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Kyiv Mathematical Festival
2019 Kyiv Mathematical Festival
2019 Kyiv Mathematical Festival
Part of
Kyiv Mathematical Festival
Subcontests
(5)
5
2
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divisibility in a table I
Is it possible to fill the cells of a table of size
2019
×
2019
2019\times2019
2019
×
2019
with pairwise distinct positive integers in such a way that in each rectangle of size
1
×
2
1\times2
1
×
2
or
2
×
1
2\times1
2
×
1
the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most
201
9
4
?
2019^4?
201
9
4
?
divisibility in a table II
Is it possible to fill the cells of a table of size
2019
×
2019
2019\times2019
2019
×
2019
with pairwise distinct positive integers in such a way that in each rectangle of size
1
×
2
1\times2
1
×
2
or
2
×
1
2\times1
2
×
1
the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most
2019
?
2019?
2019
?
4
2
Hide problems
dwarfs in hats
99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats?
an isosceles triangle II
Let
D
D
D
be the midpoint of the base
B
C
BC
BC
of an isosceles triangle
A
B
C
,
ABC,
A
BC
,
E
E
E
be the point at the side
A
C
AC
A
C
such that
∠
C
D
E
=
6
0
∘
,
\angle CDE=60^\circ,
∠
C
D
E
=
6
0
∘
,
and
M
M
M
be the midpoint of
D
E
.
DE.
D
E
.
Prove that
∠
A
M
E
=
∠
B
M
D
.
\angle AME=\angle BMD.
∠
A
ME
=
∠
BM
D
.
3
3
Hide problems
an isosceles triangle I
Let
A
B
C
ABC
A
BC
be an isosceles triangle in which
∠
B
A
C
=
12
0
∘
,
\angle BAC=120^\circ,
∠
B
A
C
=
12
0
∘
,
D
D
D
be the midpoint of
B
C
,
BC,
BC
,
D
E
DE
D
E
be the altitude of triangle
A
D
C
,
ADC,
A
D
C
,
and
M
M
M
be the midpoint of
D
E
.
DE.
D
E
.
Prove that
B
M
=
3
A
M
.
BM=3AM.
BM
=
3
A
M
.
easy inequality I
Let
a
,
b
,
c
≥
0
a,b,c\ge0
a
,
b
,
c
≥
0
and
a
+
b
+
c
≥
3.
a+b+c\ge3.
a
+
b
+
c
≥
3.
Prove that
a
4
+
b
3
+
c
2
≥
a
3
+
b
2
+
c
.
a^4+b^3+c^2\ge a^3+b^2+c.
a
4
+
b
3
+
c
2
≥
a
3
+
b
2
+
c
.
weird tournament II
There were
2
n
,
2n,
2
n
,
n
≥
2
,
n\ge2,
n
≥
2
,
teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which
n
n
n
all the teams could end up with the same non-zero number of points?
2
2
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weird tournament I
There were
n
≥
2
n\ge2
n
≥
2
teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which
n
n
n
all the teams could end up with the same number of points?
easy inequality II
Let
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
and
a
b
c
≥
1.
abc\ge1.
ab
c
≥
1.
Prove that
a
4
+
b
3
+
c
2
≥
a
3
+
b
2
+
c
.
a^4+b^3+c^2\ge a^3+b^2+c.
a
4
+
b
3
+
c
2
≥
a
3
+
b
2
+
c
.
1
1
Hide problems
a bunch of lilac
A bunch of lilac consists of flowers with 4 or 5 petals. The number of flowers and the total number of petals are perfect squares. Can the number of flowers with 4 petals be divisible by the number of flowers with 5 petals?