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International Contests
European Mathematical Cup
2021 European Mathematical Cup
2021 European Mathematical Cup
Part of
European Mathematical Cup
Subcontests
(4)
4
2
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10th EMC - Colouring the set 1, 2, ..., n
Let
n
n
n
be a positive integer. Morgane has coloured the integers
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
. Each of them is coloured in exactly one colour. It turned out that for all positive integers
a
a
a
and
b
b
b
such that
a
<
b
a<b
a
<
b
and
a
+
b
⩽
n
a+b \leqslant n
a
+
b
⩽
n
, at least two of the integers among
a
a
a
,
b
b
b
and
a
+
b
a+b
a
+
b
are of the same colour. Prove that there exists a colour that has been used for at least
2
n
/
5
2n/5
2
n
/5
integers. \\ \\ (Vincent Jugé)
P(x)^2+1=(x^2+1)Q(x^2), der(P)=?
Find all positive integers
d
d
d
for which there exist polynomials
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
with real coefficients such that degree of
P
P
P
equals
d
d
d
and
P
(
x
)
2
+
1
=
(
x
2
+
1
)
Q
(
x
)
2
.
P(x)^2+1=(x^2+1)Q(x)^2.
P
(
x
)
2
+
1
=
(
x
2
+
1
)
Q
(
x
)
2
.
3
2
Hide problems
10th EMC - Factorials and perfect squares
Let
ℓ
\ell
ℓ
be a positive integer. We say that a positive integer
k
k
k
is nice if
k
!
+
ℓ
k!+\ell
k
!
+
ℓ
is a square of an integer. Prove that for every positive integer
n
⩾
ℓ
n \geqslant \ell
n
⩾
ℓ
, the set
{
1
,
2
,
…
,
n
2
}
\{1, 2, \ldots,n^2\}
{
1
,
2
,
…
,
n
2
}
contains at most
n
2
−
n
+
ℓ
n^2-n +\ell
n
2
−
n
+
ℓ
nice integers. \\ \\ (Théo Lenoir)
f is N to N, x^2-y^2+2y(f(x)+f(y)) is a perfect square
Let
N
\mathbb{N}
N
denote the set of all positive integers. Find all functions
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
such that
x
2
−
y
2
+
2
y
(
f
(
x
)
+
f
(
y
)
)
x^2-y^2+2y(f(x)+f(y))
x
2
−
y
2
+
2
y
(
f
(
x
)
+
f
(
y
))
is a square of an integer for all positive integers
x
x
x
and
y
y
y
.
2
2
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10th EMC - Angle Trisection
Let
A
B
C
ABC
A
BC
be an acute-angled triangle such that
∣
A
B
∣
<
∣
A
C
∣
|AB|<|AC|
∣
A
B
∣
<
∣
A
C
∣
. Let
X
X
X
and
Y
Y
Y
be points on the minor arc
B
C
{BC}
BC
of the circumcircle of
A
B
C
ABC
A
BC
such that
∣
B
X
∣
=
∣
X
Y
∣
=
∣
Y
C
∣
|BX|=|XY|=|YC|
∣
BX
∣
=
∣
X
Y
∣
=
∣
Y
C
∣
. Suppose that there exists a point
N
N
N
on the segment
A
Y
‾
\overline{AY}
A
Y
such that
∣
A
B
∣
=
∣
A
N
∣
=
∣
N
C
∣
|AB|=|AN|=|NC|
∣
A
B
∣
=
∣
A
N
∣
=
∣
NC
∣
. Prove that the line
N
C
NC
NC
passes through the midpoint of the segment
A
X
‾
\overline{AX}
A
X
. \\ \\ (Ivan Novak)
Nice geometry from EMC
Let
A
B
C
ABC
A
BC
be a triangle and let
D
,
E
D, E
D
,
E
and
F
F
F
be the midpoints of sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
, respectively. Let
X
≠
A
X\ne A
X
=
A
be the intersection of
A
D
AD
A
D
with the circumcircle of
A
B
C
ABC
A
BC
. Let
Ω
\Omega
Ω
be the circle through
D
D
D
and
X
X
X
, tangent to the circumcircle of
A
B
C
ABC
A
BC
. Let
Y
Y
Y
and
Z
Z
Z
be the intersections of the tangent to
Ω
\Omega
Ω
at
D
D
D
with the perpendicular bisectors of segments
D
E
DE
D
E
and
D
F
DF
D
F
, respectively. Let
P
P
P
be the intersection of
Y
E
YE
Y
E
and
Z
F
ZF
ZF
and let
G
G
G
be the centroid of
A
B
C
ABC
A
BC
. Show that the tangents at
B
B
B
and
C
C
C
to the circumcircle of
A
B
C
ABC
A
BC
and the line
P
G
PG
PG
are concurrent.
1
2
Hide problems
10th EMC - Balanced Quadruples
We say that a quadruple of nonnegative real numbers
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
is balanced if
a
+
b
+
c
+
d
=
a
2
+
b
2
+
c
2
+
d
2
.
a+b+c+d=a^2+b^2+c^2+d^2.
a
+
b
+
c
+
d
=
a
2
+
b
2
+
c
2
+
d
2
.
Find all positive real numbers
x
x
x
such that
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
≥
0
(x-a)(x-b)(x-c)(x-d)\geq 0
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
≥
0
for every balanced quadruple
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
. \\ \\ (Ivan Novak)
min # of pairs whose labels differ by at most 1
Alice drew a regular
2021
2021
2021
-gon in the plane. Bob then labeled each vertex of the
2021
2021
2021
-gon with a real number, in such a way that the labels of consecutive vertices differ by at most
1
1
1
. Then, for every pair of non-consecutive vertices whose labels differ by at most
1
1
1
, Alice drew a diagonal connecting them. Let
d
d
d
be the number of diagonals Alice drew. Find the least possible value that
d
d
d
can obtain.