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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2022 Turkey Junior National Olympiad
2022 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
3
1
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Minimum value of exponent in diophantine equation with 4 unknowns
Let
m
,
n
,
a
,
k
m, n, a, k
m
,
n
,
a
,
k
be positive integers and
k
>
1
k>1
k
>
1
such that the equality
5
m
+
63
n
+
49
=
a
k
5^m+63n+49=a^k
5
m
+
63
n
+
49
=
a
k
holds. Find the minimum value of
k
k
k
.
4
1
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ABCD Parallelogram with circle tangent to AB
In parallellogram
A
B
C
D
ABCD
A
BC
D
, on the arc
B
C
BC
BC
of the circumcircle
(
A
B
C
)
(ABC)
(
A
BC
)
, not containing the point
A
A
A
, we take a point
P
P
P
and on the
[
A
C
[AC
[
A
C
, we take a point
Q
Q
Q
such that
∠
P
B
C
=
∠
C
D
Q
\angle PBC= \angle CDQ
∠
PBC
=
∠
C
D
Q
. Prove that
(
A
P
Q
)
(APQ)
(
A
PQ
)
is tangent to
A
B
AB
A
B
.
1
1
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Inequality with AM-GM
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive reals such that
x
≤
1
x \leq 1
x
≤
1
. Prove that
x
y
+
y
+
2
z
≥
4
x
y
z
xy+y+2z \geq 4 \sqrt{xyz}
x
y
+
y
+
2
z
≥
4
x
yz
2
1
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School principal promotes social interactions
In a school with
101
101
101
students, each student has at least one friend among the other students. Show that for every integer
1
<
n
<
101
1<n<101
1
<
n
<
101
, a group of
n
n
n
students can be selected from this school in such a way that each selected student has at least one friend among the other selected students.