MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand TST Selection Test
2018 Thailand TSTST
2018 Thailand TSTST
Part of
Thailand TST Selection Test
Subcontests
(6)
8
1
Hide problems
Color edges of a graph with red or blue
There are
n
n
n
vertices and
m
>
n
m > n
m
>
n
edges in a graph. Each edge is colored either red or blue. In each year, we are allowed to choose a vertex and flip the color of all edges incident to it. Prove that there is a way to color the edges (initially) so that they will never all have the same color
7
1
Hide problems
Sum of fractional part of binomial coefficients
Evaluate
∑
n
=
2017
2030
∑
k
=
1
n
{
(
n
k
)
2017
}
\sum_{n=2017}^{2030}\sum_{k=1}^{n}\left\{\frac{\binom{n}{k}}{2017}\right\}
∑
n
=
2017
2030
∑
k
=
1
n
{
2017
(
k
n
)
}
.Note:
{
x
}
=
x
−
⌊
x
⌋
\{x\}=x-\lfloor x\rfloor
{
x
}
=
x
−
⌊
x
⌋
for every real numbers
x
x
x
.
5
1
Hide problems
Easy Algebra
Find all triples of real numbers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
satisfying a+b+c=14, a^2+b^2+c^2=84, a^3+b^3+c^3=584.
1
3
Hide problems
Rational number can be written uniquely as sum of a_i/i!
Prove that any rational
r
∈
(
0
,
1
)
r \in (0, 1)
r
∈
(
0
,
1
)
can be written uniquely in the form
r
=
a
1
1
!
+
a
2
2
!
+
a
3
3
!
+
⋯
+
a
k
k
!
r=\frac{a_1}{1!}+\frac{a_2}{2!}+\frac{a_3}{3!}+\cdots+\frac{a_k}{k!}
r
=
1
!
a
1
+
2
!
a
2
+
3
!
a
3
+
⋯
+
k
!
a
k
where
a
i
’s
a_i\text{’s}
a
i
’s
are nonnegative integers with
a
i
≤
i
−
1
a_i\leq i-1
a
i
≤
i
−
1
for all
i
i
i
.
(P(x)+1)^2=P(x^2+1)
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients satisfying:
P
(
2017
)
=
2016
P(2017) = 2016
P
(
2017
)
=
2016
and
(
P
(
x
)
+
1
)
2
=
P
(
x
2
+
1
)
.
(P(x)+1)^2=P(x^2+1).
(
P
(
x
)
+
1
)
2
=
P
(
x
2
+
1
)
.
f(x+y)=f(x)+f(y), f(P(x))=f(x)
Let
P
P
P
be a given quadratic polynomial. Find all functions
f
:
R
→
R
f : \mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
and
f
(
P
(
x
)
)
=
f
(
x
)
for all
x
,
y
∈
R
.
f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}.
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
and
f
(
P
(
x
))
=
f
(
x
)
for all
x
,
y
∈
R
.
2
3
Hide problems
Draw 9 horizontal and 9 vertical line through a square
9
9
9
horizontal and
9
9
9
vertical lines are drawn through a square. Prove that it is possible to select
20
20
20
rectangles so that the sides of each rectangle is a segment of one of the given lines (including the sides of the square), and for any two of the
20
20
20
rectangles, it is possible to cover one of them with the other (rotations are allowed).
Three stick wth integer length
There are three sticks, each of which has an integer length which is at least
n
n
n
; the sum of their lengths is
n
(
n
+
1
)
/
2
n(n + 1)/2
n
(
n
+
1
)
/2
. Prove that it is possible to break the sticks (possibly several times) so that the resulting sticks have length
1
,
2
,
…
,
n
1, 2,\dots, n
1
,
2
,
…
,
n
.Note: a stick of length
a
+
b
a + b
a
+
b
can be broken into sticks of lengths
a
a
a
and
b
b
b
.
ME = BC wanted, AE = MD, AN = AM, <BAC = 135^o, circumcircle
In triangle
△
A
B
C
\vartriangle ABC
△
A
BC
,
∠
B
A
C
=
13
5
o
\angle BAC = 135^o
∠
B
A
C
=
13
5
o
.
M
M
M
is the midpoint of
B
C
BC
BC
, and
N
≠
M
N \ne M
N
=
M
is on
B
C
BC
BC
such that
A
N
=
A
M
AN = AM
A
N
=
A
M
. The line
A
M
AM
A
M
meets the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
at
D
D
D
. Point
E
E
E
is chosen on segment
A
N
AN
A
N
such that
A
E
=
M
D
AE = MD
A
E
=
M
D
. Show that
M
E
=
B
C
ME = BC
ME
=
BC
.
3
3
Hide problems
TR = TS for intersecting circles
Circles
O
1
,
O
2
O_1, O_2
O
1
,
O
2
intersects at
A
,
B
A, B
A
,
B
. The circumcircle of
O
1
B
O
2
O_1BO_2
O
1
B
O
2
intersects
O
1
,
O
2
O_1, O_2
O
1
,
O
2
and line
A
B
AB
A
B
at
R
,
S
,
T
R, S, T
R
,
S
,
T
respectively. Prove that
T
R
=
T
S
TR = TS
TR
=
TS
fixed point and perpendicular wanted, 2 circles, tangents
Let
B
C
BC
BC
be a chord not passing through the center of a circle
ω
\omega
ω
. Point
A
A
A
varies on the major arc
B
C
BC
BC
. Let
E
E
E
and
F
F
F
be the projection of
B
B
B
onto
A
C
AC
A
C
, and of
C
C
C
onto
A
B
AB
A
B
respectively. The tangents to the circumcircle of
△
A
E
F
\vartriangle AEF
△
A
EF
at
E
,
F
E, F
E
,
F
intersect at
P
P
P
. (a) Prove that
P
P
P
is independent of the choice of
A
A
A
. (b) Let
H
H
H
be the orthocenter of
△
A
B
C
\vartriangle ABC
△
A
BC
, and let
T
T
T
be the intersection of
E
F
EF
EF
and
B
C
BC
BC
. Prove that
T
H
⊥
A
P
TH \perp AP
T
H
⊥
A
P
.
n|1+m^(3^n)+m^(2*3^n)
Find all pairs of integers
m
,
n
≥
2
m, n \geq 2
m
,
n
≥
2
such that
n
∣
1
+
m
3
n
+
m
2
⋅
3
n
.
n\mid 1+m^{3^n}+m^{2\cdot 3^n}.
n
∣
1
+
m
3
n
+
m
2
⋅
3
n
.