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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
1996 Turkey MO (2nd round)
1996 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
3
2
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Integers on real axis
Let
n
n
n
integers on the real axis be colored. Determine for which positive integers
k
k
k
there exists a family
K
K
K
of closed intervals with the following properties: i) The union of the intervals in
K
K
K
contains all of the colored points; ii) Any two distinct intervals in
K
K
K
are disjoint; iii) For each interval
I
I
I
at
K
K
K
we have
a
I
=
k
.
b
I
{{a}_{I}}=k.{{b}_{I}}
a
I
=
k
.
b
I
, where
a
I
{{a}_{I}}
a
I
denotes the number of integers in
I
I
I
, and
b
I
{{b}_{I}}
b
I
the number of colored integers in
I
I
I
.
Proving existence of x,y such that f(x+y)<=f(x)(1+yf(x))
Show that there is no function
f
:
R
+
→
R
+
f:{{\mathbb{R}}^{+}}\to {{\mathbb{R}}^{+}}
f
:
R
+
→
R
+
such that
f
(
x
+
y
)
>
f
(
x
)
(
1
+
y
f
(
x
)
)
f(x+y)>f(x)(1+yf(x))
f
(
x
+
y
)
>
f
(
x
)
(
1
+
y
f
(
x
))
for all
x
,
y
∈
R
+
x,y\in {{\mathbb{R}}^{+}}
x
,
y
∈
R
+
.
2
2
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A geometric Inequality on a square
Let
A
B
C
D
ABCD
A
BC
D
be a square of side length 2, and let
M
M
M
and
N
N
N
be points on the sides
A
B
AB
A
B
and
C
D
CD
C
D
respectively. The lines
C
M
CM
CM
and
B
N
BN
BN
meet at
P
P
P
, while the lines
A
N
AN
A
N
and
D
M
DM
D
M
meet at
Q
Q
Q
. Prove that
∣
P
Q
∣
≥
1
\left| PQ \right|\ge 1
∣
PQ
∣
≥
1
.
n! divides product of (2^n-2^k) for k=0,..,n-1
Prove that
∏
k
=
0
n
−
1
(
2
n
−
2
k
)
\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}
k
=
0
∏
n
−
1
(
2
n
−
2
k
)
is divisible by
n
!
n!
n
!
for all positive integers
n
n
n
.
1
2
Hide problems
a linear diophantine equation for two sequences
Let
(
A
n
)
n
=
1
∞
({{A}_{n}})_{n=1}^{\infty }
(
A
n
)
n
=
1
∞
and
(
a
n
)
n
=
1
∞
({{a}_{n}})_{n=1}^{\infty }
(
a
n
)
n
=
1
∞
be sequences of positive integers. Assume that for each positive integer
x
x
x
, there is a unique positive integer
N
N
N
and a unique
N
−
t
u
p
l
e
N-tuple
N
−
t
u
pl
e
(
x
1
,
.
.
.
,
x
N
)
({{x}_{1}},...,{{x}_{N}})
(
x
1
,
...
,
x
N
)
such that
0
≤
x
k
≤
a
k
0\le {{x}_{k}}\le {{a}_{k}}
0
≤
x
k
≤
a
k
for
k
=
1
,
2
,
.
.
.
N
k=1,2,...N
k
=
1
,
2
,
...
N
,
x
N
≠
0
{{x}_{N}}\ne 0
x
N
=
0
, and
x
=
∑
k
=
1
N
A
k
x
k
x=\sum\limits_{k=1}^{N}{{{A}_{k}}{{x}_{k}}}
x
=
k
=
1
∑
N
A
k
x
k
. (a) Prove that
A
k
=
1
{{A}_{k}}=1
A
k
=
1
for some
k
k
k
; (b) Prove that
A
k
=
A
j
⇔
k
=
j
{{A}_{k}}={{A}_{j}}\Leftrightarrow k=j
A
k
=
A
j
⇔
k
=
j
; (c) Prove that if
A
k
≤
A
j
{{A}_{k}}\le {{A}_{j}}
A
k
≤
A
j
, then
A
k
∣
A
j
\left. {{A}_{k}} \right|{{A}_{j}}
A
k
∣
A
j
.
proving PL=PN
A circle is tangent to sides
A
D
,
D
C
,
C
B
AD,\text{ }DC,\text{ }CB
A
D
,
D
C
,
CB
of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
at
K
,
L
,
M
\text{K},\text{ L},\text{ M}
K
,
L
,
M
respectively. A line
l
l
l
, passing through
L
L
L
and parallel to
A
D
AD
A
D
, meets
K
M
KM
K
M
at
N
N
N
and
K
C
KC
K
C
at
P
P
P
. Prove that
P
L
=
P
N
PL=PN
P
L
=
PN
.