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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2013 Turkey Junior National Olympiad
2013 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
4
1
Hide problems
2013 balls inside boxes
Player
A
A
A
places an odd number of boxes around a circle and distributes
2013
2013
2013
balls into some of these boxes. Then the player
B
B
B
chooses one of these boxes and takes the balls in it. After that the player
A
A
A
chooses half of the remaining boxes such that none of two are consecutive and take the balls in them. If player
A
A
A
guarantees to take
k
k
k
balls, find the maximum possible value of
k
k
k
.
3
1
Hide problems
Show that three lines are concurrent
Let
A
B
C
ABC
A
BC
be a triangle such that
A
C
>
A
B
.
AC>AB.
A
C
>
A
B
.
A circle tangent to the sides
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
respectively, intersects the circumcircle of
A
B
C
ABC
A
BC
at
K
K
K
and
L
L
L
. Let
X
X
X
and
Y
Y
Y
be points on the sides
A
B
AB
A
B
and
A
C
AC
A
C
respectively, satisfying \frac{AX}{AB}=\frac{CE}{BD+CE} \text{and} \frac{AY}{AC}=\frac{BD}{BD+CE} Show that the lines
X
Y
,
B
C
XY, BC
X
Y
,
BC
and
K
L
KL
K
L
are concurrent.
2
1
Hide problems
Find all prime triples
Find all prime numbers
p
,
q
,
r
p, q, r
p
,
q
,
r
satisfying the equation
p
4
+
2
p
+
q
4
+
q
2
=
r
2
+
4
q
3
+
1
p^4+2p+q^4+q^2=r^2+4q^3+1
p
4
+
2
p
+
q
4
+
q
2
=
r
2
+
4
q
3
+
1
1
1
Hide problems
Find max of |(x-y)(y-z)(z-x)|
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be real numbers satisfying
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
and
x
2
+
y
2
+
z
2
=
6
x^2+y^2+z^2=6
x
2
+
y
2
+
z
2
=
6
. Find the maximum value of
∣
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
∣
|(x-y)(y-z)(z-x) |
∣
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
∣