, and Number Theory
1.1 Let
f
(
A
)
f(A)
f
(
A
)
denote the difference between the maximum value and the minimum value of a set
A
A
A
. Find the sum of
f
(
A
)
f(A)
f
(
A
)
as
A
A
A
ranges over the subsets of
{
1
,
2
,
…
,
n
}
\{1, 2, \dots, n\}
{
1
,
2
,
…
,
n
}
.1.2 All cells of an
8
×
8
8 × 8
8
×
8
board are initially white. A move consists of flipping the color (white to black or vice versa) of cells in a
1
×
3
1\times 3
1
×
3
or
3
×
1
3\times 1
3
×
1
rectangle. Determine whether there is a finite sequence of moves resulting in the state where all
64
64
64
cells are black.1.3 Prove that for all positive integers
m
m
m
, there exists a positive integer
n
n
n
such that the set
{
n
,
n
+
1
,
n
+
2
,
…
,
3
n
}
\{n, n + 1, n + 2, \dots , 3n\}
{
n
,
n
+
1
,
n
+
2
,
…
,
3
n
}
contains exactly
m
m
m
perfect squares.
5
3
Hide problems
P(x)-x|P^n(x)-x
Prove that for all polynomials
P
∈
R
[
x
]
P \in \mathbb{R}[x]
P
∈
R
[
x
]
and positive integers
n
n
n
,
P
(
x
)
−
x
P(x)-x
P
(
x
)
−
x
divides
P
n
(
x
)
−
x
P^n(x)-x
P
n
(
x
)
−
x
as polynomials.
collinear wanted, 2 circles and exsimilicenter center
Let
ω
1
,
ω
2
\omega_1, \omega_2
ω
1
,
ω
2
be two circles with different radii, and let
H
H
H
be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from
X
X
X
to
ω
1
\omega_1
ω
1
touch
ω
1
\omega_1
ω
1
at
P
,
Q
P, Q
P
,
Q
, and the tangents from
X
X
X
to
ω
2
\omega_2
ω
2
touch
ω
2
\omega_2
ω
2
at
R
,
S
R, S
R
,
S
. If
P
R
PR
PR
passes through
H
H
H
and is not a common tangent line of
ω
1
,
ω
2
\omega_1, \omega_2
ω
1
,
ω
2
, prove that
Q
S
QS
QS
also passes through
H
H
H
.
Another Weird Inequality
Let
a
,
b
,
c
∈
R
+
a, b, c \in \mathbb{R}^+
a
,
b
,
c
∈
R
+
such that
a
+
b
+
c
=
3
a + b + c = 3
a
+
b
+
c
=
3
. Prove that
∑
c
y
c
(
a
3
+
1
a
2
+
1
)
≥
1
27
(
a
b
+
b
c
+
c
a
)
4
.
\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)\geq\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4.
cyc
∑
(
a
2
+
1
a
3
+
1
)
≥
27
1
(
ab
+
b
c
+
c
a
)
4
.