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Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
2018 Romania Team Selection Tests
2018 Romania Team Selection Tests
Part of
Romania Team Selection Test
Subcontests
(4)
4
3
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(n+k)^2+1 divides factorial
Given an non-negative integer
k
k
k
, show that there are infinitely many positive integers
n
n
n
such that the product of any
n
n
n
consecutive integers is divisible by
(
n
+
k
)
2
+
1
(n+k)^2+1
(
n
+
k
)
2
+
1
.
Bijection from Z to dZ
Let
D
D
D
be a non-empty subset of positive integers and let
d
d
d
be the greatest common divisor of
D
D
D
, and let
d
Z
=
[
d
n
:
n
∈
Z
]
d\mathbb{Z}=[dn: n \in \mathbb{Z} ]
d
Z
=
[
d
n
:
n
∈
Z
]
. Prove that there exists a bijection
f
:
Z
→
d
Z
f: \mathbb{Z} \rightarrow d\mathbb{Z}
f
:
Z
→
d
Z
such that
∣
f
(
n
+
1
)
−
f
(
n
)
∣
| f(n+1)-f(n)|
∣
f
(
n
+
1
)
−
f
(
n
)
∣
is member of
D
D
D
for every integer
n
n
n
.
v2 of sum of divisors
Given two positives integers
m
m
m
and
n
n
n
, prove that there exists a positive integer
k
k
k
and a set
S
S
S
of at least
m
m
m
multiples of
n
n
n
such that the numbers
2
k
σ
(
s
)
s
\frac {2^k{\sigma({s})}} {s}
s
2
k
σ
(
s
)
are odd for every
s
∈
S
s \in S
s
∈
S
.
σ
(
s
)
\sigma({s})
σ
(
s
)
is the sum of all positive integers of
s
s
s
(1 and
s
s
s
included).
3
4
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2
3
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Concyclic points from mixtilinear circle
Let
A
B
C
ABC
A
BC
be a triangle, let
I
I
I
be its incenter, let
Ω
\Omega
Ω
be its circumcircle, and let
ω
\omega
ω
be the
A
A
A
- mixtilinear incircle. Let
D
,
E
D,E
D
,
E
and
T
T
T
be the intersections of
ω
\omega
ω
and
A
B
,
A
C
AB,AC
A
B
,
A
C
and
Ω
\Omega
Ω
, respectively, let the line
I
T
IT
I
T
cross
ω
\omega
ω
again at
P
P
P
, and let lines
P
D
PD
P
D
and
P
E
PE
PE
cross the line
B
C
BC
BC
at
M
M
M
and
N
N
N
respectively. Prove that points
D
,
E
,
M
,
N
D,E,M,N
D
,
E
,
M
,
N
are concyclic. What is the center of this circle?
2n^2+2n+1 is composite
Show that a number
n
(
n
+
1
)
n(n+1)
n
(
n
+
1
)
where
n
n
n
is positive integer is the sum of 2 numbers
k
(
k
+
1
)
k(k+1)
k
(
k
+
1
)
and
m
(
m
+
1
)
m(m+1)
m
(
m
+
1
)
where
m
m
m
and
k
k
k
are positive integers if and only if the number
2
n
2
+
2
n
+
1
2n^2+2n+1
2
n
2
+
2
n
+
1
is composite.
Sum [kn^(1/3)]
Given a square-free integer
n
>
2
n>2
n
>
2
, evaluate the sum
∑
k
=
1
(
n
−
2
)
(
n
−
1
)
⌊
(
k
n
)
1
/
3
⌋
\sum_{k=1}^{(n-2)(n-1)} \lfloor ({kn})^{1/3} \rfloor
∑
k
=
1
(
n
−
2
)
(
n
−
1
)
⌊(
kn
)
1/3
⌋
.
1
3
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maxima of xi^2
Find the least number
c
c
c
satisfyng the condition
∑
i
=
1
n
x
i
2
≤
c
n
\sum_{i=1}^n {x_i}^2\leq cn
∑
i
=
1
n
x
i
2
≤
c
n
and all real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
are greater than or equal to
−
1
-1
−
1
such that
∑
i
=
1
n
x
i
3
=
0
\sum_{i=1}^n {x_i}^3=0
∑
i
=
1
n
x
i
3
=
0
Izogonal points
Let
A
B
C
ABC
A
BC
be a triangle, and let
M
M
M
be a point on the side
(
A
C
)
(AC)
(
A
C
)
.The line through
M
M
M
and parallel to
B
C
BC
BC
crosses
A
B
AB
A
B
at
N
N
N
. Segments
B
M
BM
BM
and
C
N
CN
CN
cross at
P
P
P
, and the circles
B
N
P
BNP
BNP
and
C
M
P
CMP
CMP
cross again at
Q
Q
Q
. Show that angles
B
A
P
BAP
B
A
P
and
C
A
Q
CAQ
C
A
Q
are equal.
Incenters in cyclic quadrilaterals
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral and let its diagonals
A
C
AC
A
C
and
B
D
BD
B
D
cross at
X
X
X
. Let
I
I
I
be the incenter of
X
B
C
XBC
XBC
, and let
J
J
J
be the center of the circle tangent to the side
B
C
BC
BC
and the extensions of sides
A
B
AB
A
B
and
D
C
DC
D
C
beyond
B
B
B
and
C
C
C
. Prove that the line
I
J
IJ
I
J
bisects the arc
B
C
BC
BC
of circle
A
B
C
D
ABCD
A
BC
D
, not containing the vertices
A
A
A
and
D
D
D
of the quadrilateral.