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National and Regional Contests
Serbia Contests
Serbia JBMO TST
2024 Serbia JBMO TST
2024 Serbia JBMO TST
Part of
Serbia JBMO TST
Subcontests
(4)
4
1
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Geometry about a well-known lemma
Let
I
I
I
be the incenter of a triangle
A
B
C
ABC
A
BC
with
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
. Let
M
M
M
be the midpoint of
B
C
BC
BC
,
M
′
∈
B
C
M' \in BC
M
′
∈
BC
be such that
I
M
′
=
I
M
IM'=IM
I
M
′
=
I
M
and
K
K
K
be the midpoint of the arc
B
A
C
BAC
B
A
C
. If
A
K
∩
B
C
=
L
AK \cap BC=L
A
K
∩
BC
=
L
, show that
K
L
I
M
′
KLIM'
K
L
I
M
′
is cyclic.
3
1
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Checkers on a board
a) Is it possible to place
2024
2024
2024
checkers on a board
70
×
70
70 \times 70
70
×
70
so that any square
2
×
2
2 \times 2
2
×
2
contains even number of checkers?b) Is it possible to place
2023
2023
2023
checkers on a board
70
×
70
70 \times 70
70
×
70
so that any square
2
×
2
2 \times 2
2
×
2
contains odd number of checkers?
2
1
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Symmetric 3-variable inequality
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals such that
a
b
+
b
c
+
c
a
=
3
4
ab+bc+ca=\frac{3}{4}
ab
+
b
c
+
c
a
=
4
3
. Show that
(
a
+
b
+
c
)
6
≥
(
9
8
)
3
(
1
+
(
a
+
b
)
2
)
(
1
+
(
b
+
c
)
2
)
(
1
+
(
c
+
a
)
2
)
.
(a+b+c)^6 \geq (\frac{9} {8})^3(1+(a+b)^2)(1+(b+c)^2)(1+(c+a)^2).
(
a
+
b
+
c
)
6
≥
(
8
9
)
3
(
1
+
(
a
+
b
)
2
)
(
1
+
(
b
+
c
)
2
)
(
1
+
(
c
+
a
)
2
)
.
When does equality hold?
1
1
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Equation with powers
Find all non-negative integers
x
,
y
x, y
x
,
y
and primes
p
p
p
such that
3
x
+
p
2
=
7
⋅
2
y
.
3^x+p^2=7 \cdot 2^y.
3
x
+
p
2
=
7
⋅
2
y
.