MathDB
Problems
Contests
International Contests
Romanian Masters of Mathematics Collection
2016 Romanian Master of Mathematics
2016 Romanian Master of Mathematics
Part of
Romanian Masters of Mathematics Collection
Subcontests
(6)
6
1
Hide problems
as if 2D combo geo wasn't hard enough already
A set of
n
n
n
points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets
A
\mathcal{A}
A
and
B
\mathcal{B}
B
. An
A
B
\mathcal{AB}
A
B
-tree is a configuration of
n
−
1
n-1
n
−
1
segments, each of which has an endpoint in
A
\mathcal{A}
A
and an endpoint in
B
\mathcal{B}
B
, and such that no segments form a closed polyline. An
A
B
\mathcal{AB}
A
B
-tree is transformed into another as follows: choose three distinct segments
A
1
B
1
A_1B_1
A
1
B
1
,
B
1
A
2
B_1A_2
B
1
A
2
, and
A
2
B
2
A_2B_2
A
2
B
2
in the
A
B
\mathcal{AB}
A
B
-tree such that
A
1
A_1
A
1
is in
A
\mathcal{A}
A
and
∣
A
1
B
1
∣
+
∣
A
2
B
2
∣
>
∣
A
1
B
2
∣
+
∣
A
2
B
1
∣
|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|
∣
A
1
B
1
∣
+
∣
A
2
B
2
∣
>
∣
A
1
B
2
∣
+
∣
A
2
B
1
∣
, and remove the segment
A
1
B
1
A_1B_1
A
1
B
1
to replace it by the segment
A
1
B
2
A_1B_2
A
1
B
2
. Given any
A
B
\mathcal{AB}
A
B
-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.
5
1
Hide problems
Circumferences tangents
A convex hexagon
A
1
B
1
A
2
B
2
A
3
B
3
A_1B_1A_2B_2A_3B_3
A
1
B
1
A
2
B
2
A
3
B
3
it is inscribed in a circumference
Ω
\Omega
Ω
with radius
R
R
R
. The diagonals
A
1
B
2
A_1B_2
A
1
B
2
,
A
2
B
3
A_2B_3
A
2
B
3
,
A
3
B
1
A_3B_1
A
3
B
1
are concurrent in
X
X
X
. For each
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
let
ω
i
\omega_i
ω
i
tangent to the segments
X
A
i
XA_i
X
A
i
and
X
B
i
XB_i
X
B
i
and tangent to the arc
A
i
B
i
A_iB_i
A
i
B
i
of
Ω
\Omega
Ω
that does not contain the other vertices of the hexagon; let
r
i
r_i
r
i
the radius of
ω
i
\omega_i
ω
i
.
(
a
)
(a)
(
a
)
Prove that
R
≥
r
1
+
r
2
+
r
3
R\geq r_1+r_2+r_3
R
≥
r
1
+
r
2
+
r
3
(
b
)
(b)
(
b
)
If
R
=
r
1
+
r
2
+
r
3
R= r_1+r_2+r_3
R
=
r
1
+
r
2
+
r
3
, prove that the six points of tangency of the circumferences
ω
i
\omega_i
ω
i
with the diagonals
A
1
B
2
A_1B_2
A
1
B
2
,
A
2
B
3
A_2B_3
A
2
B
3
,
A
3
B
1
A_3B_1
A
3
B
1
are concyclic
4
1
Hide problems
Nice inequality
Let
x
x
x
and
y
y
y
be positive real numbers such that:
x
+
y
2016
≥
1
x+y^{2016}\geq 1
x
+
y
2016
≥
1
. Prove that
x
2016
+
y
>
1
−
1
100
x^{2016}+y> 1-\frac{1}{100}
x
2016
+
y
>
1
−
100
1
3
1
Hide problems
Cubic sequence
A
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
u
b
i
c
s
e
q
u
e
n
c
e
<
/
s
p
a
n
>
<span class='latex-italic'>cubic sequence</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
u
bi
cse
q
u
e
n
ce
<
/
s
p
an
>
is a sequence of integers given by
a
n
=
n
3
+
b
n
2
+
c
n
+
d
a_n =n^3 + bn^2 + cn + d
a
n
=
n
3
+
b
n
2
+
c
n
+
d
, where
b
,
c
b, c
b
,
c
and
d
d
d
are integer constants and
n
n
n
ranges over all integers, including negative integers.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
a
)
<
/
s
p
a
n
>
<span class='latex-bold'>(a)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
a
)
<
/
s
p
an
>
Show that there exists a cubic sequence such that the only terms of the sequence which are squares of integers are
a
2015
a_{2015}
a
2015
and
a
2016
a_{2016}
a
2016
.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
b
)
<
/
s
p
a
n
>
<span class='latex-bold'>(b)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
b
)
<
/
s
p
an
>
Determine the possible values of
a
2015
⋅
a
2016
a_{2015} \cdot a_{2016}
a
2015
⋅
a
2016
for a cubic sequence satisfying the condition in part
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
a
)
<
/
s
p
a
n
>
<span class='latex-bold'>(a)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
a
)
<
/
s
p
an
>
.
2
1
Hide problems
Dominoes placed on a rectangular board
Given positive integers
m
m
m
and
n
≥
m
n \ge m
n
≥
m
, determine the largest number of dominoes (
1
×
2
1\times2
1
×
2
or
2
×
1
2 \times 1
2
×
1
rectangles) that can be placed on a rectangular board with
m
m
m
rows and
2
n
2n
2
n
columns consisting of cells (
1
×
1
1 \times 1
1
×
1
squares) so that: (i) each domino covers exactly two adjacent cells of the board; (ii) no two dominoes overlap; (iii) no two form a
2
×
2
2 \times 2
2
×
2
square; and (iv) the bottom row of the board is completely covered by
n
n
n
dominoes.
1
1
Hide problems
Concurrent lines(or parallel)
Let
A
B
C
ABC
A
BC
be a triangle and let
D
D
D
be a point on the segment
B
C
,
D
≠
B
BC, D\neq B
BC
,
D
=
B
and
D
≠
C
D\neq C
D
=
C
. The circle
A
B
D
ABD
A
B
D
meets the segment
A
C
AC
A
C
again at an interior point
E
E
E
. The circle
A
C
D
ACD
A
C
D
meets the segment
A
B
AB
A
B
again at an interior point
F
F
F
. Let
A
′
A'
A
′
be the reflection of
A
A
A
in the line
B
C
BC
BC
. The lines
A
′
C
A'C
A
′
C
and
D
E
DE
D
E
meet at
P
P
P
, and the lines
A
′
B
A'B
A
′
B
and
D
F
DF
D
F
meet at
Q
Q
Q
. Prove that the lines
A
D
,
B
P
AD, BP
A
D
,
BP
and
C
Q
CQ
CQ
are concurrent (or all parallel).