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Contests
International Contests
European Mathematical Cup
2015 European Mathematical Cup
2015 European Mathematical Cup
Part of
European Mathematical Cup
Subcontests
(4)
4
2
Hide problems
B', A', O are collinear
Let
A
B
C
ABC
A
BC
be an acute angled triangle. Let
B
′
,
A
′
B' , A'
B
′
,
A
′
be points on the perpendicular bisectors of
A
C
,
B
C
AC, BC
A
C
,
BC
respectively such that
B
′
A
⊥
A
B
B'A \perp AB
B
′
A
⊥
A
B
and
A
′
B
⊥
A
B
A'B \perp AB
A
′
B
⊥
A
B
. Let
P
P
P
be a point on the segment
A
B
AB
A
B
and
O
O
O
the circumcenter of the triangle
A
B
C
ABC
A
BC
. Let
D
,
E
D, E
D
,
E
be points on
B
C
,
A
C
BC, AC
BC
,
A
C
respectively such that
D
P
⊥
B
O
DP \perp BO
D
P
⊥
BO
and
E
P
⊥
A
O
EP \perp AO
EP
⊥
A
O
. Let
O
′
O'
O
′
be the circumcenter of the triangle
C
D
E
CDE
C
D
E
. Prove that
B
′
,
A
′
B', A'
B
′
,
A
′
and
O
′
O'
O
′
are collinear.Steve Dinh
Making everyone content even with a brick wall in-between
A group of mathematicians is attending a conference. We say that a mathematician is
k
−
k-
k
−
content if he is in a room with at least
k
k
k
people he admires or if he is admired by at least
k
k
k
other people in the room. It is known that when all participants are in a same room then they are all at least
3
k
+
1
3k + 1
3
k
+
1
-content. Prove that you can assign everyone into one of
2
2
2
rooms in a way that everyone is at least
k
k
k
-content in his room and neither room is empty. Admiration is not necessarily mutual and no one admires himself.Matija Bucić
3
2
Hide problems
Almost perfect numbers
Let
d
(
n
)
d(n)
d
(
n
)
denote the number of positive divisors of
n
n
n
. For positive integer
n
n
n
we define
f
(
n
)
f(n)
f
(
n
)
as
f
(
n
)
=
d
(
k
1
)
+
d
(
k
2
)
+
⋯
+
d
(
k
m
)
,
f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),
f
(
n
)
=
d
(
k
1
)
+
d
(
k
2
)
+
⋯
+
d
(
k
m
)
,
where
1
=
k
1
<
k
2
<
⋯
<
k
m
=
n
1 = k_1 < k_2 < \cdots < k_m = n
1
=
k
1
<
k
2
<
⋯
<
k
m
=
n
are all divisors of the number
n
n
n
. We call an integer
n
>
1
n > 1
n
>
1
almost perfect if
f
(
n
)
=
n
f(n) = n
f
(
n
)
=
n
. Find all almost perfect numbers. Paulius Ašvydis
KP is parallel to median
Circles
k
1
k_1
k
1
and
k
2
k_2
k
2
intersect in points
A
A
A
and
B
B
B
, such that
k
1
k_1
k
1
passes through the center
O
O
O
of the circle
k
2
k_2
k
2
. The line
p
p
p
intersects
k
1
k_1
k
1
in points
K
K
K
and
O
O
O
and
k
2
k_2
k
2
in points
L
L
L
and
M
M
M
, such that the point
L
L
L
is between
K
K
K
and
O
O
O
. The point
P
P
P
is orthogonal projection of the point
L
L
L
to the line
A
B
AB
A
B
. Prove that the line
K
P
KP
K
P
is parallel to the
M
−
M-
M
−
median of the triangle
A
B
M
ABM
A
BM
.Matko Ljulj
2
2
Hide problems
Minimise x^2+y^2+z^2+ mxy + nxz + pyz
Let
m
,
n
,
p
m, n, p
m
,
n
,
p
be fixed positive real numbers which satisfy
m
n
p
=
8
mnp = 8
mn
p
=
8
. Depending on these constants, find the minimum of
x
2
+
y
2
+
z
2
+
m
x
y
+
n
x
z
+
p
y
z
,
x^2+y^2+z^2+ mxy + nxz + pyz,
x
2
+
y
2
+
z
2
+
m
x
y
+
n
x
z
+
p
yz
,
where
x
,
y
,
z
x, y, z
x
,
y
,
z
are arbitrary positive real numbers satisfying
x
y
z
=
8
xyz = 8
x
yz
=
8
. When is the equality attained? Solve the problem for: [*]
m
=
n
=
p
=
2
,
m = n = p = 2,
m
=
n
=
p
=
2
,
[*] arbitrary (but fixed) positive real numbers
m
,
n
,
p
.
m, n, p.
m
,
n
,
p
.
Stijn Cambie
(a+b+c+3)/4 is bigger than sum of fractions
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
b
c
=
1
abc=1
ab
c
=
1
. Prove that
a
+
b
+
c
+
3
4
⩾
1
a
+
b
+
1
b
+
c
+
1
c
+
a
.
\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.
4
a
+
b
+
c
+
3
⩾
a
+
b
1
+
b
+
c
1
+
c
+
a
1
.
Dimitar Trenevski
1
2
Hide problems
Travelling on a nxn board
We are given an
n
×
n
n \times n
n
×
n
board. Rows are labeled with numbers
1
1
1
to
n
n
n
downwards and columns are labeled with numbers
1
1
1
to
n
n
n
from left to right. On each field we write the number
x
2
+
y
2
x^2 + y^2
x
2
+
y
2
where
(
x
,
y
)
(x, y)
(
x
,
y
)
are its coordinates. We are given a figure and can initially place it on any field. In every step we can move the figure from one field to another if the other field has not already been visited and if at least one of the following conditions is satisfied: [*] the numbers in those
2
2
2
fields give the same remainders when divided by
n
n
n
, [*] those fields are point reflected with respect to the center of the board.Can all the fields be visited in case: [*]
n
=
4
n = 4
n
=
4
, [*]
n
=
5
n = 5
n
=
5
?Josip Pupić
10 divides difference of products of powers
A
=
{
a
,
b
,
c
}
A = \{a, b, c\}
A
=
{
a
,
b
,
c
}
is a set containing three positive integers. Prove that we can find a set
B
⊂
A
B \subset A
B
⊂
A
,
B
=
{
x
,
y
}
B = \{x, y\}
B
=
{
x
,
y
}
such that for all odd positive integers
m
,
n
m, n
m
,
n
we have
10
∣
x
m
y
n
−
x
n
y
m
.
10\mid x^my^n-x^ny^m.
10
∣
x
m
y
n
−
x
n
y
m
.
Tomi Dimovski