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Problems
Contests
National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2014 JBMO TST - Turkey
2014 JBMO TST - Turkey
Part of
JBMO TST - Turkey
Subcontests
(4)
4
2
Hide problems
Minimize (a+5)^2+(b-2)^2+(c-9)^2 subject to a constraint
Determine the smallest value of
(
a
+
5
)
2
+
(
b
−
2
)
2
+
(
c
−
9
)
2
(a+5)^2+(b-2)^2+(c-9)^2
(
a
+
5
)
2
+
(
b
−
2
)
2
+
(
c
−
9
)
2
for all real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying
a
2
+
b
2
+
c
2
−
a
b
−
b
c
−
c
a
=
3
a^2+b^2+c^2-ab-bc-ca=3
a
2
+
b
2
+
c
2
−
ab
−
b
c
−
c
a
=
3
.
A game on a complete graph
Alice and Bob play a game on a complete graph
G
G
G
with
2014
2014
2014
vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of
G
G
G
. At each move Bob chooses a positive integer number
m
,
m,
m
,
1
≤
m
≤
1000
1 \le m \le 1000
1
≤
m
≤
1000
and after that directs
m
m
m
undirected edges of
G
G
G
. The game ends when all edges are directed. If there is some directed cycle in
G
G
G
Alice wins. Determine whether Alice has a winning strategy.
3
2
Hide problems
m^6+5n^2=m+n^3
Find all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of positive integers satsifying
m
6
+
5
n
2
=
m
+
n
3
m^6+5n^2=m+n^3
m
6
+
5
n
2
=
m
+
n
3
.
Equivalence of identities in a triangle
Let a line
ℓ
\ell
ℓ
intersect the line
A
B
AB
A
B
at
F
F
F
, the sides
A
C
AC
A
C
and
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
at
D
D
D
and
E
E
E
, respectively and the internal bisector of the angle
B
A
C
BAC
B
A
C
at
P
P
P
. Suppose that
F
F
F
is at the opposite side of
A
A
A
with respect to the line
B
C
BC
BC
,
C
D
=
C
E
CD = CE
C
D
=
CE
and
P
P
P
is in the interior the triangle
A
B
C
ABC
A
BC
. Prove that
F
B
⋅
F
A
+
C
P
2
=
C
F
2
⟺
A
D
⋅
B
E
=
P
D
2
.
FB \cdot FA+CP^2 = CF^2 \iff AD \cdot BE = PD^2.
FB
⋅
F
A
+
C
P
2
=
C
F
2
⟺
A
D
⋅
BE
=
P
D
2
.
2
2
Hide problems
3m balls distributed into 8 boxes
3
m
3m
3
m
balls numbered
1
,
1
,
1
,
2
,
2
,
2
,
3
,
3
,
3
,
…
,
m
,
m
,
m
1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m
1
,
1
,
1
,
2
,
2
,
2
,
3
,
3
,
3
,
…
,
m
,
m
,
m
are distributed into
8
8
8
boxes so that any two boxes contain identical balls. Find the minimal possible value of
m
m
m
.
(a^3+b)(b^3+a)=2^c
Find all triples of positive integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
satisfying
(
a
3
+
b
)
(
b
3
+
a
)
=
2
c
(a^3+b)(b^3+a)=2^c
(
a
3
+
b
)
(
b
3
+
a
)
=
2
c
.
1
2
Hide problems
Show that the sum of the angles is equal to 180
In a triangle
A
B
C
ABC
A
BC
, the external bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersects the ray
B
C
BC
BC
at
D
D
D
. The feet of the perpendiculars from
B
B
B
and
C
C
C
to line
A
D
AD
A
D
are
E
E
E
and
F
F
F
, respectively and the foot of the perpendicular from
D
D
D
to
A
C
AC
A
C
is
G
G
G
. Show that
∠
D
G
E
+
∠
D
G
F
=
18
0
∘
\angle DGE + \angle DGF = 180^{\circ}
∠
D
GE
+
∠
D
GF
=
18
0
∘
.
x(x+1)^3=(2x+a)(x+a+1)
Find all real values of
a
a
a
for which the equation
x
(
x
+
1
)
3
=
(
2
x
+
a
)
(
x
+
a
+
1
)
x(x+1)^3=(2x+a)(x+a+1)
x
(
x
+
1
)
3
=
(
2
x
+
a
)
(
x
+
a
+
1
)
has four distinct real roots.