MathDB
Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2022 Bosnia and Herzegovina Junior BMO TST
2022 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
4
1
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JBMO TST Bosnia and Herzegovina 2022 P4
Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is
s
o
c
i
a
l
social
soc
ia
l
if it knows at least
20
20
20
other persons and at least
2
2
2
of those
20
20
20
know each other. For a person, we say that it is
s
h
y
shy
s
h
y
if it doesn't know at least
20
20
20
other persons and at least
2
2
2
of those
20
20
20
don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any
s
o
c
i
a
l
social
soc
ia
l
nor
s
h
y
shy
s
h
y
persons.
3
1
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JBMO TST Bosnia and Herzegovina 2022 P3
Let
A
B
C
ABC
A
BC
be an acute triangle. Tangents on the circumscribed circle of triangle
A
B
C
ABC
A
BC
at points
B
B
B
and
C
C
C
intersect at point
T
T
T
. Let
D
D
D
and
E
E
E
be a foot of the altitudes from
T
T
T
onto
A
B
AB
A
B
and
A
C
AC
A
C
and let
M
M
M
be the midpoint of
B
C
BC
BC
. Prove: A) Prove that
M
M
M
is the orthocenter of the triangle
A
D
E
ADE
A
D
E
. B) Prove that
T
M
TM
TM
cuts
D
E
DE
D
E
in half.
2
1
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JBMO TST Bosnia and Herzegovina 2022 P2
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive integers greater than
1
1
1
such that
p
=
a
b
+
b
c
+
a
c
p=ab+bc+ac
p
=
ab
+
b
c
+
a
c
is prime. A) Prove that
a
2
,
b
2
,
c
2
a^2, b^2, c^2
a
2
,
b
2
,
c
2
all have different reminder
m
o
d
p
mod\ p
m
o
d
p
. B) Prove that
a
3
,
b
3
,
c
3
a^3, b^3, c^3
a
3
,
b
3
,
c
3
all have different reminder
m
o
d
p
mod\ p
m
o
d
p
.
1
1
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JBMO TST- Bosnia and Herzegovina 2022 P1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers such that
a
2
−
b
c
=
b
2
−
c
a
=
c
2
−
a
b
=
2
a^2-bc=b^2-ca=c^2-ab=2
a
2
−
b
c
=
b
2
−
c
a
=
c
2
−
ab
=
2
. Find the value of
a
b
+
b
c
+
c
a
ab+bc+ca
ab
+
b
c
+
c
a
and find at least one triplet
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
that satisfy those conditions.