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Contests
International Contests
Romanian Masters of Mathematics Collection
2020 Romanian Master of Mathematics Shortlist
2020 Romanian Master of Mathematics Shortlist
Part of
Romanian Masters of Mathematics Collection
Subcontests
(11)
N2
1
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Nice Number Theory
For a positive integer
n
n
n
, let
φ
(
n
)
\varphi(n)
φ
(
n
)
and
d
(
n
)
d(n)
d
(
n
)
denote the value of the Euler phi function at
n
n
n
and the number of positive divisors of
n
n
n
, respectively. Prove that there are infinitely many positive integers
n
n
n
such that
φ
(
n
)
\varphi(n)
φ
(
n
)
and
d
(
n
)
d(n)
d
(
n
)
are both perfect squares.Finland, Olli Järviniemi
N1
1
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Number Theory FE
Determine all pairs of positive integers
(
m
,
n
)
(m, n)
(
m
,
n
)
for which there exists a bijective function
f
:
Z
m
×
Z
n
→
Z
m
×
Z
n
f : \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_m \times \mathbb{Z}_n
f
:
Z
m
×
Z
n
→
Z
m
×
Z
n
such that the vectors
f
(
v
)
+
v
f(\mathbf{v}) + \mathbf{v}
f
(
v
)
+
v
, as
v
\mathbf{v}
v
runs through all of
Z
m
×
Z
n
\mathbb{Z}_m \times \mathbb{Z}_n
Z
m
×
Z
n
, are pairwise distinct. (For any integers
a
a
a
and
b
b
b
, the vectors
[
a
,
b
]
,
[
a
+
m
,
b
]
[a, b], [a + m, b]
[
a
,
b
]
,
[
a
+
m
,
b
]
and
[
a
,
b
+
n
]
[a, b + n]
[
a
,
b
+
n
]
are treated as equal.)Poland, Wojciech Nadara
G3
1
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Hard Cyclic Quadrilateral
In the triangle
A
B
C
ABC
A
BC
with circumcircle
Γ
\Gamma
Γ
, the incircle
ω
\omega
ω
touches sides
B
C
,
C
A
BC, CA
BC
,
C
A
, and
A
B
AB
A
B
at
D
,
E
D, E
D
,
E
, and
F
F
F
, respectively. The line through
D
D
D
perpendicular to
E
F
EF
EF
meets
ω
\omega
ω
at
K
≠
D
K\neq D
K
=
D
. Line
A
K
AK
A
K
meets
Γ
\Gamma
Γ
at
L
≠
A
L\neq A
L
=
A
. Rays
K
I
KI
K
I
and
I
L
IL
I
L
meet the circumcircle of triangle
B
I
C
BIC
B
I
C
at
Q
≠
I
Q\neq I
Q
=
I
and
P
≠
I
P\neq I
P
=
I
, respectively. The circumcircles of triangles
K
F
B
KFB
K
FB
and
K
E
C
KEC
K
EC
meet
E
F
EF
EF
at
R
≠
F
R\neq F
R
=
F
and
S
≠
E
S\neq E
S
=
E
, respectively. Prove that
P
Q
R
S
PQRS
PQRS
is cyclic.India, Anant Mugdal
G2
1
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Intersections On The Euler Line
Let
A
B
C
ABC
A
BC
be an acute scalene triangle, and let
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
be the feet of the altitudes from
A
,
B
,
C
A, B, C
A
,
B
,
C
. Let
A
2
A_2
A
2
be the intersection of the tangents to the circle
A
B
C
ABC
A
BC
at
B
,
C
B, C
B
,
C
and define
B
2
,
C
2
B_2, C_2
B
2
,
C
2
similarly. Let
A
2
A
1
A_2A_1
A
2
A
1
intersect the circle
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
again at
A
3
A_3
A
3
and define
B
3
,
C
3
B_3, C_3
B
3
,
C
3
similarly. Show that the circles
A
A
1
A
3
,
B
B
1
B
3
AA_1A_3, BB_1B_3
A
A
1
A
3
,
B
B
1
B
3
, and
C
C
1
C
3
CC_1C_3
C
C
1
C
3
all have two common points,
X
1
X_1
X
1
and
X
2
X_2
X
2
which both lie on the Euler line of the triangle
A
B
C
ABC
A
BC
.United Kingdom, Joe Benton
G1
1
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Find The Angle!
The incircle of a scalene triangle
A
B
C
ABC
A
BC
touches the sides
B
C
,
C
A
BC, CA
BC
,
C
A
, and
A
B
AB
A
B
at points
D
,
E
D, E
D
,
E
, and
F
F
F
, respectively. Triangles
A
P
E
APE
A
PE
and
A
Q
F
AQF
A
QF
are constructed outside the triangle so that
A
P
=
P
E
,
A
Q
=
Q
F
,
∠
A
P
E
=
∠
A
C
B
,
and
∠
A
Q
F
=
∠
A
B
C
.
AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.
A
P
=
PE
,
A
Q
=
QF
,
∠
A
PE
=
∠
A
CB
,
and
∠
A
QF
=
∠
A
BC
.
Let
M
M
M
be the midpoint of
B
C
BC
BC
. Find
∠
Q
M
P
\angle QMP
∠
QMP
in terms of the angles of the triangle
A
B
C
ABC
A
BC
.Iran, Shayan Talaei
C4
1
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Ternary Sequences
A ternary sequence is one whose terms all lie in the set
{
0
,
1
,
2
}
\{0, 1, 2\}
{
0
,
1
,
2
}
. Let
w
w
w
be a length
n
n
n
ternary sequence
(
a
1
,
…
,
a
n
)
(a_1,\ldots,a_n)
(
a
1
,
…
,
a
n
)
. Prove that
w
w
w
can be extended leftwards and rightwards to a length
m
=
6
n
m=6n
m
=
6
n
ternary sequence (d_1,\ldots,d_m) = (b_1,\ldots,b_p,a_1,\ldots,a_n,c_1,\ldots,c_q), p,q\geqslant 0,containing no length
t
>
2
n
t > 2n
t
>
2
n
palindromic subsequence.(A sequence is called palindromic if it reads the same rightwards and leftwards. A length
t
t
t
subsequence of
(
d
1
,
…
,
d
m
)
(d_1,\ldots,d_m)
(
d
1
,
…
,
d
m
)
is a sequence of the form
(
d
i
1
,
…
,
d
i
t
)
(d_{i_1},\ldots,d_{i_t})
(
d
i
1
,
…
,
d
i
t
)
, where
1
⩽
i
1
<
⋯
<
i
t
⩽
m
1\leqslant i_1<\cdots<i_t \leqslant m
1
⩽
i
1
<
⋯
<
i
t
⩽
m
.)
C2
1
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Small Collection Wanted
Let
n
n{}
n
be a positive integer, and let
C
\mathcal{C}
C
be a collection of subsets of
{
1
,
2
,
…
,
2
n
}
\{1,2,\ldots,2^n\}
{
1
,
2
,
…
,
2
n
}
satisfying both of the following conditions: [*]Every
(
2
n
−
1
)
(2^n-1)
(
2
n
−
1
)
-element subset of
{
1
,
2
,
…
,
2
n
}
\{1,2,\ldots,2^n\}
{
1
,
2
,
…
,
2
n
}
is a member of
C
\mathcal{C}
C
, and [*]Every non-empty member
C
C
C
of
C
\mathcal{C}
C
contains an element
c
c
c
such that
C
∖
{
c
}
C\setminus\{c\}
C
∖
{
c
}
is again a member of
C
\mathcal{C}
C
. Determine the smallest size
C
\mathcal{C}
C
may have.Serbia, Pavle Martinovic ́
C3
1
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Colourful Queens
Determine the smallest positive integer
k
k{}
k
satisfying the following condition: For any configuration of chess queens on a
100
×
100
100 \times 100
100
×
100
chequered board, the queens can be coloured one of
k
k
k
colours so that no two queens of the same colour attack each other.Russia, Sergei Avgustinovich and Dmitry Khramtsov
C1
1
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Bethan Gets Stuck
Bethan is playing a game on an
n
×
n
n\times n
n
×
n
grid consisting of
n
2
n^2
n
2
cells. A move consists of placing a counter in an unoccupied cell
C
C
C
where the
2
n
−
2
2n-2
2
n
−
2
other cells in the same row or column as
C
C
C
contain an even number of counters. After making
M
M
M
moves Bethan realises she cannot make any more moves. Determine the minimum value of
M
M
M
.United Kingdom, Sam Bealing
A1
1
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Almost Certainly Irreducible!
Prove that for all sufficiently large positive integers
d
d{}
d
, at least
99
%
99\%
99%
of the polynomials of the form
∑
i
⩽
d
∑
j
⩽
d
±
x
i
y
j
\sum_{i\leqslant d}\sum_{j\leqslant d}\pm x^iy^j
i
⩽
d
∑
j
⩽
d
∑
±
x
i
y
j
are irreducible over the integers.
A2
1
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My best problem
Let
n
>
1
n>1
n
>
1
be a positive integer and
S
\mathcal S
S
be the set of
n
th
n^{\text{th}}
n
th
roots of unity. Suppose
P
P
P
is an
n
n
n
-variable polynomial with complex coefficients such that for all
a
1
,
…
,
a
n
∈
S
a_1,\ldots,a_n\in\mathcal S
a
1
,
…
,
a
n
∈
S
,
P
(
a
1
,
…
,
a
n
)
=
0
P(a_1,\ldots,a_n)=0
P
(
a
1
,
…
,
a
n
)
=
0
if and only if
a
1
,
…
,
a
n
a_1,\ldots,a_n
a
1
,
…
,
a
n
are all different. What is the smallest possible degree of
P
P
P
?Adam Ardeishar and Michael Ren