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Problems
Contests
International Contests
Middle European Mathematical Olympiad
2011 Middle European Mathematical Olympiad
2011 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(8)
8
1
Hide problems
Prove that there exist 2011 consecutive amazing numbers
We call a positive integer
n
n
n
amazing if there exist positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that the equality
n
=
(
b
,
c
)
(
a
,
b
c
)
+
(
c
,
a
)
(
b
,
c
a
)
+
(
a
,
b
)
(
c
,
a
b
)
n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)
n
=
(
b
,
c
)
(
a
,
b
c
)
+
(
c
,
a
)
(
b
,
c
a
)
+
(
a
,
b
)
(
c
,
ab
)
holds. Prove that there exist
2011
2011
2011
consecutive positive integers which are amazing.Note. By
(
m
,
n
)
(m, n)
(
m
,
n
)
we denote the greatest common divisor of positive integers
m
m
m
and
n
n
n
.
7
1
Hide problems
Show that there are elements a, b such that 11|a^3+ab^2+b^3
Let
A
A
A
and
B
B
B
be disjoint nonempty sets with
A
∪
B
=
{
1
,
2
,
3
,
…
,
10
}
A \cup B = \{1, 2,3, \ldots, 10\}
A
∪
B
=
{
1
,
2
,
3
,
…
,
10
}
. Show that there exist elements
a
∈
A
a \in A
a
∈
A
and
b
∈
B
b \in B
b
∈
B
such that the number
a
3
+
a
b
2
+
b
3
a^3 + ab^2 + b^3
a
3
+
a
b
2
+
b
3
is divisible by
11
11
11
.
6
1
Hide problems
Show that the line AX is perpendicular to BC
Let
A
B
C
ABC
A
BC
be an acute triangle. Denote by
B
0
B_0
B
0
and
C
0
C_0
C
0
the feet of the altitudes from vertices
B
B
B
and
C
C
C
, respectively. Let
X
X
X
be a point inside the triangle
A
B
C
ABC
A
BC
such that the line
B
X
BX
BX
is tangent to the circumcircle of the triangle
A
X
C
0
AXC_0
A
X
C
0
and the line
C
X
CX
CX
is tangent to the circumcircle of the triangle
A
X
B
0
AXB_0
A
X
B
0
. Show that the line
A
X
AX
A
X
is perpendicular to
B
C
BC
BC
.
5
1
Hide problems
Prove that ABCDE has a pair of parallel sides
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon with all five sides equal in length. The diagonals
A
D
AD
A
D
and
E
C
EC
EC
meet in
S
S
S
with
∠
A
S
E
=
6
0
∘
\angle ASE = 60^\circ
∠
A
SE
=
6
0
∘
. Prove that
A
B
C
D
E
ABCDE
A
BC
D
E
has a pair of parallel sides.
4
2
Hide problems
At least ceil[2n/9] of the languages can be chosen
Let
n
≥
3
n \geq 3
n
≥
3
be an integer. At a MEMO-like competition, there are
3
n
3n
3
n
participants, there are n languages spoken, and each participant speaks exactly three different languages. Prove that at least
⌈
2
n
9
⌉
\left\lceil\frac{2n}{9}\right\rceil
⌈
9
2
n
⌉
of the spoken languages can be chosen in such a way that no participant speaks more than two of the chosen languages.Note.
⌈
x
⌉
\lceil x\rceil
⌈
x
⌉
is the smallest integer which is greater than or equal to
x
x
x
.
If k^3 - m^3 | km(k^2 - m^2), show that (k - m)^3 > 3km
Let
k
k
k
and
m
m
m
, with
k
>
m
k > m
k
>
m
, be positive integers such that the number
k
m
(
k
2
−
m
2
)
km(k^2 - m^2)
km
(
k
2
−
m
2
)
is divisible by
k
3
−
m
3
k^3 - m^3
k
3
−
m
3
. Prove that
(
k
−
m
)
3
>
3
k
m
(k - m)^3 > 3km
(
k
−
m
)
3
>
3
km
.
3
2
Hide problems
Ah, don't bother me with this much point (show AE prp KL)
In a plane the circles
K
1
\mathcal K_1
K
1
and
K
2
\mathcal K_2
K
2
with centers
I
1
I_1
I
1
and
I
2
I_2
I
2
, respectively, intersect in two points
A
A
A
and
B
B
B
. Assume that
∠
I
1
A
I
2
\angle I_1AI_2
∠
I
1
A
I
2
is obtuse. The tangent to
K
1
\mathcal K_1
K
1
in
A
A
A
intersects
K
2
\mathcal K_2
K
2
again in
C
C
C
and the tangent to
K
2
\mathcal K_2
K
2
in
A
A
A
intersects
K
1
\mathcal K_1
K
1
again in
D
D
D
. Let
K
3
\mathcal K_3
K
3
be the circumcircle of the triangle
B
C
D
BCD
BC
D
. Let
E
E
E
be the midpoint of that arc
C
D
CD
C
D
of
K
3
\mathcal K_3
K
3
that contains
B
B
B
. The lines
A
C
AC
A
C
and
A
D
AD
A
D
intersect
K
3
\mathcal K_3
K
3
again in
K
K
K
and
L
L
L
, respectively. Prove that the line
A
E
AE
A
E
is perpendicular to
K
L
KL
K
L
.
max. of points with no triangles
For an integer
n
≥
3
n \geq 3
n
≥
3
, let
M
\mathcal M
M
be the set
{
(
x
,
y
)
∣
x
,
y
∈
Z
,
1
≤
x
≤
n
,
1
≤
y
≤
n
}
\{(x, y) | x, y \in \mathbb Z, 1 \leq x \leq n, 1 \leq y \leq n\}
{(
x
,
y
)
∣
x
,
y
∈
Z
,
1
≤
x
≤
n
,
1
≤
y
≤
n
}
of points in the plane. What is the maximum possible number of points in a subset
S
⊆
M
S \subseteq \mathcal M
S
⊆
M
which does not contain three distinct points being the vertices of a right triangle?
2
2
Hide problems
Determine the value of S
Let
n
≥
3
n \geq 3
n
≥
3
be an integer. John and Mary play the following game: First John labels the sides of a regular
n
n
n
-gon with the numbers
1
,
2
,
…
,
n
1, 2,\ldots, n
1
,
2
,
…
,
n
in whatever order he wants, using each number exactly once. Then Mary divides this
n
n
n
-gon into triangles by drawing
n
−
3
n-3
n
−
3
diagonals which do not intersect each other inside the
n
n
n
-gon. All these diagonals are labeled with number
1
1
1
. Into each of the triangles the product of the numbers on its sides is written. Let S be the sum of those
n
−
2
n - 2
n
−
2
products.Determine the value of
S
S
S
if Mary wants the number
S
S
S
to be as small as possible and John wants
S
S
S
to be as large as possible and if they both make the best possible choices.
An inequality with condition sum a/(1+a)=2
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
1
+
a
+
b
1
+
b
+
c
1
+
c
=
2.
\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2.
1
+
a
a
+
1
+
b
b
+
1
+
c
c
=
2.
Prove that
a
+
b
+
c
2
≥
1
a
+
1
b
+
1
c
.
\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}.
2
a
+
b
+
c
≥
a
1
+
b
1
+
c
1
.
1
2
Hide problems
Show that (sum b_i^2)/n is greater or equal to 2011
Initially, only the integer
44
44
44
is written on a board. An integer a on the board can be re- placed with four pairwise different integers
a
1
,
a
2
,
a
3
,
a
4
a_1, a_2, a_3, a_4
a
1
,
a
2
,
a
3
,
a
4
such that the arithmetic mean
1
4
(
a
1
+
a
2
+
a
3
+
a
4
)
\frac 14 (a_1 + a_2 + a_3 + a_4)
4
1
(
a
1
+
a
2
+
a
3
+
a
4
)
of the four new integers is equal to the number
a
a
a
. In a step we simultaneously replace all the integers on the board in the above way. After
30
30
30
steps we end up with
n
=
4
30
n = 4^{30}
n
=
4
30
integers
b
1
,
b
2
,
…
,
b
n
b_1, b2,\ldots, b_n
b
1
,
b
2
,
…
,
b
n
on the board. Prove that
b
1
2
+
b
2
2
+
b
3
2
+
⋯
+
b
n
2
n
≥
2011.
\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.
n
b
1
2
+
b
2
2
+
b
3
2
+
⋯
+
b
n
2
≥
2011.
The functional equation problem of MEMO 2011 (team compt.)
Find all functions
f
:
R
→
R
f : \mathbb R \to \mathbb R
f
:
R
→
R
such that the equality
y
2
f
(
x
)
+
x
2
f
(
y
)
+
x
y
=
x
y
f
(
x
+
y
)
+
x
2
+
y
2
y^2f(x) + x^2f(y) + xy = xyf(x + y) + x^2 + y^2
y
2
f
(
x
)
+
x
2
f
(
y
)
+
x
y
=
x
y
f
(
x
+
y
)
+
x
2
+
y
2
holds for all
x
,
y
∈
R
x, y \in \Bbb R
x
,
y
∈
R
, where
R
\Bbb R
R
is the set of real numbers.