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National and Regional Contests
Romania Contests
Romania Team Selection Test
2022 Romania Team Selection Test
2022 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(5)
4
1
Hide problems
Romania TST 2022 Day 2 P4
Any positive integer
N
N
N
which can be expressed as the sum of three squares can obviously be written as
N
=
a
2
+
b
2
+
c
2
+
d
2
1
+
a
b
c
d
N=\frac{a^2+b^2+c^2+d^2}{1+abcd}
N
=
1
+
ab
c
d
a
2
+
b
2
+
c
2
+
d
2
where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are nonnegative integers. Is the mutual assertion true?
3
3
Hide problems
Romania TST 2022 Day 2 P3
Let
n
≥
2
n\geq 2
n
≥
2
be an integer. Let
a
i
j
,
i
,
j
=
1
,
2
,
…
,
n
a_{ij}, \ i,j=1,2,\ldots,n
a
ij
,
i
,
j
=
1
,
2
,
…
,
n
be
n
2
n^2
n
2
positive real numbers satisfying the following conditions:[*]For all
i
=
1
,
…
,
n
i=1,\ldots,n
i
=
1
,
…
,
n
we have
a
i
i
=
1
a_{ii}=1
a
ii
=
1
and, [*]For all
j
=
2
,
…
,
n
j=2,\ldots,n
j
=
2
,
…
,
n
the numbers
a
i
j
,
i
=
1
,
…
,
j
−
1
a_{ij}, \ i=1,\ldots, j-1
a
ij
,
i
=
1
,
…
,
j
−
1
form a permutation of
1
/
a
j
i
,
i
=
1
,
…
,
j
−
1.
1/a_{ji}, \ i=1,\ldots, j-1.
1/
a
ji
,
i
=
1
,
…
,
j
−
1.
Given that
S
i
=
a
i
1
+
⋯
+
a
i
n
S_i=a_{i1}+\cdots+a_{in}
S
i
=
a
i
1
+
⋯
+
a
in
, determine the maximum value of the sum
1
/
S
1
+
⋯
+
1
/
S
n
.
1/S_1+\cdots+1/S_n.
1/
S
1
+
⋯
+
1/
S
n
.
Congruence NT
Consider a prime number
p
⩾
11
p\geqslant 11
p
⩾
11
. We call a triple
a
,
b
,
c
a,b,c
a
,
b
,
c
of natural numbers suitable if they give non-zero, pairwise distinct residues modulo
p
p{}
p
. Further, for any natural numbers
a
,
b
,
c
,
k
a,b,c,k
a
,
b
,
c
,
k
we define
f
k
(
a
,
b
,
c
)
=
a
(
b
−
c
)
p
−
k
+
b
(
c
−
a
)
p
−
k
+
c
(
a
−
b
)
p
−
k
.
f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.
f
k
(
a
,
b
,
c
)
=
a
(
b
−
c
)
p
−
k
+
b
(
c
−
a
)
p
−
k
+
c
(
a
−
b
)
p
−
k
.
Prove that there exist suitable
a
,
b
,
c
a,b,c
a
,
b
,
c
for which
p
∣
f
2
(
a
,
b
,
c
)
p\mid f_2(a,b,c)
p
∣
f
2
(
a
,
b
,
c
)
. Furthermore, for each such triple, prove that there exists
k
⩾
3
k\geqslant 3
k
⩾
3
for which
p
∤
f
k
(
a
,
b
,
c
)
p\nmid f_k(a,b,c)
p
∤
f
k
(
a
,
b
,
c
)
and determine the minimal
k
k{}
k
with this property.Călin Popescu and Marian Andronache
Romania TST 2022 Day 4 P3
Let
A
B
C
ABC
A
BC
be a triangle and let its incircle
γ
\gamma
γ
touch the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Let
P
P
P
be a point strictly in the interior of
γ
.
\gamma.
γ
.
The segments
P
A
,
P
B
,
P
C
PA,PB,PC
P
A
,
PB
,
PC
cross
γ
\gamma
γ
at
A
0
,
B
0
,
C
0
A_0,B_0,C_0
A
0
,
B
0
,
C
0
respectively. Let
S
A
,
S
B
,
S
C
S_A,S_B,S_C
S
A
,
S
B
,
S
C
be the centres of the circles
P
E
F
,
P
F
D
,
P
D
E
PEF,PFD,PDE
PEF
,
PF
D
,
P
D
E
respectively and let
T
A
,
T
B
,
T
C
T_A,T_B,T_C
T
A
,
T
B
,
T
C
be the centres of the circles
P
B
0
C
0
,
P
C
0
A
0
,
P
A
0
B
0
PB_0C_0,PC_0A_0,PA_0B_0
P
B
0
C
0
,
P
C
0
A
0
,
P
A
0
B
0
respectively. Prove that
S
A
T
A
,
S
B
T
B
S_AT_A, S_BT_B
S
A
T
A
,
S
B
T
B
and
S
C
T
C
S_CT_C
S
C
T
C
are concurrent.
2
2
Hide problems
Romania TST 2022 Day 2 P2
Let
A
B
C
ABC
A
BC
be an acute triangle and let
B
′
B'
B
′
and
C
′
C'
C
′
be the feet of the heights
B
B
B
and
C
C
C
of triangle
A
B
C
ABC
A
BC
respectively. Let
B
A
′
B_A'
B
A
′
and
B
C
′
B_C'
B
C
′
be reflections of
B
′
B'
B
′
with respect to the lines
B
C
BC
BC
and
A
B
AB
A
B
, respectively. The circle
B
B
A
′
B
C
′
BB_A'B_C'
B
B
A
′
B
C
′
, centered in
O
B
O_B
O
B
, intersects the line
A
B
AB
A
B
in
X
B
X_B
X
B
for the second time. The points
C
A
′
,
C
B
′
,
O
C
,
X
C
C_A', C_B', O_C, X_C
C
A
′
,
C
B
′
,
O
C
,
X
C
are defined analogously, by replacing the pair
(
B
,
B
′
)
(B, B')
(
B
,
B
′
)
with the pair
(
C
,
C
′
)
(C, C')
(
C
,
C
′
)
. Show that
O
B
X
B
O_BX_B
O
B
X
B
and
O
C
X
C
O_CX_C
O
C
X
C
are parallel.
Romania TST 2022 Day 4 P2
Fix a nonnegative integer
a
0
a_0
a
0
to define a sequence of integers
a
0
,
a
1
,
…
a_0,a_1,\ldots
a
0
,
a
1
,
…
by letting
a
k
,
k
≥
1
a_k,k\geq 1
a
k
,
k
≥
1
be the smallest integer (strictly) greater than
a
k
−
1
a_{k-1}
a
k
−
1
making
a
k
−
1
+
a
k
a_{k-1}+a_k{}
a
k
−
1
+
a
k
into a perfect square. Let
S
S{}
S
be the set of positive integers not expressible as the difference of two terms of the sequence
(
a
k
)
k
≥
0
.
(a_k)_{k\geq 0}.
(
a
k
)
k
≥
0
.
Prove that
S
S
S
is finite and determine its size in terms of
a
0
.
a_0.
a
0
.
5
2
Hide problems
All distances between points are >1 or \leq 2
Given is an integer
k
≥
2
k\geq 2
k
≥
2
. Determine the smallest positive integer
n
n
n
, such that, among any
n
n
n
points in the plane, there exist
k
k
k
points among them, such that all distances between them are less than or equal to
2
2
2
, or all distances are strictly greater than
1
1
1
.
Romania TST 2022 Day 2 P5
Let
m
,
n
≥
2
m,n\geq 2
m
,
n
≥
2
be positive integers and
S
⊆
[
1
,
m
]
×
[
1
,
n
]
S\subseteq [1,m]\times [1,n]
S
⊆
[
1
,
m
]
×
[
1
,
n
]
be a set of lattice points. Prove that if
∣
S
∣
≥
m
+
n
+
⌊
m
+
n
4
−
1
2
⌋
|S|\geq m+n+\bigg\lfloor\frac{m+n}{4}-\frac{1}{2}\bigg\rfloor
∣
S
∣
≥
m
+
n
+
⌊
4
m
+
n
−
2
1
⌋
then there exists a circle which passes through at least four distinct points of
S
.
S.
S
.
1
3
Hide problems
n-variable inequality with a weird condition
Given are positive reals
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,..., x_n
x
1
,
x
2
,
...
,
x
n
such that
∑
1
1
+
x
i
2
=
1
\sum\frac {1}{1+x_i^2}=1
∑
1
+
x
i
2
1
=
1
. Find the minimal value of the expression
∑
x
i
∑
1
x
i
\frac{\sum x_i}{\sum \frac{1}{x_i}}
∑
x
i
1
∑
x
i
and find when it is achieved.
Romania TST 2022 Day 2 P1
Let
A
B
C
ABC
A
BC
be an acute scalene triangle and let
ω
\omega
ω
be its Euler circle. The tangent
t
A
t_A
t
A
of
ω
\omega
ω
at the foot of the height
A
A
A
of the triangle ABC, intersects the circle of diameter
A
B
AB
A
B
at the point
K
A
K_A
K
A
for the second time. The line determined by the feet of the heights
A
A
A
and
C
C
C
of the triangle
A
B
C
ABC
A
BC
intersects the lines
A
K
A
AK_A
A
K
A
and
B
K
A
BK_A
B
K
A
at the points
L
A
L_A
L
A
and
M
A
M_A
M
A
, respectively, and the lines
t
A
t_A
t
A
and
C
M
A
CM_A
C
M
A
intersect at the point
N
A
N_A
N
A
.Points
K
B
,
L
B
,
M
B
,
N
B
K_B, L_B, M_B, N_B
K
B
,
L
B
,
M
B
,
N
B
and
K
C
,
L
C
,
M
C
,
N
C
K_C, L_C, M_C, N_C
K
C
,
L
C
,
M
C
,
N
C
are defined similarly for
(
B
,
C
,
A
)
(B, C, A)
(
B
,
C
,
A
)
and
(
C
,
A
,
B
)
(C, A, B)
(
C
,
A
,
B
)
respectively. Show that the lines
L
A
N
A
,
L
B
N
B
,
L_AN_A, L_BN_B,
L
A
N
A
,
L
B
N
B
,
and
L
C
N
C
L_CN_C
L
C
N
C
are concurrent.
Romania TST 2022 Day 4 P1
A finite set
L
\mathcal{L}
L
of coplanar lines, no three of which are concurrent, is called odd if, for every line
ℓ
\ell
ℓ
in
L
\mathcal{L}
L
the total number of lines in
L
\mathcal{L}
L
crossed by
ℓ
\ell
ℓ
is odd.[*]Prove that every finite set of coplanar lines, no three of which are concurrent, extends to an odd set of coplanar lines. [*]Given a positive integer
n
n
n
determine the smallest nonnegative integer
k
k
k
satisfying the following condition: Every set of
n
n
n
coplanar lines, no three of which are concurrent, extends to an odd set of
n
+
k
n+k
n
+
k
coplanar lines.