MathDB
Problems
Contests
International Contests
Mathematical Excellence Olympiad
2019 IMEO
2019 IMEO
Part of
Mathematical Excellence Olympiad
Subcontests
(6)
6
1
Hide problems
Hard Geometry on circle intersections
Let
A
B
C
ABC
A
BC
be a scalene triangle with incenter
I
I
I
and circumcircle
ω
\omega
ω
. The internal and external bisectors of angle
∠
B
A
C
\angle BAC
∠
B
A
C
intersect
B
C
BC
BC
at
D
D
D
and
E
E
E
, respectively. Let
M
M
M
be the point on segment
A
C
AC
A
C
such that
M
C
=
M
B
MC = MB
MC
=
MB
. The tangent to
ω
\omega
ω
at
B
B
B
meets
M
D
MD
M
D
at
S
S
S
. The circumcircles of triangles
A
D
E
ADE
A
D
E
and
B
I
C
BIC
B
I
C
intersect each other at
P
P
P
and
Q
Q
Q
. If
A
S
AS
A
S
meets
ω
\omega
ω
at a point
K
K
K
other than
A
A
A
, prove that
K
K
K
lies on
P
Q
PQ
PQ
.Proposed by Alexandru Lopotenco (Moldova)
5
1
Hide problems
Interesting divisibility condition on exponential expressions
Find all pairs of positive integers
(
s
,
t
)
(s, t)
(
s
,
t
)
, so that for any two different positive integers
a
a
a
and
b
b
b
there exists some positive integer
n
n
n
, for which
a
s
+
b
t
∣
a
n
+
b
n
+
1
.
a^s + b^t | a^n + b^{n+1}.
a
s
+
b
t
∣
a
n
+
b
n
+
1
.
Proposed by Oleksii Masalitin (Ukraine)
4
1
Hide problems
Polynomial on primes
Call a two-element subset of
N
\mathbb{N}
N
cute if it contains exactly one prime number and one composite number. Determine all polynomials
f
∈
Z
[
x
]
f \in \mathbb{Z}[x]
f
∈
Z
[
x
]
such that for every cute subset
{
p
,
q
}
\{ p,q \}
{
p
,
q
}
, the subset
{
f
(
p
)
+
q
,
f
(
q
)
+
p
}
\{ f(p) + q, f(q) + p \}
{
f
(
p
)
+
q
,
f
(
q
)
+
p
}
is cute as well.Proposed by Valentio Iverson (Indonesia)
3
1
Hide problems
Challenging Functional Equation
Find all functions
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that for all real
x
,
y
x, y
x
,
y
, the following relation holds:
(
x
+
y
)
⋅
f
(
x
+
y
)
=
f
(
f
(
x
)
+
y
)
⋅
f
(
x
+
f
(
y
)
)
.
(x+y) \cdot f(x+y)= f(f(x)+y) \cdot f(x+f(y)).
(
x
+
y
)
⋅
f
(
x
+
y
)
=
f
(
f
(
x
)
+
y
)
⋅
f
(
x
+
f
(
y
))
.
Proposed by Vadym Koval (Ukraine)
2
1
Hide problems
Evaluation and Example for graphs!
Consider some graph
G
G
G
with
2019
2019
2019
nodes. Let's define inverting a vertex
v
v
v
the following process: for every other vertex
u
u
u
, if there was an edge between
v
v
v
and
u
u
u
, it is deleted, and if there wasn't, it is added. We want to minimize the number of edges in the graph by several invertings (we are allowed to invert the same vertex several times). Find the smallest number
M
M
M
such that we can always make the number of edges in the graph not larger than
M
M
M
, for any initial choice of
G
G
G
.Proposed by Arsenii Nikolaev, Anton Trygub (Ukraine)
1
1
Hide problems
Easy Problem 1 Geometry
Let
A
B
C
ABC
A
BC
be a scalene triangle with circumcircle
ω
\omega
ω
. The tangent to
ω
\omega
ω
at
A
A
A
meets
B
C
BC
BC
at
D
D
D
. The
A
A
A
-median of triangle
A
B
C
ABC
A
BC
intersects
B
C
BC
BC
and
ω
\omega
ω
at
M
M
M
and
N
N
N
, respectively. Suppose that
K
K
K
is a point such that
A
D
M
K
ADMK
A
D
M
K
is a parallelogram. Prove that
K
A
=
K
N
KA = KN
K
A
=
K
N
. Proposed by Alexandru Lopotenco (Moldova)