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National and Regional Contests
Romania Contests
Romania Team Selection Test
2021 Romania Team Selection Test
2021 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(4)
4
1
Hide problems
Romania TST 2021 Day 1 P4
Determine all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
which satisfy the following relationship for all real numbers
x
x
x
and
y
y
y
f
(
x
f
(
y
)
−
f
(
x
)
)
=
2
f
(
x
)
+
x
y
.
f(xf(y)-f(x))=2f(x)+xy.
f
(
x
f
(
y
)
−
f
(
x
))
=
2
f
(
x
)
+
x
y
.
3
3
Hide problems
Romania TST 2021 Day 1 P3
The external bisectors of the angles of the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect each other in
E
,
F
,
G
E,F,G
E
,
F
,
G
and
H
H
H
such that
A
∈
E
H
,
B
∈
E
F
,
C
∈
F
G
,
D
∈
G
H
A\in EH, \ B\in EF, \ C\in FG, \ D\in GH
A
∈
E
H
,
B
∈
EF
,
C
∈
FG
,
D
∈
G
H
. We know that the perpendiculars from
E
E
E
to
A
B
AB
A
B
, from
F
F
F
to
B
C
BC
BC
and from
G
G
G
to
C
D
CD
C
D
are concurrent. Prove that
A
B
C
D
ABCD
A
BC
D
is cyclic.
Romania TST 2021 Day 2 P3
Let
P
\mathcal{P}
P
be a convex quadrilateral. Consider a point
X
X
X
inside
P
.
\mathcal{P}.
P
.
Let
M
,
N
,
P
,
Q
M,N,P,Q
M
,
N
,
P
,
Q
be the projections of
X
X
X
on the sides of
P
.
\mathcal{P}.
P
.
We know that
M
,
N
,
P
,
Q
M,N,P,Q
M
,
N
,
P
,
Q
all sit on a circle of center
L
.
L.
L
.
Let
J
J
J
and
K
K
K
be the midpoints of the diagonals of
P
.
\mathcal{P}.
P
.
Prove that
J
,
K
J,K
J
,
K
and
L
L
L
lie on a line.
Romania TST 2021 Day 3 P3
Let
α
\alpha
α
be a real number in the interval
(
0
,
1
)
.
(0,1).
(
0
,
1
)
.
Prove that there exists a sequence
(
ε
n
)
n
≥
1
(\varepsilon_n)_{n\geq 1}
(
ε
n
)
n
≥
1
where each term is either
0
0
0
or
1
1
1
such that the sequence
(
s
n
)
n
≥
1
(s_n)_{n\geq 1}
(
s
n
)
n
≥
1
s
n
=
ε
1
n
(
n
+
1
)
+
ε
2
(
n
+
1
)
(
n
+
2
)
+
.
.
.
+
ε
n
(
2
n
−
1
)
2
n
s_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}
s
n
=
n
(
n
+
1
)
ε
1
+
(
n
+
1
)
(
n
+
2
)
ε
2
+
...
+
(
2
n
−
1
)
2
n
ε
n
verifies the inequality
0
≤
α
−
2
n
s
n
≤
2
n
+
1
0\leq \alpha-2ns_n\leq\frac{2}{n+1}
0
≤
α
−
2
n
s
n
≤
n
+
1
2
for any
n
≥
2.
n\geq 2.
n
≥
2.
2
3
Hide problems
Romania TST 2021 Day 1 P2
For any positive integer
n
>
1
n>1
n
>
1
, let
p
(
n
)
p(n)
p
(
n
)
be the greatest prime factor of
n
n
n
. Find all the triplets of distinct positive integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
which satisfy the following properties:
x
,
y
x,y
x
,
y
and
z
z
z
form an arithmetic progression, and
p
(
x
y
z
)
≤
3.
p(xyz)\leq 3.
p
(
x
yz
)
≤
3.
Romania TST 2021 Day 2 P2
Consider the set
M
=
{
1
,
2
,
3
,
.
.
.
,
2020
}
.
M=\{1,2,3,...,2020\}.
M
=
{
1
,
2
,
3
,
...
,
2020
}
.
Find the smallest positive integer
k
k
k
such that for any subset
A
A
A
of
M
M
M
with
k
k
k
elements, there exist
3
3
3
distinct numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
from
M
M
M
such that
a
+
b
,
b
+
c
a+b, b+c
a
+
b
,
b
+
c
and
c
+
a
c+a
c
+
a
are all in
A
.
A.
A
.
Romania TST 2021 Day 3 P2
Let
N
≥
4
N\geq 4
N
≥
4
be a fixed positive integer. Two players,
A
A
A
and
B
B
B
are forming an ordered set
{
x
1
,
x
2
,
.
.
.
}
,
\{x_1,x_2,...\},
{
x
1
,
x
2
,
...
}
,
adding elements alternatively.
A
A
A
chooses
x
1
x_1
x
1
to be
1
1
1
or
−
1
,
-1,
−
1
,
then
B
B
B
chooses
x
2
x_2
x
2
to be
2
2
2
or
−
2
,
-2,
−
2
,
then
A
A
A
chooses
x
3
x_3
x
3
to be
3
3
3
or
−
3
,
-3,
−
3
,
and so on. (at the
k
t
h
k^{th}
k
t
h
step, the chosen number must always be
k
k
k
or
−
k
-k
−
k
)The winner is the first player to make the sequence sum up to a multiple of
N
.
N.
N
.
Depending on
N
,
N,
N
,
find out, with proof, which player has a winning strategy.
1
3
Hide problems
Romania TST 2021 Day 1 P1
Let
k
>
1
k>1
k
>
1
be a positive integer. A set
S
S{}
S
is called good if there exists a colouring of the positive integers with
k
k{}
k
colours, such that no element from
S
S{}
S
can be written as the sum of two distinct positive integers having the same colour. Find the greatest positive integer
t
t{}
t
(in terms of
k
k{}
k
) for which the set
S
=
{
a
+
1
,
a
+
2
,
…
,
a
+
t
}
S=\{a+1,a+2,\ldots,a+t\}
S
=
{
a
+
1
,
a
+
2
,
…
,
a
+
t
}
is good, for any positive integer
a
a{}
a
.
Divisibility
Find all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of positive odd integers, such that
n
∣
3
m
+
1
n \mid 3m+1
n
∣
3
m
+
1
and
m
∣
n
2
+
3
m \mid n^2+3
m
∣
n
2
+
3
.
Romania TST 2021 Day 3 P1
Consider a fixed triangle
A
B
C
ABC
A
BC
such that
A
B
=
A
C
.
AB=AC.
A
B
=
A
C
.
Let
M
M
M
be the midpoint of
B
C
.
BC.
BC
.
Let
P
P
P
be a variable point inside
△
A
B
C
,
\triangle ABC,
△
A
BC
,
such that
∠
P
B
C
=
∠
P
C
A
.
\angle PBC=\angle PCA.
∠
PBC
=
∠
PC
A
.
Prove that the sum of the measures of
∠
B
P
M
\angle BPM
∠
BPM
and
∠
A
P
C
\angle APC
∠
A
PC
is constant.