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National and Regional Contests
Romania Contests
Romania Team Selection Test
2019 Romania Team Selection Test
2019 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(4)
4
2
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Divisibility mingled with combinatorics
Let be two natural numbers
m
,
n
,
m,n,
m
,
n
,
and
m
m
m
pairwise disjoint sets of natural numbers
A
0
,
A
1
,
…
,
A
m
−
1
,
A_0,A_1,\ldots ,A_{m-1},
A
0
,
A
1
,
…
,
A
m
−
1
,
each having
n
n
n
elements, such that no element of
A
i
(
m
o
d
m
)
A_{i\pmod m}
A
i
(
mod
m
)
is divisible by an element of
A
i
+
1
(
m
o
d
m
)
,
A_{i+1\pmod m} ,
A
i
+
1
(
mod
m
)
,
for any natural number
i
.
i.
i
.
Determine the number of ordered pairs
(
a
,
b
)
∈
⋃
0
≤
j
<
m
A
j
×
⋃
0
≤
j
<
m
A
j
(a,b)\in\bigcup_{0\le j < m} A_j\times\bigcup_{0\le j < m} A_j
(
a
,
b
)
∈
0
≤
j
<
m
⋃
A
j
×
0
≤
j
<
m
⋃
A
j
such that
a
∣
b
a|b
a
∣
b
and such that
{
a
,
b
}
∉
A
k
,
\{ a,b \}\not\in A_k,
{
a
,
b
}
∈
A
k
,
for any
k
∈
{
0
,
1
,
…
,
m
−
1
}
.
k\in\{ 0,1,\ldots ,m-1 \} .
k
∈
{
0
,
1
,
…
,
m
−
1
}
.
Radu Bumbăcea
Strings of binary numbers (pure math solution required, not informatical ones)
For a natural number
n
,
n,
n
,
a string
s
s
s
of
n
n
n
binary digits and a natural number
k
≤
n
,
k\le n,
k
≤
n
,
define an
n
,
s
,
k
n,s,k
n
,
s
,
k
-block as a string of
k
k
k
consecutive elements from
s
.
s.
s
.
We say that two
n
,
s
,
k
-blocks
,
n,s,k\text{-blocks} ,
n
,
s
,
k
-blocks
,
namely,
a
1
a
2
…
a
k
,
b
1
b
2
…
b
k
,
a_1a_2\ldots a_k,b_1b_2\ldots b_k,
a
1
a
2
…
a
k
,
b
1
b
2
…
b
k
,
are incompatible if there exists an
i
∈
{
1
,
2
,
…
,
k
}
i\in\{1,2,\ldots ,k\}
i
∈
{
1
,
2
,
…
,
k
}
such that
a
i
≠
b
i
.
a_i\neq b_i.
a
i
=
b
i
.
Also, for two natural numbers
r
≤
n
,
l
,
r\le n, l,
r
≤
n
,
l
,
we say that
s
s
s
is
r
,
l
r,l
r
,
l
-typed if there are, at most,
l
l
l
pairwise incompatible
n
,
s
,
r
-blocks
.
n,s,r\text{-blocks} .
n
,
s
,
r
-blocks
.
Let be a
3
,
7
-typed
3,7\text{-typed}
3
,
7
-typed
string
t
t
t
consisting of
10000
10000
10000
binary digits. Determine the maximum number
M
M
M
that satisfies the condition that
t
t
t
is
10
,
M
-typed
.
10,M\text{-typed} .
10
,
M
-typed
.
Cătălin Gherghe
3
2
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Number of fixed points of certain condition functions
Let be three positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
and a function
f
:
N
⟶
N
f:\mathbb{N}\longrightarrow\mathbb{N}
f
:
N
⟶
N
defined as
f
(
n
)
=
{
n
−
a
,
n
>
c
f
(
f
(
n
+
b
)
)
,
n
≤
c
.
f(n)=\left\{ \begin{matrix} n-a, & n>c\\ f\left( f(n+b) \right) ,& n\le c \end{matrix} \right. .
f
(
n
)
=
{
n
−
a
,
f
(
f
(
n
+
b
)
)
,
n
>
c
n
≤
c
.
Determine the number of fixed points this function has.
Special Rectangular Grids
Given an integer
n
≥
2
,
n\geq 2,
n
≥
2
,
colour red exactly
n
n
n
cells of an infinite sheet of grid paper. A rectangular grid array is called special if it contains at least two red opposite corner cells; single red cells and 1-row or 1-column grid arrays whose end-cells are both red are special. Given a configuration of exactly
n
n
n
red cells, let
N
N
N
be the largest number of red cells a special rectangular grid array may contain. Determine the least value
N
N
N
may take over all possible configurations of exactly
n
n
n
red cells
2
1
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An acute triangle
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
B
C
AB<BC
A
B
<
BC
. Let
I
I
I
be the incenter of
A
B
C
ABC
A
BC
, and let
ω
\omega
ω
be the circumcircle of
A
B
C
ABC
A
BC
. The incircle of
A
B
C
ABC
A
BC
is tangent to the side
B
C
BC
BC
at
K
K
K
. The line
A
K
AK
A
K
meets
ω
\omega
ω
again at
T
T
T
. Let
M
M
M
be the midpoint of the side
B
C
BC
BC
, and let
N
N
N
be the midpoint of the arc
B
A
C
BAC
B
A
C
of
ω
\omega
ω
. The segment
N
T
NT
NT
intersects the circumcircle of
B
I
C
BIC
B
I
C
at
P
P
P
. Prove that
P
M
∥
A
K
PM\parallel AK
PM
∥
A
K
.
1
5
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