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Serbia Contests
Serbia JBMO TST
2021 Serbia JBMO TSTs
2021 Serbia JBMO TSTs
Part of
Serbia JBMO TST
Subcontests
(4)
4
1
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Serbia JBMO TST 2021 problem 4
On sides
A
B
AB
A
B
and
A
C
AC
A
C
of an acute triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
, with orthocenter
H
H
H
and circumcenter
O
O
O
, are given points
P
P
P
and
Q
Q
Q
respectively such that
A
P
H
Q
APHQ
A
P
H
Q
is a parallelogram. Prove the following equality: \begin{align*} \frac{PB\cdot PQ}{QA\cdot QO}=2 \end{align*}
3
1
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Serbia JBMO TST 2021 problem 3
Two players play the following game: alternatively they write numbers
1
1
1
or
0
0
0
in the vertices of an
n
n
n
-gon. First player starts the game and wins if after any of his moves there exists a triangle, whose vertices are three consecutive vertices of the
n
n
n
-gon, such that the sum of numbers in it's vertices is divisible by
3
3
3
. Second player wins if he prevents this. Determine which player has a winning strategy if: a)
n
=
2019
n=2019
n
=
2019
b)
n
=
2020
n=2020
n
=
2020
c)
n
=
2021
n=2021
n
=
2021
2
1
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Serbia JBMO TST 2021 problem 2
Solve the following equation in natural numbers: \begin{align*} x^2=2^y+2021^z \end{align*}
1
1
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Serbia JBMO TST 2021 problem 1
Prove that for positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
the following inequality holds: \begin{align*} \frac{a}{9bc+1}+\frac{b}{9ca+1}+\frac{c}{9ab+1}\geq \frac{a+b+c}{1+(a+b+c)^2} \end{align*} When does equality occur?