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Problems
Contests
International Contests
Middle European Mathematical Olympiad
2015 Middle European Mathematical Olympiad
2015 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(8)
8
1
Hide problems
Number of residues whose square is -1
Let
n
≥
2
n\ge 2
n
≥
2
be an integer. Determine the number of positive integers
m
m
m
such that
m
≤
n
m\le n
m
≤
n
and
m
2
+
1
m^2+1
m
2
+
1
is divisible by
n
n
n
.
7
1
Hide problems
Factorial and exponential diophantine equation
Find all pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
a
!
+
b
!
=
a
b
+
b
a
.
a!+b!=a^b + b^a.
a
!
+
b
!
=
a
b
+
b
a
.
6
1
Hide problems
Two isogonals in incenter diagram
Let
I
I
I
be the incentre of triangle
A
B
C
ABC
A
BC
with
A
B
>
A
C
AB>AC
A
B
>
A
C
and let the line
A
I
AI
A
I
intersect the side
B
C
BC
BC
at
D
D
D
. Suppose that point
P
P
P
lies on the segment
B
C
BC
BC
and satisfies
P
I
=
P
D
PI=PD
P
I
=
P
D
. Further, let
J
J
J
be the point obtained by reflecting
I
I
I
over the perpendicular bisector of
B
C
BC
BC
, and let
Q
Q
Q
be the other intersection of the circumcircles of the triangles
A
B
C
ABC
A
BC
and
A
P
D
APD
A
P
D
. Prove that
∠
B
A
Q
=
∠
C
A
J
\angle BAQ=\angle CAJ
∠
B
A
Q
=
∠
C
A
J
.
5
1
Hide problems
XY passes through D for certain X, Y
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
>
A
C
AB>AC
A
B
>
A
C
. Prove that there exists a point
D
D
D
with the following property: whenever two distinct points
X
X
X
and
Y
Y
Y
lie in the interior of
A
B
C
ABC
A
BC
such that the points
B
B
B
,
C
C
C
,
X
X
X
, and
Y
Y
Y
lie on a circle and
∠
A
X
B
−
∠
A
C
B
=
∠
C
Y
A
−
∠
C
B
A
\angle AXB-\angle ACB=\angle CYA-\angle CBA
∠
A
XB
−
∠
A
CB
=
∠
C
Y
A
−
∠
CB
A
holds, the line
X
Y
XY
X
Y
passes through
D
D
D
.
4
2
Hide problems
For which m,n is (a^m+b^m) / (a^n+b^n) an integer
Find all pairs of positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
for which there exist relatively prime integers
a
a
a
and
b
b
b
greater than
1
1
1
such that
a
m
+
b
m
a
n
+
b
n
\frac{a^m+b^m}{a^n+b^n}
a
n
+
b
n
a
m
+
b
m
is an integer.
Number of regions in NxN board with diagonals
Let
N
N
N
be a positive integer. In each of the
N
2
N^2
N
2
unit squares of an
N
×
N
N\times N
N
×
N
board, one of the two diagonals is drawn. The drawn diagonals divide the
N
×
N
N\times N
N
×
N
board into
K
K
K
regions. For each
N
N
N
, determine the smallest and the largest possible values of
K
K
K
.[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((1,0)--(1,3), dotted); draw((2,0)--(2,3), dotted); draw((0,1)--(3,1), dotted); draw((0,2)--(3,2), dotted); draw((1,0)--(0,1)--(2,3)--(3,2)--(2,1)--(0,3)); draw((1,1)--(2,0)--(3,1)); label("
1
1
1
",(0.35,2)); label("
2
2
2
",(1,2.65)); label("
3
3
3
",(2,2)); label("
4
4
4
",(2.65,2.65)); label("
5
5
5
",(0.35,0.35)); label("
6
6
6
",(1.3,1.3)); label("
7
7
7
",(2.65,0.35)); label("Example with
N
=
3
N=3
N
=
3
,
K
=
7
K=7
K
=
7
",(0,-0.3)--(3,-0.3),S); [/asy]
3
2
Hide problems
Lots of cyclic quadrilaterals
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. Let
E
E
E
be the intersection of lines parallel to
A
C
AC
A
C
and
B
D
BD
B
D
passing through points
B
B
B
and
A
A
A
, respectively. The lines
E
C
EC
EC
and
E
D
ED
E
D
intersect the circumcircle of
A
E
B
AEB
A
EB
again at
F
F
F
and
G
G
G
, respectively. Prove that points
C
C
C
,
D
D
D
,
F
F
F
, and
G
G
G
lie on a circle.
Maximal number of steps for students to permute
There are
n
n
n
students standing in line positions
1
1
1
to
n
n
n
. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position
i
i
i
is now in position
j
j
j
, we say the student moved for
∣
i
−
j
∣
|i-j|
∣
i
−
j
∣
steps. Determine the maximal sum of steps of all students that they can achieve.
2
2
Hide problems
f(x^2yf(x)) functional equation on nonzero reals
Determine all functions
f
:
R
∖
{
0
}
→
R
∖
{
0
}
f:\mathbb{R}\setminus\{0\}\to \mathbb{R}\setminus\{0\}
f
:
R
∖
{
0
}
→
R
∖
{
0
}
such that
f
(
x
2
y
f
(
x
)
)
+
f
(
1
)
=
x
2
f
(
x
)
+
f
(
y
)
f(x^2yf(x))+f(1)=x^2f(x)+f(y)
f
(
x
2
y
f
(
x
))
+
f
(
1
)
=
x
2
f
(
x
)
+
f
(
y
)
holds for all nonzero real numbers
x
x
x
and
y
y
y
.
Minimal number of inner diagonals iff none of them intersect
Let
n
≥
3
n\ge 3
n
≥
3
be an integer. An inner diagonal of a simple
n
n
n
-gon is a diagonal that is contained in the
n
n
n
-gon. Denote by
D
(
P
)
D(P)
D
(
P
)
the number of all inner diagonals of a simple
n
n
n
-gon
P
P
P
and by
D
(
n
)
D(n)
D
(
n
)
the least possible value of
D
(
Q
)
D(Q)
D
(
Q
)
, where
Q
Q
Q
is a simple
n
n
n
-gon. Prove that no two inner diagonals of
P
P
P
intersect (except possibly at a common endpoint) if and only if
D
(
P
)
=
D
(
n
)
D(P)=D(n)
D
(
P
)
=
D
(
n
)
.Remark: A simple
n
n
n
-gon is a non-self-intersecting polygon with
n
n
n
vertices. A polygon is not necessarily convex.
1
2
Hide problems
Surjective functions on N satisfying special condition
Find all surjective functions
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
such that for all positive integers
a
a
a
and
b
b
b
, exactly one of the following equations is true: \begin{align*} f(a)&=f(b),
\\ f(a+b)&=\min\{f(a),f(b)\}. \end{align*} Remarks:
N
\mathbb{N}
N
denotes the set of all positive integers. A function
f
:
X
→
Y
f:X\to Y
f
:
X
→
Y
is said to be surjective if for every
y
∈
Y
y\in Y
y
∈
Y
there exists
x
∈
X
x\in X
x
∈
X
such that
f
(
x
)
=
y
f(x)=y
f
(
x
)
=
y
.
a/(2b+c^2) cyclic sum is at most (a^2+b^2+c^2)/3
Prove that for all positive real numbers
a
a
a
,
b
b
b
,
c
c
c
such that
a
b
c
=
1
abc=1
ab
c
=
1
the following inequality holds:
a
2
b
+
c
2
+
b
2
c
+
a
2
+
c
2
a
+
b
2
≤
a
2
+
b
2
+
c
2
3
.
\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.
2
b
+
c
2
a
+
2
c
+
a
2
b
+
2
a
+
b
2
c
≤
3
a
2
+
b
2
+
c
2
.