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Problems
Contests
International Contests
European Mathematical Cup
2022 European Mathematical Cup
2022 European Mathematical Cup
Part of
European Mathematical Cup
Subcontests
(4)
4
2
Hide problems
Difficult abstraction with sets
A collection
F
F
F
of distinct (not necessarily non-empty) subsets of
X
=
{
1
,
2
,
…
,
300
}
X = \{1,2,\ldots,300\}
X
=
{
1
,
2
,
…
,
300
}
is lovely if for any three (not necessarily distinct) sets
A
A
A
,
B
B
B
and
C
C
C
in
F
F
F
at most three out of the following eight sets are non-empty \begin{align*}A \cap B \cap C, \ \ \ \overline{A} \cap B \cap C, \ \ \ A \cap \overline{B} \cap C, \ \ \ A \cap B \cap \overline{C}, \\ \overline{A} \cap \overline{B} \cap C, \ \ \ \overline{A} \cap B \cap \overline {C}, \ \ \ A \cap \overline{B} \cap \overline{C}, \ \ \ \overline{A} \cap \overline{B} \cap \overline{C} \end{align*} where
S
‾
\overline{S}
S
denotes the set of all elements of
X
X
X
which are not in
S
S
S
. What is the greatest possible number of sets in a lovely collection?
So few points, so many tangencies
Five points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
and
E
E
E
lie on a circle
τ
\tau
τ
clockwise in that order such that
A
B
∥
C
E
AB \parallel CE
A
B
∥
CE
and
∠
A
B
C
>
9
0
∘
\angle ABC > 90^{\circ}
∠
A
BC
>
9
0
∘
. Let
k
k
k
be a circle tangent to
A
D
AD
A
D
,
C
E
CE
CE
and
τ
\tau
τ
such that
k
k
k
and
τ
\tau
τ
touch on the arc
D
E
^
\widehat{DE}
D
E
not containing
A
A
A
,
B
B
B
and
C
C
C
. Let
F
≠
A
F \neq A
F
=
A
be the intersection of
τ
\tau
τ
and the tangent line to
k
k
k
passing through
A
A
A
different from
A
D
AD
A
D
. Prove that there exists a circle tangent to
B
D
BD
B
D
,
B
F
BF
BF
,
C
E
CE
CE
and
τ
\tau
τ
.
3
2
Hide problems
Looks like some madness geo again
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
C
>
B
C
AC > BC
A
C
>
BC
, with incircle
τ
\tau
τ
centered at
I
I
I
which touches
B
C
BC
BC
and
A
C
AC
A
C
at points
D
D
D
and
E
E
E
, respectively. The point
M
M
M
on
τ
\tau
τ
is such that
B
M
∥
D
E
BM \parallel DE
BM
∥
D
E
and
M
M
M
and
B
B
B
lie on the same halfplane with respect to the angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
. Let
F
F
F
and
H
H
H
be the intersections of
τ
\tau
τ
with
B
M
BM
BM
and
C
M
CM
CM
different from
M
M
M
, respectively. Let
J
J
J
be a point on the line
A
C
AC
A
C
such that
J
M
∥
E
H
JM \parallel EH
J
M
∥
E
H
. Let
K
K
K
be the intersection of
J
F
JF
J
F
and
τ
\tau
τ
different from
F
F
F
. Prove that
M
E
∥
K
H
ME \parallel KH
ME
∥
KH
.
Slippery functional equation
Determine all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
3
)
+
f
(
y
)
3
+
f
(
z
)
3
=
3
x
y
z
f(x^3) + f(y)^3 + f(z)^3 = 3xyz
f
(
x
3
)
+
f
(
y
)
3
+
f
(
z
)
3
=
3
x
yz
for all real numbers
x
x
x
,
y
y
y
and
z
z
z
with
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
.
2
2
Hide problems
Floor function is back, yay!
Find all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive real numbers such that
x
y
xy
x
y
is an integer and
x
+
y
=
⌊
x
2
−
y
2
⌋
x+y = \lfloor x^2 - y^2 \rfloor
x
+
y
=
⌊
x
2
−
y
2
⌋
.
As some nations like to say "Heavy theorems mostly do not help"
We say that a positive integer
n
n
n
is lovely if there exist a positive integer
k
k
k
and (not necessarily distinct) positive integers
d
1
d_1
d
1
,
d
2
d_2
d
2
,
…
\ldots
…
,
d
k
d_k
d
k
such that
n
=
d
1
d
2
⋯
d
k
n = d_1d_2\cdots d_k
n
=
d
1
d
2
⋯
d
k
and
d
i
2
∣
n
+
d
i
d_i^2 \mid n + d_i
d
i
2
∣
n
+
d
i
for
i
=
1
,
2
,
…
,
k
i=1,2,\ldots,k
i
=
1
,
2
,
…
,
k
.a) Are there infinitely many lovely numbers?b) Is there a lovely number, greater than
1
1
1
, which is a perfect square of an integer?
1
2
Hide problems
A little play with divisors
Determine all positive integers
n
n
n
for which there exist positive divisors
a
a
a
,
b
b
b
,
c
c
c
of
n
n
n
such that
a
>
b
>
c
a>b>c
a
>
b
>
c
and
a
2
−
b
2
a^2 - b^2
a
2
−
b
2
,
b
2
−
c
2
b^2 - c^2
b
2
−
c
2
,
a
2
−
c
2
a^2 - c^2
a
2
−
c
2
are also divisors of
n
n
n
.
A little game with symmetries
Let
n
≥
3
n\geq 3
n
≥
3
be a positive integer. Alice and Bob are playing a game in which they take turns colouring the vertices of a regular
n
n
n
-gon. Alice plays the first move. Initially, no vertex is coloured. Both players start the game with
0
0
0
points.In their turn, a player colours a vertex
V
V
V
which has not been coloured and gains
k
k
k
points where
k
k
k
is the number of already coloured neighbouring vertices of
V
V
V
. (Thus,
k
k
k
is either
0
0
0
,
1
1
1
or
2
2
2
.)The game ends when all vertices have been coloured and the player with more points wins; if they have the same number of points, no one wins. Determine all
n
≥
3
n\geq 3
n
≥
3
for which Alice has a winning strategy and all
n
≥
3
n\geq 3
n
≥
3
for which Bob has a winning strategy.