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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2014 Turkey Junior National Olympiad
2014 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
4
1
Hide problems
Two equations hold then two segments have equal length
A
B
C
ABC
A
BC
is an acute triangle with orthocenter
H
H
H
. Points
D
D
D
and
E
E
E
lie on segment
B
C
BC
BC
. Circumcircle of
△
B
H
C
\triangle BHC
△
B
H
C
instersects with segments
A
D
AD
A
D
,
A
E
AE
A
E
at
P
P
P
and
Q
Q
Q
, respectively. Prove that if
B
D
2
+
C
D
2
=
2
D
P
⋅
D
A
BD^2+CD^2=2DP\cdot DA
B
D
2
+
C
D
2
=
2
D
P
⋅
D
A
and
B
E
2
+
C
E
2
=
2
E
Q
⋅
E
A
BE^2+CE^2=2EQ\cdot EA
B
E
2
+
C
E
2
=
2
EQ
⋅
E
A
, then
B
P
=
C
Q
BP=CQ
BP
=
CQ
.
3
1
Hide problems
2014 balls arranged around a circle.
There are
2014
2014
2014
balls with
106
106
106
different colors,
19
19
19
of each color. Determine the least possible value of
n
n
n
so that no matter how these balls are arranged around a circle, one can choose
n
n
n
consecutive balls so that amongst them, there are
53
53
53
balls with different colors.
2
1
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Min number of distinct prime divisors of 19^{4n}+4
Determine the minimum possible amount of distinct prime divisors of
1
9
4
n
+
4
19^{4n}+4
1
9
4
n
+
4
, for a positive integer
n
n
n
.
1
1
Hide problems
Inequality with a+b+c+abc=4
Prove that for positive reals
a
a
a
,
b
b
b
,
c
c
c
so that
a
+
b
+
c
+
a
b
c
=
4
a+b+c+abc=4
a
+
b
+
c
+
ab
c
=
4
,
(
1
+
a
b
+
c
a
)
(
1
+
b
c
+
a
b
)
(
1
+
c
a
+
b
c
)
≥
27
\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27
(
1
+
b
a
+
c
a
)
(
1
+
c
b
+
ab
)
(
1
+
a
c
+
b
c
)
≥
27
holds.