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National and Regional Contests
Romania Contests
JBMO TST - Romania
2024 Junior Balkan Team Selection Tests - Romania
2024 Junior Balkan Team Selection Tests - Romania
Part of
JBMO TST - Romania
Subcontests
(5)
P5
1
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Maximal walk in a special array
An
n
n
n
-type triangle where
n
⩾
2
n\geqslant 2
n
⩾
2
is formed by the cells of a
(
2
n
+
1
)
×
(
2
n
+
1
)
(2n+1)\times(2n+1)
(
2
n
+
1
)
×
(
2
n
+
1
)
board, situated under both main diagonals. For instance, a
3
3
3
-type triangle looks like this:https://i.ibb.co/k4fmwWY/Screenshot-2024-07-31-153932.pngDetermine the maximal length of a sequence with pairwise distinct cells in an
n
n
n
-type triangle, such that, beggining with the second one, any cell of the sequence has a common side with the previous one.Cristi Săvescu
P4
4
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P3
3
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Six numbers with many equal ratios
Determine all positive integers
a
,
b
,
c
,
d
,
e
,
f
a,b,c,d,e,f
a
,
b
,
c
,
d
,
e
,
f
satisfying the following condition: for any two of them,
x
x{}
x
and
y
,
y{},
y
,
two of the remaining numbers,
z
z{}
z
and
t
,
t{},
t
,
satisfy
x
/
y
=
z
/
t
.
x/y=z/t.
x
/
y
=
z
/
t
.
Cristi Săvescu
Composition of two arithmetic functions
Let
σ
(
⋅
)
\sigma(\cdot)
σ
(
⋅
)
denote the divisor sum function and
d
(
⋅
)
d(\cdot)
d
(
⋅
)
denote the divisor counting function. Find all positve integers
n
n
n
such that
σ
(
d
(
n
)
)
=
n
.
\sigma(d(n))=n.
σ
(
d
(
n
))
=
n
.
Andrei Bâra
Right angles and orthocentre
In the exterior of the acute-angles triangle
A
B
C
ABC
A
BC
we construct the isosceles triangles
D
A
B
DAB
D
A
B
and
E
A
C
EAC
E
A
C
with bases
A
B
AB{}
A
B
and
A
C
AC{}
A
C
respectively such that
∠
D
B
C
=
∠
E
C
B
=
9
0
∘
.
\angle DBC=\angle ECB=90^\circ.
∠
D
BC
=
∠
ECB
=
9
0
∘
.
Let
M
M
M
and
N
N
N
be the reflections of
A
A
A
with respect to
D
D
D
and
E
E
E
respectively. Prove that the line
M
N
MN
MN
passes through the orthocentre of the triangle
A
B
C
.
ABC.
A
BC
.
Florin Bojor
P2
3
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Square and equilateral triangles
Let
M
M
M
be the midpoint of the side
A
D
AD
A
D
of the square
A
B
C
D
.
ABCD.
A
BC
D
.
Consider the equilateral triangles
D
F
M
DFM{}
D
FM
and
B
F
E
BFE{}
BFE
such that
F
F
F
lies in the interior of
A
B
C
D
ABCD
A
BC
D
and the lines
E
F
EF
EF
and
B
C
BC
BC
are concurrent. Denote by
P
P{}
P
the midpoint of
M
E
.
ME.
ME
.
Prove that"[*]The point
P
P
P
lies on the line
A
C
.
AC.
A
C
.
[*]The halfline
P
M
PM
PM
is the bisector of the angle
A
P
F
.
APF.
A
PF
.
Adrian Bud
Unusual fractional part
For any positive integer
n
n{}
n
define
a
n
=
{
n
/
s
(
n
)
}
a_n=\{n/s(n)\}
a
n
=
{
n
/
s
(
n
)}
where
s
(
⋅
)
s(\cdot)
s
(
⋅
)
denotes the sum of the digits and
{
⋅
}
\{\cdot\}
{
⋅
}
denotes the fractional part. [*]Prove that there exist infinitely many positive integers
n
n
n
such that
a
n
=
1
/
2.
a_n=1/2.
a
n
=
1/2.
[*]Determine the smallest positive integer
n
n
n
such that
a
n
=
1
/
6.
a_n=1/6.
a
n
=
1/6.
Marius Burtea
Random geo
Let
A
B
C
ABC
A
BC
be a scalene triangle, with circumcircle
ω
\omega
ω
and incentre
I
.
I.{}
I
.
The tangent line at
C
C
C
to
ω
\omega
ω
intersects the line
A
B
AB
A
B
at
D
.
D.{}
D
.
The angle bisector of
B
D
C
BDC
B
D
C
meets
B
I
BI
B
I
at
P
P{}
P
and
A
I
AI{}
A
I
at
Q
.
Q{}.
Q
.
Let
M
M{}
M
be the midpoint of the segment
P
Q
.
PQ.
PQ
.
Prove that the line
I
M
IM
I
M
passes through the middle of the arc
A
C
B
ACB
A
CB
of
ω
.
\omega.
ω
.
Dana Heuberger
P1
3
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Modular NT
Find all the positive integers
a
a{}
a
and
b
b{}
b
such that
(
7
a
−
5
b
)
/
8
(7^a-5^b)/8
(
7
a
−
5
b
)
/8
is a prime number.Cosmin Manea and Dragoș Petrică
Lots of bashing for a 7x7 table
The integers from 1 to 49 are written in a
7
×
7
7\times 7
7
×
7
table, such that for any
k
∈
{
1
,
2
,
…
,
7
}
k\in\{1,2,\ldots,7\}
k
∈
{
1
,
2
,
…
,
7
}
the product of the numbers in the
k
k
k
-th row equals the product of the numbers in the
(
8
−
k
)
(8-k)
(
8
−
k
)
-th row.[*]Prove that there exists a row such that the sum of the numbers written on it is a prime number. [*]Give an example of such a table.Cristi Săvescu
Permutation yields equal ratios
Let
n
⩾
3
n\geqslant 3
n
⩾
3
be an integer and
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
be pairwise distinct positive real numbers with the property that there exists a permutation
b
1
,
b
2
,
…
,
b
n
b_1,b_2,\ldots,b_n
b
1
,
b
2
,
…
,
b
n
of these numbers such that
a
1
b
1
=
a
2
b
2
=
⋯
=
a
n
−
1
b
n
−
1
≠
1.
\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_{n-1}}{b_{n-1}}\neq 1.
b
1
a
1
=
b
2
a
2
=
⋯
=
b
n
−
1
a
n
−
1
=
1.
Prove that there exist
a
,
b
>
0
a,b>0
a
,
b
>
0
such that
{
a
1
,
a
2
,
…
,
a
n
}
=
{
a
b
,
a
b
2
,
…
,
a
b
n
}
.
\{a_1,a_2,\ldots,a_n\}=\{ab,ab^2,\ldots,ab^n\}.
{
a
1
,
a
2
,
…
,
a
n
}
=
{
ab
,
a
b
2
,
…
,
a
b
n
}
.
Cristi Săvescu