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Contests
International Contests
Romanian Masters of Mathematics Collection
2021 Romanian Master of Mathematics Shortlist
2021 Romanian Master of Mathematics Shortlist
Part of
Romanian Masters of Mathematics Collection
Subcontests
(12)
A2
1
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Mini interpolations give a big interpolation
Let
n
n
n
be a positive integer and let
x
1
,
…
,
x
n
,
y
1
,
…
,
y
n
x_1,\ldots,x_n,y_1,\ldots,y_n
x
1
,
…
,
x
n
,
y
1
,
…
,
y
n
be integers satisfying the following condition: the numbers
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
are pairwise distinct and for every positive integer
m
m
m
there exists a polynomial
P
m
P_m
P
m
with integer coefficients such that
P
m
(
x
i
)
−
y
i
P_m(x_i) - y_i
P
m
(
x
i
)
−
y
i
,
i
=
1
,
…
,
n
i=1,\ldots,n
i
=
1
,
…
,
n
, are all divisible by
m
m
m
. Prove that there exists a polynomial
P
P
P
with integer coefficients such that
P
(
x
i
)
=
y
i
P(x_i) = y_i
P
(
x
i
)
=
y
i
for all
i
=
1
,
…
,
n
i=1,\ldots,n
i
=
1
,
…
,
n
.
C2
1
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Graphs with complex assignments
Fix a positive integer
n
n
n
and a fi nite graph with at least one edge; the endpoints of each edge are distinct, and any two vertices are joined by at most one edge. Vertices and edges are assigned (not necessarily distinct) numbers in the range from
0
0
0
to
n
−
1
n-1
n
−
1
, one number each. A vertex assignment and an edge assignment are compatible if the following condition is satisfi ed at each vertex
v
v
v
: The number assigned to
v
v
v
is congruent modulo
n
n
n
to the sum of the numbers assigned to the edges incident to
v
v
v
. Fix a vertex assignment and let
N
N
N
be the total number of compatible edge assignments; compatibility refers, of course, to the fixed vertex assignment. Prove that, if
N
≠
0
N \neq 0
N
=
0
, then the prime divisors of
N
N
N
are all at most
n
n
n
.
C1
1
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Graph triangles with arithmetic progressions
Determine the largest integer
n
≥
3
n\geq 3
n
≥
3
for which the edges of the complete graph on
n
n
n
vertices can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.
N2
1
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Analytic Number Theory again from its pleasant side
We call a set of positive integers suitable if none of its elements is coprime to the sum of all elements of that set. Given a real number
ε
∈
(
0
,
1
)
\varepsilon \in (0,1)
ε
∈
(
0
,
1
)
, prove that, for all large enough positive integers
N
N
N
, there exists a suitable set of size at least
ε
N
\varepsilon N
εN
, each element of which is at most
N
N
N
.
N1
1
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Divisors and Unit Fractions
Given a positive integer
N
N
N
, determine all positive integers
n
n
n
, satisfying the following condition: for any list
d
1
,
d
2
,
…
,
d
k
d_1,d_2,\ldots,d_k
d
1
,
d
2
,
…
,
d
k
of (not necessarily distinct) divisors of
n
n
n
such that
1
d
1
+
1
d
2
+
…
+
1
d
k
>
N
\frac{1}{d_1} + \frac{1}{d_2} + \ldots + \frac{1}{d_k} > N
d
1
1
+
d
2
1
+
…
+
d
k
1
>
N
, some of the fractions
1
d
1
,
1
d
2
,
…
,
1
d
k
\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_k}
d
1
1
,
d
2
1
,
…
,
d
k
1
add up to exactly
N
N
N
.
G4
1
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A geo shortlist is not a shortlist without a config
Let
A
B
C
ABC
A
BC
be an acute triangle, let
H
H
H
and
O
O
O
be its orthocentre and circumcentre, respectively, and let
S
S
S
and
T
T
T
be the feet of the altitudes from
B
B
B
to
A
C
AC
A
C
and from
C
C
C
to
A
B
AB
A
B
, respectively. Let
M
M
M
be the midpoint of the segment
S
T
ST
ST
, and let
N
N
N
be the midpoint of the segment
A
H
AH
A
H
. The line through
O
O
O
, parallel to
B
C
BC
BC
, crosses the sides
A
C
AC
A
C
and
A
B
AB
A
B
at
F
F
F
and
G
G
G
, respectively. The line
N
G
NG
NG
meets the circle
B
G
O
BGO
BGO
again at
K
K
K
, and the line
N
F
NF
NF
meets the circle
C
F
O
CFO
CFO
again at
L
L
L
. Prove that the triangles
B
C
M
BCM
BCM
and
K
L
N
KLN
K
L
N
are similar.
G3
1
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Russian geo showing its reputation again
Let
Ω
\Omega
Ω
be the circumcircle of a triangle
A
B
C
ABC
A
BC
with
∠
B
A
C
>
9
0
∘
\angle BAC > 90^{\circ}
∠
B
A
C
>
9
0
∘
and
A
B
>
A
C
AB > AC
A
B
>
A
C
. The tangents of
Ω
\Omega
Ω
at
B
B
B
and
C
C
C
cross at
D
D
D
and the tangent of
Ω
\Omega
Ω
at
A
A
A
crosses the line
B
C
BC
BC
at
E
E
E
. The line through
D
D
D
, parallel to
A
E
AE
A
E
, crosses the line
B
C
BC
BC
at
F
F
F
. The circle with diameter
E
F
EF
EF
meets the line
A
B
AB
A
B
at
P
P
P
and
Q
Q
Q
and the line
A
C
AC
A
C
at
X
X
X
and
Y
Y
Y
. Prove that one of the angles
∠
A
E
B
\angle AEB
∠
A
EB
,
∠
P
E
Q
\angle PEQ
∠
PEQ
,
∠
X
E
Y
\angle XEY
∠
XE
Y
is equal to the sum of the other two.
G2
1
Hide problems
Geometry is objectively cooler when "it turns out that"
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
. The line through
I
I
I
, perpendicular to
A
I
AI
A
I
, intersects the circumcircle of
A
B
C
ABC
A
BC
at points
P
P
P
and
Q
Q
Q
. It turns out there exists a point
T
T
T
on the side
B
C
BC
BC
such that
A
B
+
B
T
=
A
C
+
C
T
AB + BT = AC + CT
A
B
+
BT
=
A
C
+
CT
and
A
T
2
=
A
B
⋅
A
C
AT^2 = AB \cdot AC
A
T
2
=
A
B
⋅
A
C
. Determine all possible values of the ratio
I
P
/
I
Q
IP/IQ
I
P
/
I
Q
.
G1
1
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Innovatively formulated geometry
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. A line through
C
C
C
crosses the side
A
B
AB
A
B
at an interior point
X
X
X
, and the line
A
D
AD
A
D
at
Y
Y
Y
. The tangents of the circle
A
X
Y
AXY
A
X
Y
at
X
X
X
and
Y
Y
Y
, respectively, cross at
T
T
T
. Prove that the circumcircles of triangles
A
B
D
ABD
A
B
D
and
T
X
Y
TXY
TX
Y
intersect at two points, one lying on the line
A
T
AT
A
T
and the other one lying on the line
C
T
CT
CT
.
A4
1
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RMM and Analysis? Why not bro!
Let
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
be a non-decreasing function such that
f
(
y
)
−
f
(
x
)
<
y
−
x
f(y) - f(x) < y - x
f
(
y
)
−
f
(
x
)
<
y
−
x
for all real numbers
x
x
x
and
y
>
x
y > x
y
>
x
. The sequence
u
1
,
u
2
,
…
u_1,u_2,\ldots
u
1
,
u
2
,
…
of real numbers is such that
u
n
+
2
=
f
(
u
n
+
1
)
−
f
(
u
n
)
u_{n+2} = f(u_{n+1}) - f(u_n)
u
n
+
2
=
f
(
u
n
+
1
)
−
f
(
u
n
)
for all
n
≥
1
n\geq 1
n
≥
1
. Prove that for any
ε
>
0
\varepsilon > 0
ε
>
0
there exists a positive integer
N
N
N
such that
∣
u
n
∣
<
ε
|u_n| < \varepsilon
∣
u
n
∣
<
ε
for all
n
≥
N
n\geq N
n
≥
N
.
A3
1
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Reals modulo 1 arise
A tile
T
T
T
is a union of finitely many pairwise disjoint arcs of a unit circle
K
K
K
. The size of
T
T
T
, denoted by
∣
T
∣
|T|
∣
T
∣
, is the sum of the lengths of the arcs
T
T
T
consists of, divided by
2
π
2\pi
2
π
. A copy of
T
T
T
is a tile
T
′
T'
T
′
obtained by rotating
T
T
T
about the centre of
K
K
K
through some angle. Given a positive real number
ε
<
1
\varepsilon < 1
ε
<
1
, does there exist an infinite sequence of tiles
T
1
,
T
2
,
…
,
T
n
,
…
T_1,T_2,\ldots,T_n,\ldots
T
1
,
T
2
,
…
,
T
n
,
…
satisfying the following two conditions simultaneously: 1)
∣
T
n
∣
>
1
−
ε
|T_n| > 1 - \varepsilon
∣
T
n
∣
>
1
−
ε
for all
n
n
n
; 2) The union of all
T
n
′
T_n'
T
n
′
(as
n
n
n
runs through the positive integers) is a proper subset of
K
K
K
for any choice of the copies
T
1
′
T_1'
T
1
′
,
T
2
′
T_2'
T
2
′
,
…
\ldots
…
,
T
n
′
,
…
T_n', \ldots
T
n
′
,
…
? In the extralist the problem statement had the clause "three conditions" rather than two, but only two are presented, the ones you see. I am quite confident this is a typo or that the problem might have been reformulated after submission.
A1
1
Hide problems
Exercisable functional equation
Determine all functions
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
y
+
f
(
x
)
)
+
f
(
y
)
=
x
f
(
y
)
+
f
(
x
+
y
)
f(xy+f(x)) + f(y) = xf(y) + f(x+y)
f
(
x
y
+
f
(
x
))
+
f
(
y
)
=
x
f
(
y
)
+
f
(
x
+
y
)
for all real numbers
x
x
x
and
y
y
y
.