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National and Regional Contests
Romania Contests
JBMO TST - Romania
2021 Junior Balkan Team Selection Tests - Romania
2021 Junior Balkan Team Selection Tests - Romania
Part of
JBMO TST - Romania
Subcontests
(5)
P5
1
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Romania Junior TST 2021 Day 1 P5
Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
. The circle of centre
A
A
A
and radius
A
I
AI
A
I
intersects the circumcircle of triangle
A
B
C
ABC
A
BC
in
M
M
M
and
N
N
N
. Prove that the line
M
N
MN
MN
is tangent to the incircle of triangle
A
B
C
ABC
A
BC
P4
3
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Romania Junior TST 2021 Day 1 P4
Let
n
≥
2
n\geq 2
n
≥
2
be a positive integer. On an
n
×
n
n\times n
n
×
n
board,
n
n
n
rooks are placed in such a manner that no two attack each other. All rooks move at the same time and are only allowed to move in a square adjacent to the one in which they are located. Determine all the values of
n
n
n
for which there is a placement of the rooks so that, after a move, the rooks still do not attack each other.Note: Two squares are adjacent if they share a common side.
Romania Junior TST 2021 Day 2 P4
Let
n
≥
2
n\geq 2
n
≥
2
be a positive integer. Prove that there exists a positive integer
m
m
m
, such that
n
∣
m
,
m
<
n
4
n\mid m, \ m<n^4
n
∣
m
,
m
<
n
4
and at most four distinct digits are used in the decimal representation of
m
m
m
.
Romania Junior TST 2021 Day 3 P4
Let
M
M
M
be a set of
13
13
13
positive integers with the property that
∀
m
∈
M
,
100
≤
m
≤
999
\forall \ m\in M, \ 100\leq m\leq 999
∀
m
∈
M
,
100
≤
m
≤
999
. Prove that there exists a subset
S
⊂
M
S\subset M
S
⊂
M
and a combination of arithmetic operations (addition, subtraction, multiplication, division – without using parentheses) between the elements of
S
S
S
, such that the value of the resulting expression is a rational number in the interval
(
3
,
4
)
(3,4)
(
3
,
4
)
.
P3
3
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Romania Junior TST 2021 Day 1 P3
The incircle of triangle
A
B
C
ABC
A
BC
is tangent to the sides
A
B
,
A
C
AB,AC
A
B
,
A
C
and
B
C
BC
BC
at the points
M
,
N
M,N
M
,
N
and
K
K
K
respectively. The median
A
D
AD
A
D
of the triangle
A
B
C
ABC
A
BC
intersects
M
N
MN
MN
at the point
L
L
L
. Prove that
K
,
I
K,I
K
,
I
and
L
L
L
are collinear, where
I
I
I
is the incenter of the triangle
A
B
C
ABC
A
BC
.
Romania Junior TST 2021 Day 3 P3
Let
p
,
q
p,q
p
,
q
be positive integers. For any
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
define the sets
P
(
a
)
=
{
a
n
=
a
+
n
⋅
1
p
:
n
∈
N
}
and
Q
(
b
)
=
{
b
n
=
b
+
n
⋅
1
q
:
n
∈
N
}
.
P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.
P
(
a
)
=
{
a
n
=
a
+
n
⋅
p
1
:
n
∈
N
}
and
Q
(
b
)
=
{
b
n
=
b
+
n
⋅
q
1
:
n
∈
N
}
.
The distance between
P
(
a
)
P(a)
P
(
a
)
and
Q
(
b
)
Q(b)
Q
(
b
)
is the minimum value of
∣
x
−
y
∣
|x-y|
∣
x
−
y
∣
as
x
∈
P
(
a
)
,
y
∈
Q
(
b
)
x\in P(a), y\in Q(b)
x
∈
P
(
a
)
,
y
∈
Q
(
b
)
. Find the maximum value of the distance between
P
(
a
)
P(a)
P
(
a
)
and
Q
(
b
)
Q(b)
Q
(
b
)
as
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
.
Romania Junior TST 2021 Day 2 P3
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with angles
∢
A
,
∢
C
≥
9
0
∘
\sphericalangle A, \sphericalangle C\geq90^{\circ}
∢
A
,
∢
C
≥
9
0
∘
. On sides
A
B
,
B
C
,
C
D
AB,BC,CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
, consider the points
K
,
L
,
M
K,L,M
K
,
L
,
M
and
N
N
N
respectively. Prove that the perimeter of
K
L
M
N
KLMN
K
L
MN
is greater than or equal to
2
⋅
A
C
2\cdot AC
2
⋅
A
C
.
P2
3
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Romania Junior TST 2021 Day 1 P2
Find all the pairs of positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
x
≤
y
x\leq y
x
≤
y
and
(
x
+
y
)
(
x
y
−
1
)
x
y
+
1
=
p
,
\frac{(x+y)(xy-1)}{xy+1}=p,
x
y
+
1
(
x
+
y
)
(
x
y
−
1
)
=
p
,
where
p
p
p
is a prime number.
Romania Junior TST 2021 Day 2 P2
For any non-empty subset
X
X
X
of
M
=
{
1
,
2
,
3
,
.
.
.
,
2021
}
M=\{1,2,3,...,2021\}
M
=
{
1
,
2
,
3
,
...
,
2021
}
, let
a
X
a_X
a
X
be the sum of the greatest and smallest elements of
X
X
X
. Determine the arithmetic mean of all the values of
a
X
a_X
a
X
, as
X
X
X
covers all the non-empty subsets of
M
M
M
.
Romania Junior TST 2021 Day 3 P2
Let
O
O
O
be the circumcenter of triangle
A
B
C
ABC
A
BC
and let
A
D
AD
A
D
be the height from
A
A
A
(
D
∈
B
C
D\in BC
D
∈
BC
). Let
M
,
N
,
P
M,N,P
M
,
N
,
P
and
Q
Q
Q
be the midpoints of
A
B
,
A
C
,
B
D
AB,AC,BD
A
B
,
A
C
,
B
D
and
C
D
CD
C
D
respectively. Let
C
1
\mathcal{C}_1
C
1
and
C
2
\mathcal{C}_2
C
2
be the circumcircles of triangles
A
M
N
AMN
A
MN
and
P
O
Q
POQ
POQ
. Prove that
C
1
∩
C
2
∩
A
D
≠
∅
\mathcal{C}_1\cap \mathcal{C}_2\cap AD\neq \emptyset
C
1
∩
C
2
∩
A
D
=
∅
.
P1
3
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Romania Junior TST 2021 Day 1 P1
Let
n
≥
2
n\geq 2
n
≥
2
be a positive integer and let
a
1
,
a
2
,
.
.
.
,
a
n
∈
[
0
,
1
]
a_1,a_2,...,a_n\in[0,1]
a
1
,
a
2
,
...
,
a
n
∈
[
0
,
1
]
be real numbers. Find the maximum value of the smallest of the numbers:
a
1
−
a
1
a
2
,
a
2
−
a
2
a
3
,
.
.
.
,
a
n
−
a
n
a
1
.
a_1-a_1a_2, \ a_2-a_2a_3,...,a_n-a_na_1.
a
1
−
a
1
a
2
,
a
2
−
a
2
a
3
,
...
,
a
n
−
a
n
a
1
.
Romania Junior TST 2021 Day 2 P1
On a board, Ana and Bob start writing
0
0
0
s and
1
1
1
s alternatively until each of them has written
2021
2021
2021
digits. Ana starts this procedure and each of them always adds a digit to the right of the already existing ones.Ana wins the game if, after they stop writing, the resulting number (in binary) can be written as the sum of two squares. Otherwise, Bob wins. Determine who has a winning strategy.
Romania Junior TST 2021 Day 3 P1
Let
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
be real numbers with the property that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that
1
a
+
b
c
+
1
b
+
c
a
+
1
c
+
a
b
≥
7
1
+
a
b
c
.
\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq\frac{7}{1+abc}.
a
+
b
c
1
+
b
+
c
a
1
+
c
+
ab
1
≥
1
+
ab
c
7
.