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Contests
National and Regional Contests
Thailand Contests
Mathcenter Contest
2010 Mathcenter Contest
2010 Mathcenter Contest
Part of
Mathcenter Contest
Subcontests
(6)
6
1
Hide problems
f(f(f(...f(x)))...)=g(x)+a where 1-1 f appears 2009 times
Find all
a
∈
N
a\in\mathbb{N}
a
∈
N
such that exists a bijective function
g
:
N
→
N
g :\mathbb{N} \to \mathbb{N}
g
:
N
→
N
and a function
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
, such that for all
x
∈
N
x\in\mathbb{N}
x
∈
N
,
f
(
f
(
f
(
.
.
.
f
(
x
)
)
)
.
.
.
)
=
g
(
x
)
+
a
f(f(f(...f(x)))...)=g(x)+a
f
(
f
(
f
(
...
f
(
x
)))
...
)
=
g
(
x
)
+
a
where
f
f
f
appears
2009
2009
2009
times.(tatari/nightmare)
5
1
Hide problems
only one of a+b,| a-b| is a member of set of itnegers X
The set
X
X
X
of integers is called good If for each pair
a
,
b
∈
X
a,b\in X
a
,
b
∈
X
, only one of
a
+
b
,
∣
a
−
b
∣
a+b,\mid a-b\mid
a
+
b
,
∣
a
−
b
∣
is a member of
X
X
X
(
a
,
b
a,b
a
,
b
may be equal). Find the total number of sets with
2008
2008
2008
as member.(tatari/nightmare)
1
1
Hide problems
P^a+Q^b=R^c and 1/a+1/b+1/c>1, polynomials
Let
a
,
b
,
c
∈
N
a,b,c\in\mathbb{N}
a
,
b
,
c
∈
N
prove that if there is a polynomial
P
,
Q
,
R
∈
C
[
x
]
P,Q,R\in\mathbb{C}[x]
P
,
Q
,
R
∈
C
[
x
]
, which have no common factors and satisfy
P
a
+
Q
b
=
R
c
P^a+Q^b=R^c
P
a
+
Q
b
=
R
c
and \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}>1.(tatari/nightmare)
2
2
Hide problems
exists n such that the k-th digit from the right of 2^n is d
Let
k
k
k
and
d
d
d
be integers such that
k
>
1
k>1
k
>
1
and
0
≤
d
<
9
0\leq d<9
0
≤
d
<
9
. Prove that there exists some integer
n
n
n
such that the
k
k
k
th digit from the right of
2
n
2^n
2
n
is
d
d
d
.(tatari/nightmare)
|{x^n}-a|<=1/ 2(p+q)
A positive rational number
x
x
x
is called banzai if the following conditions are met:
∙
\bullet
∙
x=\frac{p}{q}>1 where
p
,
q
p,q
p
,
q
are comprime natural numbers
∙
\bullet
∙
exist constants
α
,
N
\alpha,N
α
,
N
such that for all integers
n
≥
N
n\geq N
n
≥
N
,
∣
{
x
n
}
−
α
∣
≤
1
2
(
p
+
q
)
.
\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}.
∣
{
x
n
}
−
α
∣≤
2
(
p
+
q
)
1
.
Find the total number of banzai numbers.Note:
{
x
}
\left\{\,x\right\}
{
x
}
means fractional part of
x
x
x
(tatari/nightmare)
3
2
Hide problems
<KDB=<LDC if <KDA=<BCD
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral, and the point
K
K
K
is a point on side
A
B
AB
A
B
, where
∠
K
D
A
=
∠
B
C
D
\angle KDA=\angle BCD
∠
KD
A
=
∠
BC
D
, let
L
L
L
be a point on the diagonal
A
C
AC
A
C
, where
K
L
KL
K
L
is parallel to
B
C
BC
BC
. Prove that
∠
K
D
B
=
∠
L
D
C
.
\angle KDB=\angle LDC.
∠
KD
B
=
∠
L
D
C
.
(tatari/nightmare)
(CP+CD)/PF wanted, touchpoints of incircle related
Let triangle
A
B
C
ABC
A
BC
be a triangle right at
B
B
B
. The inscribed circle is tangent to sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at points
D
,
E
,
F
D,E,F
D
,
E
,
F
, respectively. Let
C
F
CF
CF
intersect the circle at the point
P
P
P
. If
∠
A
P
B
=
9
0
∘
\angle APB=90^{\circ}
∠
A
PB
=
9
0
∘
, find the value of
C
P
+
C
D
P
F
\dfrac{CP+CD}{PF}
PF
CP
+
C
D
.(tatari/nightmare)
4
2
Hide problems
f :P -> P, ABCD convex , f(A),f(B),f(C),f (D) vertices of concave
Let
P
P
P
be a plane. Prove that there is no function
f
:
P
→
P
f :P\rightarrow P
f
:
P
→
P
where, for any convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, the points
f
(
A
)
f(A)
f
(
A
)
,
f
(
B
)
f(B)
f
(
B
)
,
f
(
C
)
f(C)
f
(
C
)
,
f
(
D
)
f (D)
f
(
D
)
are the vertices of a concave quadrilateral.(tatari/nightmare)
point construction, non intersecting chords related, PE/EQ=1/2
In a circle, two non-intersecting chords
A
B
,
C
D
AB,CD
A
B
,
C
D
are drawn.On the chord
A
B
AB
A
B
,a point
E
E
E
(different from
A
A
A
,
B
B
B
) is taken Consider the arc
A
B
AB
A
B
that does not contain the points
C
,
D
C,D
C
,
D
. With a compass and a straighthedge, find all possible point
F
F
F
on that arc such that
P
E
E
Q
=
1
2
\dfrac{PE}{EQ}=\dfrac{1}{2}
EQ
PE
=
2
1
, where
P
P
P
and
Q
Q
Q
are the points in which the chord
A
B
AB
A
B
meets the segment
F
C
FC
FC
and
F
D
FD
F
D
.(tatari/nightmare)