MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand National Olympiad
2008 Thailand Mathematical Olympiad
2008 Thailand Mathematical Olympiad
Part of
Thailand National Olympiad
Subcontests
(10)
10
1
Hide problems
max no of triangles by 17 points on sides of ABC plus 3 vetrices
On the sides of triangle
△
A
B
C
\vartriangle ABC
△
A
BC
,
17
17
17
points are added, so that there are
20
20
20
points in total (including the vertices of
△
A
B
C
\vartriangle ABC
△
A
BC
.) What is the maximum possible number of (nondegenerate) triangles that can be formed by these points.
9
1
Hide problems
no of pairs od subsets (A,B) of A \subseteq B \subseteq {1, 2, ...,10}
Find the number of pairs of sets
(
A
,
B
)
(A, B)
(
A
,
B
)
satisfying
A
⊆
B
⊆
{
1
,
2
,
.
.
.
,
10
}
A \subseteq B \subseteq \{1, 2, ...,10\}
A
⊆
B
⊆
{
1
,
2
,
...
,
10
}
8
1
Hide problems
2551 x 543^n -2008 x 7^n
Prove that
2551
⋅
54
3
n
−
2008
⋅
7
n
2551 \cdot 543^n -2008\cdot 7^n
2551
⋅
54
3
n
−
2008
⋅
7
n
is never a perfect square, where
n
n
n
varies over the set of positive integers
7
1
Hide problems
m^2 + n^2 = 3789 and gcd(m, n) + lcm(m, n) = 633
Two positive integers
m
,
n
m, n
m
,
n
satisfy the two equations
m
2
+
n
2
=
3789
m^2 + n^2 = 3789
m
2
+
n
2
=
3789
and
g
c
d
(
m
,
n
)
+
l
c
m
(
m
,
n
)
=
633
gcd (m, n) + lcm (m, n) = 633
g
c
d
(
m
,
n
)
+
l
c
m
(
m
,
n
)
=
633
. Compute
m
+
n
m + n
m
+
n
.
4
2
Hide problems
\sqrt{a^2 + b^2 -\sqrt2 ab} +\sqrt{b^2 + c^2 -\sqrt2 bc} \ge \sqrt{a^2 + c^2}
Prove that
a
2
+
b
2
−
2
a
b
+
b
2
+
c
2
−
2
b
c
≥
a
2
+
c
2
\sqrt{a^2 + b^2 -\sqrt2 ab} +\sqrt{b^2 + c^2 -\sqrt2 bc} \ge \sqrt{a^2 + c^2}
a
2
+
b
2
−
2
ab
+
b
2
+
c
2
−
2
b
c
≥
a
2
+
c
2
for all real numbers
a
,
b
,
c
>
0
a, b, c > 0
a
,
b
,
c
>
0
sum (-1)^{2k+1}{2n+1 \choose 2k+1}2008^k not divisible by 19
Let
n
n
n
be a positive integer. Show that
(
2
n
+
1
1
)
−
(
2
n
+
1
3
)
2008
+
(
2
n
+
1
5
)
200
8
2
−
.
.
.
+
(
−
1
)
2
n
+
1
(
2
n
+
1
2
n
+
1
)
200
8
n
{2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{2n+1}{2n+1 \choose 2n+1}2008^n
(
1
2
n
+
1
)
−
(
3
2
n
+
1
)
2008
+
(
5
2
n
+
1
)
200
8
2
−
...
+
(
−
1
)
2
n
+
1
(
2
n
+
1
2
n
+
1
)
200
8
n
is not divisible by
19
19
19
.
2
2
Hide problems
ratio of inradii, equal circles related
Let
A
D
AD
A
D
be the common chord of two equal-sized circles
O
1
O_1
O
1
and
O
2
O_2
O
2
. Let
B
B
B
and
C
C
C
be points on
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, so that
D
D
D
lies on the segment
B
C
BC
BC
. Assume that
A
B
=
15
,
A
D
=
13
AB = 15, AD = 13
A
B
=
15
,
A
D
=
13
and
B
C
=
18
BC = 18
BC
=
18
, what is the ratio between the inradii of
△
A
B
D
\vartriangle ABD
△
A
B
D
and
△
A
C
D
\vartriangle ACD
△
A
C
D
?
N =d_1^2 + d_2 ^2+ d_3 ^2+ d_4^2 where d_i are smallest divisors of N
Find all positive integers
N
N
N
with the following properties: (i)
N
N
N
has at least two distinct prime factors, and (ii) if
d
1
<
d
2
<
d
3
<
d
4
d_1 < d_2 < d_3 < d_4
d
1
<
d
2
<
d
3
<
d
4
are the four smallest divisors of
N
N
N
then
N
=
d
1
2
+
d
2
2
+
d
3
2
+
d
4
2
N =d_1^2 + d_2 ^2+ d_3 ^2+ d_4^2
N
=
d
1
2
+
d
2
2
+
d
3
2
+
d
4
2
3
2
Hide problems
x + [x/3]= [2x/3] + [3x/5]
Find all positive real solutions to the equation
x
+
⌊
x
3
⌋
=
⌊
2
x
3
⌋
+
⌊
3
x
5
⌋
x+\left\lfloor\frac x3\right\rfloor=\left\lfloor\frac{2x}3\right\rfloor+\left\lfloor\frac{3x}5\right\rfloor
x
+
⌊
3
x
⌋
=
⌊
3
2
x
⌋
+
⌊
5
3
x
⌋
n^{1/a_1}+n^{1/a_3}+n^{1/a_5}+...+n^{1/a_{2n-1} >= n^{a_{3n+2}/a_{3n+1}
For each positive integer
n
n
n
, define
a
n
=
n
(
n
+
1
)
a_n = n(n + 1)
a
n
=
n
(
n
+
1
)
. Prove that
n
1
/
a
1
+
n
1
/
a
3
+
n
1
/
a
5
+
.
.
.
+
n
1
/
a
2
n
−
1
≥
n
a
3
n
+
2
/
a
3
n
+
1
n^{1/a_1} + n^{1/a_3} + n^{1/a_5} + ...+ n^{1/a_{2n-1}} \ge n^{a_{3n+2}/a_{3n+1}}
n
1/
a
1
+
n
1/
a
3
+
n
1/
a
5
+
...
+
n
1/
a
2
n
−
1
≥
n
a
3
n
+
2
/
a
3
n
+
1
.
6
2
Hide problems
| f (x/2008 ) - f(x) / 2008 | < 1 if |f(x + y) - f(x) - f(y)| < 1
Let
f
:
R
→
R
f : R \to R
f
:
R
→
R
be a function satisfying the inequality
∣
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
∣
<
1
|f(x + y) -f(x) - f(y)| < 1
∣
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
∣
<
1
for all reals
x
,
y
x, y
x
,
y
. Show that \left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1 for all real numbers
x
x
x
.
f(xy)^2 = f(x^2)f(y^2) for all x,y if it is valid for x^2y^3 > 2008
Let
f
:
R
+
→
R
+
f : R^+ \to R^+
f
:
R
+
→
R
+
satisfy
f
(
x
y
)
2
=
f
(
x
2
)
f
(
y
2
)
f(xy)^2 = f(x^2)f(y^2)
f
(
x
y
)
2
=
f
(
x
2
)
f
(
y
2
)
for all positive reals
x
,
y
x, y
x
,
y
with
x
2
y
3
>
2008.
x^2y^3 > 2008.
x
2
y
3
>
2008.
Prove that
f
(
x
y
)
2
=
f
(
x
2
)
f
(
y
2
)
f(xy)^2 = f(x^2)f(y^2)
f
(
x
y
)
2
=
f
(
x
2
)
f
(
y
2
)
for all positive reals
x
,
y
x, y
x
,
y
.
5
2
Hide problems
P must have a repeated root, if P(a) = 0 then P(a+ 1) = 1
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial of degree
2008
2008
2008
with the following property: all roots of
P
P
P
are real, and for all real
a
a
a
, if
P
(
a
)
=
0
P(a) = 0
P
(
a
)
=
0
then
P
(
a
+
1
)
=
1
P(a+ 1) = 1
P
(
a
+
1
)
=
1
. Prove that P must have a repeated root.
students in a class consisting of m boys and n girls line up.
Students in a class consisting of
m
m
m
boys and
n
n
n
girls line up. Over all possible ways of lining up, compute the average number of pairs of two boys or two girls who are next to each other.
1
2
Hide problems
cos< CAE wanted, 90-60-30 triangle, BE : EC = 3 : 2
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle with
∠
B
A
C
=
9
0
o
\angle BAC = 90^o
∠
B
A
C
=
9
0
o
and
∠
A
B
C
=
6
0
o
\angle ABC = 60^o
∠
A
BC
=
6
0
o
. Point
E
E
E
is chosen on side
B
C
BC
BC
so that
B
E
:
E
C
=
3
:
2
BE : EC = 3 : 2
BE
:
EC
=
3
:
2
. Compute
cos
∠
C
A
E
\cos\angle CAE
cos
∠
C
A
E
.
rhombus wanted, tangents from an exterior point to circle related
Let
P
P
P
be a point outside a circle
ω
\omega
ω
. The tangents from
P
P
P
to
ω
\omega
ω
are drawn touching
ω
\omega
ω
at points
A
A
A
and
B
B
B
. Let
M
M
M
and
N
N
N
be the midpoints of
A
P
AP
A
P
and
A
B
AB
A
B
, respectively. Line
M
N
MN
MN
is extended to cut
ω
\omega
ω
at
C
C
C
so that
N
N
N
lies between
M
M
M
and
C
C
C
. Line
P
C
PC
PC
intersects
ω
\omega
ω
again at
D
D
D
, and lines
N
D
ND
N
D
and
P
B
PB
PB
intersect at
O
O
O
. Prove that
M
N
O
P
MNOP
MNOP
is a rhombus.