MathDB
Problems
Contests
International Contests
Romanian Masters of Mathematics Collection
2013 Romanian Master of Mathematics
2013 Romanian Master of Mathematics
Part of
Romanian Masters of Mathematics Collection
Subcontests
(6)
6
1
Hide problems
RMM 2013 Problem 6
A token is placed at each vertex of a regular
2
n
2n
2
n
-gon. A move consists in choosing an edge of the
2
n
2n
2
n
-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.
5
1
Hide problems
RMM 2013 Problem 5
Given a positive integer
k
≥
2
k\geq2
k
≥
2
, set
a
1
=
1
a_1=1
a
1
=
1
and, for every integer
n
≥
2
n\geq 2
n
≥
2
, let
a
n
a_n
a
n
be the smallest solution of equation
x
=
1
+
∑
i
=
1
n
−
1
⌊
x
a
i
k
⌋
x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor
x
=
1
+
i
=
1
∑
n
−
1
⌊
k
a
i
x
⌋
that exceeds
a
n
−
1
a_{n-1}
a
n
−
1
. Prove that all primes are among the terms of the sequence
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
4
1
Hide problems
RMM 2013 Problem 4
Suppose two convex quadrangles in the plane
P
P
P
and
P
′
P'
P
′
, share a point
O
O
O
such that, for every line
l
l
l
trough
O
O
O
, the segment along which
l
l
l
and
P
P
P
meet is longer then the segment along which
l
l
l
and
P
′
P'
P
′
meet. Is it possible that the ratio of the area of
P
′
P'
P
′
to the area of
P
P
P
is greater then
1.9
1.9
1.9
?
3
1
Hide problems
RMM 2013 Problem 3
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribed in a circle
ω
\omega
ω
. The lines
A
B
AB
A
B
and
C
D
CD
C
D
meet at
P
P
P
, the lines
A
D
AD
A
D
and
B
C
BC
BC
meet at
Q
Q
Q
, and the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
meet at
R
R
R
. Let
M
M
M
be the midpoint of the segment
P
Q
PQ
PQ
, and let
K
K
K
be the common point of the segment
M
R
MR
MR
and the circle
ω
\omega
ω
. Prove that the circumcircle of the triangle
K
P
Q
KPQ
K
PQ
and
ω
\omega
ω
are tangent to one another.
2
1
Hide problems
RMM 2013 Problem 2
Does there exist a pair
(
g
,
h
)
(g,h)
(
g
,
h
)
of functions
g
,
h
:
R
→
R
g,h:\mathbb{R}\rightarrow\mathbb{R}
g
,
h
:
R
→
R
such that the only function
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
satisfying
f
(
g
(
x
)
)
=
g
(
f
(
x
)
)
f(g(x))=g(f(x))
f
(
g
(
x
))
=
g
(
f
(
x
))
and
f
(
h
(
x
)
)
=
h
(
f
(
x
)
)
f(h(x))=h(f(x))
f
(
h
(
x
))
=
h
(
f
(
x
))
for all
x
∈
R
x\in\mathbb{R}
x
∈
R
is identity function
f
(
x
)
≡
x
f(x)\equiv x
f
(
x
)
≡
x
?
1
1
Hide problems
RMM 2013 Problem 1
For a positive integer
a
a
a
, define a sequence of integers
x
1
,
x
2
,
…
x_1,x_2,\ldots
x
1
,
x
2
,
…
by letting
x
1
=
a
x_1=a
x
1
=
a
and
x
n
+
1
=
2
x
n
+
1
x_{n+1}=2x_n+1
x
n
+
1
=
2
x
n
+
1
for
n
≥
1
n\geq 1
n
≥
1
. Let
y
n
=
2
x
n
−
1
y_n=2^{x_n}-1
y
n
=
2
x
n
−
1
. Determine the largest possible
k
k
k
such that, for some positive integer
a
a
a
, the numbers
y
1
,
…
,
y
k
y_1,\ldots,y_k
y
1
,
…
,
y
k
are all prime.