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Problems
Contests
National and Regional Contests
Thailand Contests
Thailand National Olympiad
2011 Thailand Mathematical Olympiad
2011 Thailand Mathematical Olympiad
Part of
Thailand National Olympiad
Subcontests
(12)
12
1
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there are 7662 chairs , 2554 students and find the numbers of lin-ping labelling
7662
7662
7662
chairs are placed in a circle around the city of Chiang Mai. They are also marked with a label for either
1
1
1
st,
2
2
2
nd, or
3
3
3
rd grade students, so that there are
2554
2554
2554
chairs labeled with each label. The following situations happen, in order[*]
2554
2554
2554
students each from the
1
1
1
st,
2
2
2
nd, and
3
3
3
rd grades are given a ball as follows:
1
1
1
st grade students receive footballs,
2
2
2
nd grade students receive basketballs, and
3
3
3
rd grade students receive volleyballs. [*] The students go sit in chairs labeled for their grade [*] The students simultaneously send their balls to the student to their left, and this happens some positive number of times.A labelling of the chairs is called lin-ping if it is possible for all
1
1
1
st,
2
2
2
nd, and
3
3
3
rd grade students to now hold volleyballs, footballs, and basketballs respectively. Compute the number of lin-ping labellings
11
1
Hide problems
Prove that incenter, centroid of excentral triangle and orthocenter of contact
In
Δ
A
B
C
\Delta ABC
Δ
A
BC
, Let the Incircle touch
B
C
‾
,
C
A
‾
,
A
B
‾
\overline{BC}, \overline{CA}, \overline{AB}
BC
,
C
A
,
A
B
at
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
. Let
I
A
,
I
B
,
I
C
I_A,I_B,I_C
I
A
,
I
B
,
I
C
be
A
A
A
,
B
B
B
,
C
−
C-
C
−
excenters, respectively. Prove that Incenter of
Δ
A
B
C
\Delta ABC
Δ
A
BC
, orthocenter of
Δ
X
Y
Z
\Delta XYZ
Δ
X
Y
Z
and centroid of
Δ
I
A
I
B
I
C
\Delta I_AI_BI_C
Δ
I
A
I
B
I
C
are collinear.
10
1
Hide problems
Does there exists a function from positive integers to itself satisfying the ...
Does there exists a function
f
:
N
⟶
N
f : \mathbb{N} \longrightarrow \mathbb{N}
f
:
N
⟶
N
\begin{align*} f \left( m+ f(n) \right) = f(m) +f(n) + f(n+1) \end{align*} for all
m
,
n
∈
N
m,n \in \mathbb{N}
m
,
n
∈
N
?
9
1
Hide problems
Prove for all natural numbers that the given fraction has no integral resultant
Prove that, for all
n
∈
N
n \in \mathbb{N}
n
∈
N
\begin{align*} \frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{2n+1} \not\in \mathbb{Z} \end{align*}
8
1
Hide problems
If AC is tangen to (ABG) ( G is centroid ), then prove the given inequality
Given
Δ
A
B
C
\Delta ABC
Δ
A
BC
and its centroid
G
G
G
, If line
A
C
AC
A
C
is tangent to
⊙
(
A
B
G
)
\odot (ABG)
⊙
(
A
BG
)
. Prove that, \begin{align*} AB+BC \leq 2AC \end{align*}
7
1
Hide problems
An inequality on 4 variables where the given polynomial has all roots real
Let
a
,
b
,
c
,
d
∈
R
+
a,b,c,d \in \mathbb{R}^+
a
,
b
,
c
,
d
∈
R
+
and suppose that all roots of the equation \begin{align*} x^5-ax^4+bx^3-cx^2+dx=1 \end{align*} are real. Prove \begin{align*} \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \le \frac{3}{5} \end{align*}
6
1
Hide problems
Find the maximum value of the given form where m is the arithmetic mean
For any
0
≤
x
1
,
x
2
,
…
,
x
2011
≤
1
0\leq x_1,x_2,\ldots,x_{2011} \leq 1
0
≤
x
1
,
x
2
,
…
,
x
2011
≤
1
, Find the maximum value of \begin{align*} \sum_{k=1}^{2011}(x_k-m)^2 \end{align*} where
m
m
m
is the arithmetic mean of
x
1
,
x
2
,
…
,
x
2011
x_1,x_2,\ldots,x_{2011}
x
1
,
x
2
,
…
,
x
2011
.
4
1
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900 students at IS, 3 conditions are given to prove there exists 2 classrooms
There are
900
900
900
students in an International School. There are
59
59
59
international boys and
59
59
59
international girls. The Students are partitioned into
30
30
30
classrooms (each classrooms have equal number of student) and in each of the classrooms, the student will labelled number from
1
1
1
to
30
30
30
. The Partition must satisfy at least one follow condition:[*] Any Two international boys in same classroom can't have consecutive numbers. [*] For every classroom, the student who is labelled
1
1
1
must be a boy.Prove that there are
2
2
2
classrooms, each of which has
2
2
2
international boys with their labels difference equal.
3
1
Hide problems
Prove that if \angle BRP =\angle PRC then MR=MC
Given a
Δ
A
B
C
\Delta ABC
Δ
A
BC
where
∠
C
=
9
0
∘
\angle C = 90^{\circ}
∠
C
=
9
0
∘
,
D
D
D
is a point in the interior of
Δ
A
B
C
\Delta ABC
Δ
A
BC
and lines
A
D
AD
A
D
,
,
,
B
D
BD
B
D
and
C
D
CD
C
D
intersect
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
at points
P
P
P
,
Q
Q
Q
and
R
R
R
,respectively. Let
M
M
M
be the midpoint of
P
Q
‾
\overline{PQ}
PQ
. Prove that, if
∠
B
R
P
\angle BRP
∠
BRP
=
=
=
∠
P
R
C
\angle PRC
∠
PRC
then
M
R
=
M
C
MR=MC
MR
=
MC
.
1
1
Hide problems
Prove p is smaller than q , where both are primes
Given a natural number
n
n
n
≥
3
\geq 3
≥
3
. If
p
,
q
p,q
p
,
q
are primes, such that,
p
∣
n
!
p \mid n!
p
∣
n
!
and
q
∣
(
n
−
1
)
!
−
1
q \mid (n-1)!-1
q
∣
(
n
−
1
)!
−
1
. Prove that,
p
<
q
p<q
p
<
q
5
1
Hide problems
Find all n such that n = d (n) ^ 4
Find all
n
n
n
such that
n
=
d
(
n
)
4
n = d (n) ^ 4
n
=
d
(
n
)
4
Where
d
(
n
)
d (n)
d
(
n
)
is the number of divisors of
n
n
n
, for example
n
=
2
⋅
3
⋅
5
⟹
d
(
n
)
=
2
⋅
2
⋅
2
n = 2 \cdot 3\cdot 5\implies d (n) = 2 \cdot 2\cdot 2
n
=
2
⋅
3
⋅
5
⟹
d
(
n
)
=
2
⋅
2
⋅
2
.
2
1
Hide problems
Functionnal Equation .
Find all functions
f
:
N
→
N
f : \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
such that
f
(
2
m
+
2
n
)
=
f
(
m
)
f
(
n
)
f(2m+2n)=f(m)f(n)
f
(
2
m
+
2
n
)
=
f
(
m
)
f
(
n
)
for all natural numbers
m
,
n
m,n
m
,
n
.