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Problems
Contests
National and Regional Contests
Thailand Contests
Thailand National Olympiad
2012 Thailand Mathematical Olympiad
2012 Thailand Mathematical Olympiad
Part of
Thailand National Olympiad
Subcontests
(11)
1
1
Hide problems
CP^2 = AC x CQ - AP x PR , starting with a right triangle
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a right triangle with
∠
B
=
9
0
o
\angle B = 90^o
∠
B
=
9
0
o
. Let
P
P
P
be a point on side
B
C
BC
BC
, and let
ω
\omega
ω
be the circle with diameter
C
P
CP
CP
. Suppose that
ω
\omega
ω
intersects
A
C
AC
A
C
and
A
P
AP
A
P
again at
Q
Q
Q
and
R
R
R
, respectively. Show that
C
P
2
=
A
C
⋅
C
Q
−
A
P
⋅
P
R
CP^2 = AC \cdot CQ - AP \cdot P R
C
P
2
=
A
C
⋅
CQ
−
A
P
⋅
PR
.
4
1
Hide problems
area of O_1O_2O_3O_4 <=1, incenters of right triangles outside a square
Let
A
B
C
D
ABCD
A
BC
D
be a unit square. Points
E
,
F
,
G
,
H
E, F, G, H
E
,
F
,
G
,
H
are chosen outside
A
B
C
D
ABCD
A
BC
D
so that
∠
A
E
B
=
∠
B
F
C
=
∠
C
G
D
=
∠
D
H
A
=
9
0
o
\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o
∠
A
EB
=
∠
BFC
=
∠
CG
D
=
∠
DH
A
=
9
0
o
. Let
O
1
,
O
2
,
O
3
,
O
4
O_1, O_2, O_3, O_4
O
1
,
O
2
,
O
3
,
O
4
, respectively, be the incenters of
△
A
B
E
,
△
B
C
F
,
△
C
D
G
,
△
D
A
H
\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH
△
A
BE
,
△
BCF
,
△
C
D
G
,
△
D
A
H
. Show that the area of
O
1
O
2
O
3
O
4
O_1O_2O_3O_4
O
1
O
2
O
3
O
4
is at most
1
1
1
.
11
1
Hide problems
QS bisects < DSE wanted, cirvle with diameter BC related
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an acute triangle, and let
P
P
P
be the foot of altitude from
C
C
C
to
A
B
AB
A
B
. Let
ω
\omega
ω
be the circle with diameter
B
C
BC
BC
. The tangents from
A
A
A
to
ω
\omega
ω
are drawn touching
ω
\omega
ω
at
D
D
D
and
E
E
E
. Lines
A
D
AD
A
D
and
A
E
AE
A
E
intersect line
B
C
BC
BC
at
M
M
M
and
N
N
N
respectively, so that
B
B
B
lies between
M
M
M
and
C
C
C
. Let
C
P
CP
CP
intersect
D
E
DE
D
E
at
Q
,
M
E
Q, ME
Q
,
ME
intersect
N
D
ND
N
D
at
R
R
R
, and let
Q
R
QR
QR
intersect
B
C
BC
BC
at
S
S
S
. Show that
Q
S
QS
QS
bisects
∠
D
S
E
\angle DSE
∠
D
SE
12
1
Hide problems
abc is perfect cube when a/b + b/c + c/a is an integer
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers. Show that if
a
b
+
b
c
+
c
a
\frac{a}{b} +\frac{b}{c} +\frac{c}{a}
b
a
+
c
b
+
a
c
is an integer then
a
b
c
abc
ab
c
is a perfect cube.
10
1
Hide problems
1/ 2555 < mx + n <1/ 2012 where x is irrational
Let
x
x
x
be an irrational number. Show that there are integers
m
m
m
and
n
n
n
such that
1
2555
<
m
x
+
n
<
1
2012
\frac{1}{2555}< mx + n <\frac{1}{2012}
2555
1
<
m
x
+
n
<
2012
1
9
1
Hide problems
P(x) >= (x + 1)^n for all x > 0, all roots are real
Let
n
n
n
be a positive integer and let
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
1
P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
1
be a polynomial with positive real coefficients. Under the assumption that the roots of
P
P
P
are all real, show that
P
(
x
)
≥
(
x
+
1
)
n
P(x) \ge (x + 1)^n
P
(
x
)
≥
(
x
+
1
)
n
for all
x
>
0
x > 0
x
>
0
.
8
1
Hide problems
no of matches in a draw / total no of matches, 4n students tournament
4
n
4n
4
n
first grade students at Songkhla Primary School, including
2
n
2n
2
n
boys and
2
n
2n
2
n
girls, participate in a taekwondo tournament where every pair of students compete against each other exactly once. The tournament is scored as follows:
∙
\bullet
∙
In a match between two boys or between two girls, a win is worth
3
3
3
points, a draw
1
1
1
point, and a loss
0
0
0
points.
∙
\bullet
∙
In a math between a boy and a girl, if the boy wins, he receives
2
2
2
points, else he receives
0
0
0
points. If the girl wins, she receives
3
3
3
points, if she draws, she receives
2
2
2
points, and if she loses, she receives
0
0
0
points. After the tournament, the total score of each student is calculated. Let
P
P
P
be the number of matches ending in a draw, and let
Q
Q
Q
be the total number of matches. Suppose that the maximum total score is
4
n
−
1
4n - 1
4
n
−
1
. Find
P
/
Q
P/Q
P
/
Q
.
7
1
Hide problems
5 | m - n^2 if gcd (a, b) = 1 and 5 | ma^2 + b^2
Let
a
,
b
,
m
a, b, m
a
,
b
,
m
be integers such that gcd
(
a
,
b
)
=
1
(a, b) = 1
(
a
,
b
)
=
1
and
5
∣
m
a
2
+
b
2
5 | ma^2 + b^2
5∣
m
a
2
+
b
2
. Show that there exists an integer
n
n
n
such that
5
∣
m
−
n
2
5 | m - n^2
5∣
m
−
n
2
.
3
1
Hide problems
[ (m^{\phi (n)+1} + n^{\phi (m)+1} ) /mn} ] is even integer
Let
m
,
n
>
1
m, n > 1
m
,
n
>
1
be coprime odd integers. Show that
⌊
m
ϕ
(
n
)
+
1
+
n
ϕ
(
m
)
+
1
m
n
⌋
\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor
⌊
mn
m
ϕ
(
n
)
+
1
+
n
ϕ
(
m
)
+
1
⌋
is an even integer, where
ϕ
\phi
ϕ
is Euler’s totient function.
5
1
Hide problems
f(f(x) + xf(y))= 3f(x) + 4xy
Determine all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
satisfying
f
(
f
(
x
)
+
x
f
(
y
)
)
=
3
f
(
x
)
+
4
x
y
f(f(x) + xf(y))= 3f(x) + 4xy
f
(
f
(
x
)
+
x
f
(
y
))
=
3
f
(
x
)
+
4
x
y
for all real numbers
x
,
y
x,y
x
,
y
.
2
1
Hide problems
(x -a_1)(x - a_2)...(x - a_{2012}) = (1006!)^2 has at most one integral solution
Let
a
1
,
a
2
,
.
.
.
,
a
2012
a_1, a_2, ..., a_{2012}
a
1
,
a
2
,
...
,
a
2012
be pairwise distinct integers. Show that the equation
(
x
−
a
1
)
(
x
−
a
2
)
.
.
.
(
x
−
a
2012
)
=
(
1006
!
)
2
(x -a_1)(x - a_2)...(x - a_{2012}) = (1006!)^2
(
x
−
a
1
)
(
x
−
a
2
)
...
(
x
−
a
2012
)
=
(
1006
!
)
2
has at most one integral solution.