MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Akdeniz University MO
2011 Akdeniz University MO
2011 Akdeniz University MO
Part of
Akdeniz University MO
Subcontests
(5)
5
2
Hide problems
number theory
For all
n
∈
Z
+
n \in {\mathbb Z^+}
n
∈
Z
+
we define
I
n
=
{
0
n
,
1
n
,
2
n
,
⋯
,
n
−
1
n
,
n
n
,
n
+
1
n
,
⋯
}
I_n=\{\frac{0}{n},\frac{1}{n},\frac{2}{n},\dotsm,\frac{n-1}{n},\frac{n}{n},\frac{n+1}{n},\dotsm\}
I
n
=
{
n
0
,
n
1
,
n
2
,
⋯
,
n
n
−
1
,
n
n
,
n
n
+
1
,
⋯
}
infinite cluster. For whichever
x
x
x
and
y
y
y
real number, we say
∣
x
−
y
∣
\mid{x-y}\mid
∣
x
−
y
∣
is between distance of the
x
x
x
and
y
y
y
.a) For all
n
n
n
's we find a number in
I
n
I_n
I
n
such that, the between distance of the this number and
2
\sqrt 2
2
is less than
1
2
n
\frac{1}{2n}
2
n
1
b) We find a
n
n
n
such that, between distance of the a number in
I
n
I_n
I
n
and
2
\sqrt 2
2
is less than
1
2011
n
\frac{1}{2011n}
2011
n
1
geometry
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
H
H
H
orthocenter,
O
O
O
circumcenter.
[
A
H
]
[AH]
[
A
H
]
's perpendicular bisector intersects with
[
A
B
]
[AB]
[
A
B
]
and
[
A
C
]
[AC]
[
A
C
]
at
D
D
D
and
E
E
E
respectively. Prove that
∠
A
D
E
=
∠
B
D
O
\angle ADE =\angle BDO
∠
A
D
E
=
∠
B
D
O
4
2
Hide problems
Geometry (Classic)
Let an acute-angled triangle
A
B
C
ABC
A
BC
's circumcircle is
S
S
S
.
S
S
S
's tangent from
B
B
B
and
C
C
C
intersects at point
M
M
M
. A line, lies
M
M
M
and parallel to
[
A
B
]
[AB]
[
A
B
]
intersects with
S
S
S
at points
D
D
D
and
E
E
E
, intersect with
[
A
C
]
[AC]
[
A
C
]
at point
F
F
F
. Prove that
[
D
F
]
=
[
F
E
]
[DF]=[FE]
[
D
F
]
=
[
FE
]
Arithmetic sequence
a
n
a_n
a
n
sequence is a arithmetic sequence with all terms be positive integers. (for
a
n
a_n
a
n
non-constant sequence) Let
p
n
p_n
p
n
is greatest prime divisor of
a
n
a_n
a
n
. Prove that
(
a
n
p
n
)
(\frac{a_n}{p_n})
(
p
n
a
n
)
sequence is infinity.Note: If we find a
M
>
0
M>0
M
>
0
constant such that
x
n
≤
M
x_n \leq M
x
n
≤
M
for all
n
∈
N
n \in {\mathbb N}
n
∈
N
's,
(
x
n
)
(x_n)
(
x
n
)
sequence is non-infinite, but we can't find
M
M
M
,
(
x
n
)
(x_n)
(
x
n
)
sequence is infinity
3
2
Hide problems
inequality
For all
x
≥
2
x \geq 2
x
≥
2
,
y
≥
2
y \geq 2
y
≥
2
real numbers, prove that
x
(
4
x
y
−
1
+
1
2
y
+
x
)
+
y
(
y
6
x
−
9
+
1
2
x
+
y
)
>
26
3
x(\frac{4x}{y-1}+\frac{1}{2y+x})+y(\frac{y}{6x-9}+\frac{1}{2x+y}) > \frac{26}{3}
x
(
y
−
1
4
x
+
2
y
+
x
1
)
+
y
(
6
x
−
9
y
+
2
x
+
y
1
)
>
3
26
inequality...
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
positive reals such that
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
. Show that following expression's minimum value is
2
2
2
.
a
+
b
+
c
a
b
+
b
c
+
c
a
+
1
1
+
2
a
b
+
1
1
+
2
b
c
+
1
1
+
2
c
a
\frac{\sqrt a +\sqrt b +\sqrt c}{ab+bc+ca} + \frac{1}{1+2\sqrt {ab}} + \frac {1}{1+ 2\sqrt {bc}} + \frac{1}{1+ 2\sqrt {ca}}
ab
+
b
c
+
c
a
a
+
b
+
c
+
1
+
2
ab
1
+
1
+
2
b
c
1
+
1
+
2
c
a
1
2
2
Hide problems
number theory
Let
a
a
a
and
b
b
b
is roots of the
x
2
−
6
x
+
1
x^2-6x+1
x
2
−
6
x
+
1
equation.a) Show that, for all
n
∈
Z
+
n \in{\mathbb Z^+}
n
∈
Z
+
,
a
n
+
b
n
a^n+b^n
a
n
+
b
n
is a integer. b) Show that, for all
n
∈
Z
+
n \in{\mathbb Z^+}
n
∈
Z
+
,
5
5
5
isn't divide
a
n
+
b
n
a^n+b^n
a
n
+
b
n
question
Let
O
O
O
is a point in a plane
P
P
P
and let
[
O
X
,
[
O
Y
,
[
O
Z
[OX,[OY,[OZ
[
OX
,
[
O
Y
,
[
OZ
is distinct ray in
P
P
P
. Prove that, if
A
∈
[
O
X
A \in [OX
A
∈
[
OX
,
B
∈
[
O
Y
B \in [OY
B
∈
[
O
Y
and
C
∈
[
O
Z
C \in [OZ
C
∈
[
OZ
points such that
△
O
A
B
\triangle OAB
△
O
A
B
,
△
O
B
C
\triangle OBC
△
OBC
and
△
O
C
A
\triangle OCA
△
OC
A
's perimeter is 2, there is only one
(
A
,
B
,
C
)
(A,B,C)
(
A
,
B
,
C
)
triple
1
2
Hide problems
number theory
Let
a
a
a
be a positive number, and we show decimal part of the
a
a
a
with
{
a
}
\left\{a\right\}
{
a
}
.For a positive number
x
x
x
with
2
<
x
<
3
\sqrt 2< x <\sqrt 3
2
<
x
<
3
such that,
{
1
x
}
\left\{\frac{1}{x}\right\}
{
x
1
}
=
{
x
2
}
\left\{x^2\right\}
{
x
2
}
.Find value of the
x
(
x
7
−
21
)
x(x^7-21)
x
(
x
7
−
21
)
Number theory
Let
m
,
n
m,n
m
,
n
positive integers and
p
p
p
prime number with
p
=
3
k
+
2
p=3k+2
p
=
3
k
+
2
. If
p
∣
(
m
+
n
)
2
−
m
n
p \mid {(m+n)^2-mn}
p
∣
(
m
+
n
)
2
−
mn
, prove that
p
∣
m
,
n
p \mid m,n
p
∣
m
,
n