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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2015 Turkey Junior National Olympiad
2015 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
4
1
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Reflection point is on the median
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
be the midpoint of the segment
B
C
BC
BC
. The circle that passes through
D
D
D
and tangent to
A
B
AB
A
B
at
B
B
B
, and the circle that passes through
D
D
D
and tangent to
A
C
AC
A
C
at
C
C
C
intersect at
M
≠
D
M\neq D
M
=
D
. Let
M
′
M'
M
′
be the reflection of
M
M
M
with respect to
B
C
BC
BC
. Prove that
M
′
M'
M
′
is on
A
D
AD
A
D
.
3
1
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Diophantine equation with prime number and power of 3
Find all pairs
(
p
,
n
)
(p,n)
(
p
,
n
)
so that
p
p
p
is a prime number,
n
n
n
is a positive integer and
p
3
−
2
p
2
+
p
+
1
=
3
n
p^3-2p^2+p+1=3^n
p
3
−
2
p
2
+
p
+
1
=
3
n
holds.
2
1
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20 paintings have a common color but not all paintings do
In an exhibition there are
100
100
100
paintings each of which is made with exactly
k
k
k
colors. Find the minimum possible value of
k
k
k
if any
20
20
20
paintings have a common color but there is no color that is used in all paintings.
1
1
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Functional inequality
For a non-constant function
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
prove that there exist real numbers
x
,
y
x,y
x
,
y
satisfying
f
(
x
+
y
)
<
f
(
x
y
)
f(x+y)<f(xy)
f
(
x
+
y
)
<
f
(
x
y
)