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Problems
Contests
International Contests
Final Mathematical Cup
2021 Final Mathematical Cup
2021 Final Mathematical Cup
Part of
Final Mathematical Cup
Subcontests
(4)
4
2
Hide problems
coloring sides of regular (2n+1)-gon, external points see points on sides of P
Let
P
P
P
is a regular
(
2
n
+
1
)
(2n+1)
(
2
n
+
1
)
-gon in the plane, where
n
n
n
is a positive integer. We say that a point
S
S
S
on one of the sides of
P
P
P
can be seen from a point
E
E
E
that is external to
P
P
P
, if the line segment
E
S
‾
\overline{ES}
ES
contains no other points that lie on the sides of
P
P
P
except
S
S
S
. We want to color the sides of
P
P
P
in
3
3
3
colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to
P
P
P
, at most
2
2
2
different colors on
P
P
P
can be seen (ignore the vertices of
P
P
P
, we consider them colorless). Find the largest positive integer
n
n
n
for which such a coloring is possible.
n lamps at n vertices of a regular n-gon
A number of
n
n
n
lamps (
n
≥
3
n\ge 3
n
≥
3
) are put at
n
n
n
vertices of a regular
n
n
n
-gon. Initially, all the lamps are off. In each step. Lisa will choose three lamps that are located at three vertices of an isosceles triangle and change their states (from off to on and vice versa). Her aim is to turn on all the lamps. At least how many steps are required to do so?
3
2
Hide problems
S(1)S(3)...S(2n-1) >=S(2)S(4)...S(2n) , sum of digits inequality
For every positive integer
n
n
n
,
s
(
n
)
s(n)
s
(
n
)
denotes the sum of the digits in the decimal representation of
n
n
n
. Prove that for every integer
n
≥
5
n \ge 5
n
≥
5
, we have
S
(
1
)
S
(
3
)
.
.
.
S
(
2
n
−
1
)
≥
S
(
2
)
S
(
4
)
.
.
.
S
(
2
n
)
S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)
S
(
1
)
S
(
3
)
...
S
(
2
n
−
1
)
≥
S
(
2
)
S
(
4
)
...
S
(
2
n
)
max X_1^{X_2^{...^{X_k}}} for n = X_1X_2...X_k
For a positive integer
n
n
n
we define
f
(
n
)
=
max
X
1
X
2
.
.
.
X
k
f (n) = \max X_1^{X_2^{...^{X_k}}}
f
(
n
)
=
max
X
1
X
2
..
.
X
k
where the maximum is taken over all possible decompositions of natural numbers
n
=
X
1
X
2
.
.
.
X
k
n = X_1X_2...X_k
n
=
X
1
X
2
...
X
k
. Determine
f
(
n
)
f(n)
f
(
n
)
.
2
2
Hide problems
D circumcenter of MNK wanted, < CAP = < CBP = <ACB
Let
A
B
C
ABC
A
BC
be an acute triangle, where
A
B
AB
A
B
is the smallest side and let
D
D
D
be the midpoint of
A
B
AB
A
B
. Let
P
P
P
be a point in the interior of the triangle
A
B
C
ABC
A
BC
such that
∠
C
A
P
=
∠
C
B
P
=
∠
A
C
B
\angle CAP = \angle CBP = \angle ACB
∠
C
A
P
=
∠
CBP
=
∠
A
CB
. From the point
P
P
P
, we draw perpendicular lines on
B
C
BC
BC
and
A
C
AC
A
C
where the intersection point with
B
C
BC
BC
is
M
M
M
, and with
A
C
AC
A
C
is
N
N
N
. Through the point
M
M
M
we draw a line parallel to
A
C
AC
A
C
, and through
N
N
N
parallel to
B
C
BC
BC
. These lines intercept at the point
K
K
K
. Prove that
D
D
D
is the center of the circumscribed circle for the triangle
M
N
K
MNK
MN
K
.
ZA tangent to (AXY), altitudes related
The altitudes
B
B
1
BB_1
B
B
1
and
C
C
1
CC_1
C
C
1
, are drawn in an acute triangle
A
B
C
ABC
A
BC
. Let
X
X
X
and
Y
Y
Y
be the points, which are symmetrical to the points
B
1
B_1
B
1
and
C
1
C_1
C
1
, with respect to the midpoints of the sides
A
B
AB
A
B
and
A
C
AC
A
C
of the triangle
A
B
C
ABC
A
BC
respectively. Let's denote with
Z
Z
Z
the point of intersection of the lines
B
C
BC
BC
and
X
Y
XY
X
Y
. Prove that the line
Z
A
ZA
Z
A
is tangent to the circumscribed circle of the triangle
A
X
Y
AXY
A
X
Y
.
1
2
Hide problems
diophantine 2x^2 + 5xy + 2y^2 = n
Find all integer
n
n
n
such that the equation
2
x
2
+
5
x
y
+
2
y
2
=
n
2x^2 + 5xy + 2y^2 = n
2
x
2
+
5
x
y
+
2
y
2
=
n
has integer solution for
x
x
x
and
y
y
y
.
f(n)f(m)=f ((nm)^{2021})
Let
N
N
N
is the set of all positive integers. Determine all mappings
f
:
N
−
{
1
}
→
N
f: N-\{1\} \to N
f
:
N
−
{
1
}
→
N
such that for every
n
≠
m
n \ne m
n
=
m
the following equation is true
f
(
n
)
f
(
m
)
=
f
(
(
n
m
)
2021
)
f(n)f(m)=f\left((nm)^{2021}\right)
f
(
n
)
f
(
m
)
=
f
(
(
nm
)
2021
)