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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2005 Turkey MO (2nd round)
2005 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
5
1
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Geometric inequality with a,b,c,r
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are the sides of a triangle and
r
r
r
the inradius of the triangle, prove that
1
a
2
+
1
b
2
+
1
c
2
≤
1
4
r
2
\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le \frac{1}{4r^2}
a
2
1
+
b
2
1
+
c
2
1
≤
4
r
2
1
6
1
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Sequence determined by indices condition
Suppose that a sequence
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
of integers has the following property: For all
n
n
n
large enough (i.e.
n
≥
N
n \ge N
n
≥
N
for some
N
N
N
),
a
n
a_n
a
n
equals the number of indices
i
i
i
,
1
≤
i
<
n
1 \le i < n
1
≤
i
<
n
, such that
a
i
+
i
≥
n
a_i + i \ge n
a
i
+
i
≥
n
. Find the maximum possible number of integers which occur infinitely many times in the sequence.
4
1
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5^m + 7^n=k^3
Find all triples of nonnegative integers
(
m
,
n
,
k
)
(m,n,k)
(
m
,
n
,
k
)
satisfying
5
m
+
7
n
=
k
3
5^m+7^n=k^3
5
m
+
7
n
=
k
3
.
3
1
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Airlines and cities
Some of the
n
+
1
n + 1
n
+
1
cities in a country (including the capital city) are connected by one-way or two-way airlines. No two cities are connected by both a one-way airline and a two-way airline, but there may be more than one two-way airline between two cities. If
d
A
d_A
d
A
denotes the number of airlines from a city
A
A
A
, then
d
A
≤
n
d_A \le n
d
A
≤
n
for any city
A
A
A
other than the capital city and
d
A
+
d
B
≤
n
d_A + d_B \le n
d
A
+
d
B
≤
n
for any two cities
A
A
A
and
B
B
B
other than the capital city which are not connected by a two-way airline. Every airline has a return, possibly consisting of several connected flights. Find the largest possible number of two-way airlines and all configurations of airlines for which this largest number is attained.
2
1
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Circumcentres make a parallelogram
In a triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
, the perpendicular bisectors of
A
C
AC
A
C
and
B
C
BC
BC
intersect
B
C
BC
BC
and
A
C
AC
A
C
at
K
K
K
and
L
L
L
, respectively. Let
O
O
O
,
O
1
O_1
O
1
, and
O
2
O_2
O
2
be the circumcentres of triangles
A
B
C
ABC
A
BC
,
C
K
L
CKL
C
K
L
, and
O
A
B
OAB
O
A
B
, respectively. Prove that
O
C
O
1
O
2
OCO_1O_2
OC
O
1
O
2
is a parallelogram.
1
1
Hide problems
a,b,c,d radical inequality
For all positive real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
prove the inequality
a
4
+
c
4
+
a
4
+
d
4
+
b
4
+
c
4
+
b
4
+
d
4
≥
2
2
(
a
d
+
b
c
)
\sqrt{a^4+c^4}+\sqrt{a^4+d^4}+\sqrt{b^4+c^4}+\sqrt{b^4+d^4} \ge 2\sqrt{2}(ad+bc)
a
4
+
c
4
+
a
4
+
d
4
+
b
4
+
c
4
+
b
4
+
d
4
≥
2
2
(
a
d
+
b
c
)