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Problems
Contests
National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2012 JBMO TST - Turkey
2012 JBMO TST - Turkey
Part of
JBMO TST - Turkey
Subcontests
(4)
1
2
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Find the greatest positive integer such that
Find the greatest positive integer
n
n
n
for which
n
n
n
is divisible by all positive integers whose cube is not greater than
n
.
n.
n
.
Geometric Inequality
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the side-lengths of a triangle,
r
r
r
be the inradius and
r
a
,
r
b
,
r
c
r_a, r_b, r_c
r
a
,
r
b
,
r
c
be the corresponding exradius. Show that
a
+
b
+
c
a
2
+
b
2
+
c
2
≤
2
⋅
r
a
2
+
r
b
2
+
r
c
2
r
a
+
r
b
+
r
c
−
3
r
\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r}
a
2
+
b
2
+
c
2
a
+
b
+
c
≤
2
⋅
r
a
+
r
b
+
r
c
−
3
r
r
a
2
+
r
b
2
+
r
c
2
2
2
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Partition of a set
Let
S
=
{
1
,
2
,
3
,
…
,
2012
}
.
S=\{1,2,3,\ldots,2012\}.
S
=
{
1
,
2
,
3
,
…
,
2012
}
.
We want to partition
S
S
S
into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of
2.
2.
2.
Find the number of such partitions.
Find all numbers satisfying a condition
Find all positive integers
m
,
n
m,n
m
,
n
and prime numbers
p
p
p
for which
5
m
+
2
n
p
5
m
−
2
n
p
\frac{5^m+2^np}{5^m-2^np}
5
m
−
2
n
p
5
m
+
2
n
p
is a perfect square.
3
2
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Show that all lines pass through a fixed point
Let
[
A
B
]
[AB]
[
A
B
]
be a chord of the circle
Γ
\Gamma
Γ
not passing through its center and let
M
M
M
be the midpoint of
[
A
B
]
.
[AB].
[
A
B
]
.
Let
C
C
C
be a variable point on
Γ
\Gamma
Γ
different from
A
A
A
and
B
B
B
and
P
P
P
be the point of intersection of the tangent lines at
A
A
A
of circumcircle of
C
A
M
CAM
C
A
M
and at
B
B
B
of circumcircle of
C
B
M
.
CBM.
CBM
.
Show that all
C
P
CP
CP
lines pass through a fixed point.
2-variable symmetric inequality
Show that for all real numbers
x
,
y
x, y
x
,
y
satisfying
x
+
y
≥
0
x+y \geq 0
x
+
y
≥
0
(
x
2
+
y
2
)
3
≥
32
(
x
3
+
y
3
)
(
x
y
−
x
−
y
)
(x^2+y^2)^3 \geq 32(x^3+y^3)(xy-x-y)
(
x
2
+
y
2
)
3
≥
32
(
x
3
+
y
3
)
(
x
y
−
x
−
y
)
4
2
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Symmetric 3-variable inequality
Find the greatest real number
M
M
M
for which
a
2
+
b
2
+
c
2
+
3
a
b
c
≥
M
(
a
b
+
b
c
+
c
a
)
a^2+b^2+c^2+3abc \geq M(ab+bc+ca)
a
2
+
b
2
+
c
2
+
3
ab
c
≥
M
(
ab
+
b
c
+
c
a
)
for all non-negative real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying
a
+
b
+
c
=
4.
a+b+c=4.
a
+
b
+
c
=
4.
Connected simple graph
Let
G
G
G
be a connected simple graph. When we add an edge to
G
G
G
(between two unconnected vertices), then using at most
17
17
17
edges we can reach any vertex from any other vertex. Find the maximum number of edges to be used to reach any vertex from any other vertex in the original graph, i.e. in the graph before we add an edge.