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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia JBMO TST
2023 Serbia JBMO TST
2023 Serbia JBMO TST
Part of
Serbia JBMO TST
Subcontests
(3)
4
1
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Seemingly ugly NT with 4 conditions
Find all triples
(
k
,
m
,
n
)
(k, m, n)
(
k
,
m
,
n
)
of positive integers such that
m
m
m
is a prime and:(1)
k
n
kn
kn
is a perfect square;(2)
k
(
k
−
1
)
2
+
n
\frac{k(k-1)}{2}+n
2
k
(
k
−
1
)
+
n
is a fourth power of a prime;(3)
k
−
m
2
=
p
k-m^2=p
k
−
m
2
=
p
where
p
p
p
is a prime;(4)
n
+
2
m
2
=
p
4
\frac{n+2}{m^2}=p^4
m
2
n
+
2
=
p
4
.
3
1
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min and max of a^3+b^3+c^3
Let
a
,
b
,
c
∈
[
0
;
1
]
a, b, c \in [0;1]
a
,
b
,
c
∈
[
0
;
1
]
be reals such that
a
b
+
b
c
+
c
a
=
1
ab+bc+ca=1
ab
+
b
c
+
c
a
=
1
. Find the minimal and maximal value of
a
3
+
b
3
+
c
3
a^3+b^3+c^3
a
3
+
b
3
+
c
3
.
1
1
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Isosceles triangle geo
Given is an isosceles triangle
A
B
C
ABC
A
BC
with
C
A
=
C
B
CA=CB
C
A
=
CB
and angle bisector
B
D
BD
B
D
,
D
∈
A
C
D \in AC
D
∈
A
C
. The line through the center
O
O
O
of
(
A
B
C
)
(ABC)
(
A
BC
)
, perpendicular to
B
D
BD
B
D
, meets
B
C
BC
BC
at
E
E
E
. The line through
E
E
E
, parallel to
B
D
BD
B
D
, meets
A
C
AC
A
C
at
F
F
F
. Prove that
C
E
=
D
F
CE=DF
CE
=
D
F
.