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Contests
International Contests
Romanian Masters of Mathematics Collection
2023 Romanian Master of Mathematics Shortlist
2023 Romanian Master of Mathematics Shortlist
Part of
Romanian Masters of Mathematics Collection
Subcontests
(9)
N2
1
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Integer polynomial and decimal digits
For every non-negative integer
k
k
k
let
S
(
k
)
S(k)
S
(
k
)
denote the sum of decimal digits of
k
k
k
. Let
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
be polynomials with non-negative integer coecients such that
S
(
P
(
n
)
)
=
S
(
Q
(
n
)
)
S(P(n)) = S(Q(n))
S
(
P
(
n
))
=
S
(
Q
(
n
))
for all non-negative integers
n
n
n
. Prove that there exists an integer
t
t
t
such that
P
(
x
)
−
1
0
t
Q
(
x
)
P(x) - 10^tQ(x)
P
(
x
)
−
1
0
t
Q
(
x
)
is a constant polynomial.
N1
1
Hide problems
Number theory on Gaussian integers at RMM almost
Let
n
n
n
be a positive integer. Let
S
S
S
be a set of ordered pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
1
≤
x
≤
n
1\leq x \leq n
1
≤
x
≤
n
and
0
≤
y
≤
n
0 \leq y \leq n
0
≤
y
≤
n
in each pair, and there are no pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
and
(
c
,
d
)
(c, d)
(
c
,
d
)
of different elements in
S
S
S
such that
a
2
+
b
2
a^2+b^2
a
2
+
b
2
divides both
a
c
+
b
d
ac+bd
a
c
+
b
d
and
a
d
−
b
c
ad - bc
a
d
−
b
c
. In terms of
n
n
n
, determine the size of the largest possible set
S
S
S
.
G3
1
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Very hard isogonality
A point
P
P
P
is chosen inside a triangle
A
B
C
ABC
A
BC
with circumcircle
Ω
\Omega
Ω
. Let
Γ
\Gamma
Γ
be the circle passing through the circumcenters of the triangles
A
P
B
APB
A
PB
,
B
P
C
BPC
BPC
, and
C
P
A
CPA
CP
A
. Let
Ω
\Omega
Ω
and
Γ
\Gamma
Γ
intersect at points
X
X
X
and
Y
Y
Y
. Let
Q
Q
Q
be the reflection of
P
P
P
in the line
X
Y
XY
X
Y
. Prove that
∠
B
A
P
=
∠
C
A
Q
\angle BAP = \angle CAQ
∠
B
A
P
=
∠
C
A
Q
.
G2
1
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Geo the great
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. Let
D
A
DA
D
A
and
B
C
BC
BC
intersect at
E
E
E
and let
A
B
AB
A
B
and
C
D
CD
C
D
intersect at
F
F
F
. Assume that
A
,
E
,
F
A, E, F
A
,
E
,
F
all lie on the same side of
B
D
BD
B
D
. Let
P
P
P
be on segment
D
A
DA
D
A
such that
∠
C
P
D
=
∠
C
B
P
\angle CPD = \angle CBP
∠
CP
D
=
∠
CBP
, and let
Q
Q
Q
be on segment
C
D
CD
C
D
such that
∠
D
Q
A
=
∠
Q
B
A
\angle DQA = \angle QBA
∠
D
Q
A
=
∠
QB
A
. Let
A
C
AC
A
C
and
P
Q
PQ
PQ
meet at
X
X
X
. Prove that, if
E
X
=
E
P
EX = EP
EX
=
EP
, then
E
F
EF
EF
is perpendicular to
A
C
AC
A
C
.
G1
1
Hide problems
Nice trig + angle chase problem for lecturing and practice
Let
A
B
C
ABC
A
BC
be a triangle with incentre
I
I
I
and circumcircle
ω
\omega
ω
. The incircle of the triangle
A
B
C
ABC
A
BC
touches the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
at
D
D
D
,
E
E
E
and
F
F
F
, respectively. The circumcircle of triangle
A
D
I
ADI
A
D
I
crosses
ω
\omega
ω
again at
P
P
P
, and the lines
P
E
PE
PE
and
P
F
PF
PF
cross
ω
\omega
ω
again at
X
X
X
and
Y
Y
Y
, respectively. Prove that the lines
A
I
AI
A
I
,
B
X
BX
BX
and
C
Y
CY
C
Y
are concurrent.
C2
1
Hide problems
Great Grid Combo
For positive integers
m
,
n
≥
2
m,n \geq 2
m
,
n
≥
2
, let
S
m
,
n
=
{
(
i
,
j
)
:
i
∈
{
1
,
2
,
…
,
m
}
,
j
∈
{
1
,
2
,
…
,
n
}
}
S_{m,n} = \{(i,j): i \in \{1,2,\ldots,m\}, j\in \{1,2,\ldots,n\}\}
S
m
,
n
=
{(
i
,
j
)
:
i
∈
{
1
,
2
,
…
,
m
}
,
j
∈
{
1
,
2
,
…
,
n
}}
be a grid of
m
n
mn
mn
lattice points on the coordinate plane. Determine all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
for which there exists a simple polygon
P
P
P
with vertices in
S
m
,
n
S_{m,n}
S
m
,
n
such that all points in
S
m
,
n
S_{m,n}
S
m
,
n
are on the boundary of
P
P
P
, all interior angles of
P
P
P
are either
9
0
∘
90^{\circ}
9
0
∘
or
27
0
∘
270^{\circ}
27
0
∘
and all side lengths of
P
P
P
are
1
1
1
or
3
3
3
.
C1
1
Hide problems
All triangles but one have the same orientation
Determine all integers
n
≥
3
n \geq 3
n
≥
3
for which there exists a con guration of
n
n
n
points in the plane, no three collinear, that can be labelled
1
1
1
through
n
n
n
in two different ways, so that the following condition be satis fied: For every triple
(
i
,
j
,
k
)
,
1
≤
i
<
j
<
k
≤
n
(i,j,k), 1 \leq i < j < k \leq n
(
i
,
j
,
k
)
,
1
≤
i
<
j
<
k
≤
n
, the triangle
i
j
k
ijk
ijk
in one labelling has the same orientation as the triangle labelled
i
j
k
ijk
ijk
in the other, except for
(
i
,
j
,
k
)
=
(
1
,
2
,
3
)
(i,j,k) = (1,2,3)
(
i
,
j
,
k
)
=
(
1
,
2
,
3
)
.
A2
1
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Consecutive roots and interpolation
Fix an integer
n
≥
2
n \geq 2
n
≥
2
and let
a
1
,
…
,
a
n
a_1, \ldots, a_n
a
1
,
…
,
a
n
be integers, where
a
1
=
1
a_1 = 1
a
1
=
1
. Let
f
(
x
)
=
∑
m
=
1
n
a
m
m
x
.
f(x) = \sum_{m=1}^n a_mm^x.
f
(
x
)
=
m
=
1
∑
n
a
m
m
x
.
Suppose that
f
(
x
)
=
0
f(x) = 0
f
(
x
)
=
0
for some
K
K
K
consecutive positive integer values of
x
x
x
. In terms of
n
n
n
, determine the maximum possible value of
K
K
K
.
A1
1
Hide problems
Addition respects rationality in polynomials
Determine all polynomials
P
P
P
with real coefficients satisfying the following condition: whenever
x
x
x
and
y
y
y
are real numbers such that
P
(
x
)
P(x)
P
(
x
)
and
P
(
y
)
P(y)
P
(
y
)
are both rational, so is
P
(
x
+
y
)
P(x + y)
P
(
x
+
y
)
.