MathDB
Problems
Contests
International Contests
Romanian Masters of Mathematics Collection
2019 Romanian Master of Mathematics
2019 Romanian Master of Mathematics
Part of
Romanian Masters of Mathematics Collection
Subcontests
(4)
6
1
Hide problems
(not so) small set of residues generates all of F_p upon applying Q many times
Find all pairs of integers
(
c
,
d
)
(c, d)
(
c
,
d
)
, both greater than 1, such that the following holds:For any monic polynomial
Q
Q
Q
of degree
d
d
d
with integer coefficients and for any prime
p
>
c
(
2
c
+
1
)
p > c(2c+1)
p
>
c
(
2
c
+
1
)
, there exists a set
S
S
S
of at most
(
2
c
−
1
2
c
+
1
)
p
\big(\tfrac{2c-1}{2c+1}\big)p
(
2
c
+
1
2
c
−
1
)
p
integers, such that
⋃
s
∈
S
{
s
,
Q
(
s
)
,
Q
(
Q
(
s
)
)
,
Q
(
Q
(
Q
(
s
)
)
)
,
…
}
\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}
s
∈
S
⋃
{
s
,
Q
(
s
)
,
Q
(
Q
(
s
))
,
Q
(
Q
(
Q
(
s
)))
,
…
}
contains a complete residue system modulo
p
p
p
(i.e., intersects with every residue class modulo
p
p
p
).
5
1
Hide problems
Functional equation wrapped in f's
Determine all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
satisfying
f
(
x
+
y
f
(
x
)
)
+
f
(
x
y
)
=
f
(
x
)
+
f
(
2019
y
)
,
f(x + yf(x)) + f(xy) = f(x) + f(2019y),
f
(
x
+
y
f
(
x
))
+
f
(
x
y
)
=
f
(
x
)
+
f
(
2019
y
)
,
for all real numbers
x
x
x
and
y
y
y
.
4
1
Hide problems
RMM P4 - playing around with figures
Prove that for every positive integer
n
n
n
there exists a (not necessarily convex) polygon with no three collinear vertices, which admits exactly
n
n
n
diffferent triangulations.(A triangulation is a dissection of the polygon into triangles by interior diagonals which have no common interior points with each other nor with the sides of the polygon)
2
1
Hide problems
RMM 2019 Problem 2
Let
A
B
C
D
ABCD
A
BC
D
be an isosceles trapezoid with
A
B
∥
C
D
AB\parallel CD
A
B
∥
C
D
. Let
E
E
E
be the midpoint of
A
C
AC
A
C
. Denote by
ω
\omega
ω
and
Ω
\Omega
Ω
the circumcircles of the triangles
A
B
E
ABE
A
BE
and
C
D
E
CDE
C
D
E
, respectively. Let
P
P
P
be the crossing point of the tangent to
ω
\omega
ω
at
A
A
A
with the tangent to
Ω
\Omega
Ω
at
D
D
D
. Prove that
P
E
PE
PE
is tangent to
Ω
\Omega
Ω
.Jakob Jurij Snoj, Slovenia