MathDB
Problems
Contests
National and Regional Contests
Romania Contests
JBMO TST - Romania
2023 Junior Balkan Team Selection Tests - Romania
2023 Junior Balkan Team Selection Tests - Romania
Part of
JBMO TST - Romania
Subcontests
(5)
P4
4
Show problems
P3
4
Show problems
P2
3
Hide problems
Junior geo with incenter
Given is a triangle
A
B
C
ABC
A
BC
. Let the points
P
P
P
and
Q
Q
Q
be on the sides
A
B
,
A
C
AB, AC
A
B
,
A
C
, respectively, so that
A
P
=
A
Q
AP=AQ
A
P
=
A
Q
, and
P
Q
PQ
PQ
passes through the incenter
I
I
I
. Let
(
B
P
I
)
(BPI)
(
BP
I
)
meet
(
C
Q
I
)
(CQI)
(
CQ
I
)
at
M
M
M
,
P
M
PM
PM
meets
B
I
BI
B
I
at
D
D
D
and
Q
M
QM
QM
meets
C
I
CI
C
I
at
E
E
E
. Prove that the line
M
I
MI
M
I
passes through the midpoint of
D
E
DE
D
E
.
Nice geo problem
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
B
C
>
A
B
BC > AB
BC
>
A
B
, such that the points
A
A
A
,
H
H
H
,
I
I
I
and
C
C
C
are concyclic (where
H
H
H
is the orthocenter and
I
I
I
is the incenter of triangle
A
B
C
ABC
A
BC
). The line
A
C
AC
A
C
intersects the circumcircle of triangle
B
H
C
BHC
B
H
C
at point
T
T
T
, and the line
B
C
BC
BC
intersects the circumcircle of triangle
A
H
C
AHC
A
H
C
at point
P
P
P
. If the lines
P
T
PT
PT
and
H
I
HI
H
I
are parallel, determine the measures of the angles of triangle
A
B
C
ABC
A
BC
.
Difference of descending and ascending triples
Given is a positive integer
n
≥
2
n \geq 2
n
≥
2
and three pairwise disjoint sets
A
,
B
,
C
A, B, C
A
,
B
,
C
, each of
n
n
n
distinct real numbers. Denote by
a
a
a
the number of triples
(
x
,
y
,
z
)
∈
A
×
B
×
C
(x, y, z) \in A \times B \times C
(
x
,
y
,
z
)
∈
A
×
B
×
C
satisfying
x
<
y
<
z
x<y<z
x
<
y
<
z
and let
b
b
b
denote the number of triples
(
x
,
y
,
z
)
∈
A
×
B
×
C
(x, y, z) \in A \times B \times C
(
x
,
y
,
z
)
∈
A
×
B
×
C
such that
x
>
y
>
z
x>y>z
x
>
y
>
z
. Prove that
n
n
n
divides
a
−
b
a-b
a
−
b
.
P1
4
Show problems
P5
1
Hide problems
Trapezoid and squares
Outside of the trapezoid
A
B
C
D
ABCD
A
BC
D
with the smaller base
A
B
AB
A
B
are constructed the squares
A
D
E
F
ADEF
A
D
EF
and
B
C
G
H
BCGH
BCG
H
. Prove that the perpendicular bisector of
A
B
AB
A
B
passes through the midpoint of
F
H
FH
F
H
.