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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2012 Turkey Junior National Olympiad
2012 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
4
1
Hide problems
Balls inside pockets which are inside boxes
We want to place
2012
2012
2012
pockets, including variously colored balls, into
k
k
k
boxes such thati) For any box, all pockets in this box must include a ball with the same color or ii) For any box, all pockets in this box must include a ball having a color which is not included in any other pocket in this boxFind the smallest value of
k
k
k
for which we can always do this placement whatever the number of balls in the pockets and whatever the colors of balls.
3
1
Hide problems
A symmetric inequality with a condition
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers satisfying
a
3
+
b
3
+
c
3
=
a
4
+
b
4
+
c
4
a^3+b^3+c^3=a^4+b^4+c^4
a
3
+
b
3
+
c
3
=
a
4
+
b
4
+
c
4
. Show that
a
a
2
+
b
3
+
c
3
+
b
a
3
+
b
2
+
c
3
+
c
a
3
+
b
3
+
c
2
≥
1
\frac{a}{a^2+b^3+c^3}+\frac{b}{a^3+b^2+c^3}+\frac{c}{a^3+b^3+c^2} \geq 1
a
2
+
b
3
+
c
3
a
+
a
3
+
b
2
+
c
3
b
+
a
3
+
b
3
+
c
2
c
≥
1
2
1
Hide problems
Show that m(BCD)=90
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, the diagonals are perpendicular to each other and they intersect at
E
E
E
. Let
P
P
P
be a point on the side
A
D
AD
A
D
which is different from
A
A
A
such that
P
E
=
E
C
.
PE=EC.
PE
=
EC
.
The circumcircle of triangle
B
C
D
BCD
BC
D
intersects the side
A
D
AD
A
D
at
Q
Q
Q
where
Q
Q
Q
is also different from
A
A
A
. The circle, passing through
A
A
A
and tangent to line
E
P
EP
EP
at
P
P
P
, intersects the line segment
A
C
AC
A
C
at
R
R
R
. If the points
B
,
R
,
Q
B, R, Q
B
,
R
,
Q
are concurrent then show that
∠
B
C
D
=
9
0
∘
\angle BCD=90^{\circ}
∠
BC
D
=
9
0
∘
.
1
1
Hide problems
A diophantine equation
Let
x
,
y
x, y
x
,
y
be integers and
p
p
p
be a prime for which
x
2
−
3
x
y
+
p
2
y
2
=
12
p
x^2-3xy+p^2y^2=12p
x
2
−
3
x
y
+
p
2
y
2
=
12
p
Find all triples
(
x
,
y
,
p
)
(x,y,p)
(
x
,
y
,
p
)
.