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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2021 Turkey Junior National Olympiad
2021 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
4
1
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parallel lines given, proving tangency
Let
X
X
X
be a point on the segment
[
B
C
]
[BC]
[
BC
]
of an equilateral triangle
A
B
C
ABC
A
BC
and let
Y
Y
Y
and
Z
Z
Z
be points on the rays
[
B
A
[BA
[
B
A
and
[
C
A
[CA
[
C
A
such that the lines
A
X
,
B
Z
,
C
Y
AX, BZ, CY
A
X
,
BZ
,
C
Y
are parallel. If the intersection of
X
Y
XY
X
Y
and
A
C
AC
A
C
is
M
M
M
and the intersection of
X
Z
XZ
XZ
and
A
B
AB
A
B
is
N
N
N
, prove that
M
N
MN
MN
is tangent to the incenter of
A
B
C
ABC
A
BC
.
3
1
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max x-y where Σx=2 and Σxy=1
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be real numbers such that
x
+
y
+
z
=
2
,
x
y
+
y
z
+
z
x
=
1
x+y+z=2, \;\;\;\; xy+yz+zx=1
x
+
y
+
z
=
2
,
x
y
+
yz
+
z
x
=
1
Find the maximum possible value of
x
−
y
x-y
x
−
y
.
2
1
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Selecting squares from a 29x29 table
We are numbering the rows and columns of a
29
x
29
29 \text{x} 29
29
x
29
chess table with numbers
1
,
2
,
.
.
.
,
29
1, 2, ..., 29
1
,
2
,
...
,
29
in order (Top row is numbered with
1
1
1
and first columns is numbered with
1
1
1
as well). We choose some of the squares in this chess table and for every selected square, we know that there exist at most one square having a row number greater than or equal to this selected square's row number and a column number greater than or equal to this selected square's column number. How many squares can we choose at most?
1
1
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Making integer from m, n again?
Find all
(
m
,
n
)
(m, n)
(
m
,
n
)
positive integer pairs such that both
3
n
2
m
\frac{3n^2}{m}
m
3
n
2
and
n
2
+
m
\sqrt{n^2+m}
n
2
+
m
are integers.