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National and Regional Contests
Ukraine Contests
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Kyiv Mathematical Festival
2009 Kyiv Mathematical Festival
2009 Kyiv Mathematical Festival
Part of
Kyiv Mathematical Festival
Subcontests
(6)
5
3
Hide problems
sum of distances of P from sides <1, prove that (ABC)<= 1 /\sqrt3
Assume that a triangle
A
B
C
ABC
A
BC
satisfies the following property: For any point from the triangle, the sum of distances from
D
D
D
to the lines
A
B
,
B
C
AB,BC
A
B
,
BC
and
C
A
CA
C
A
is less than
1
1
1
. Prove that the area of the triangle is less than or equal to
1
3
\frac{1}{\sqrt3}
3
1
a_n<= a_{n+1}<= a_n+5, n >=1 and n divides a_n
The sequence of positive integers
{
a
n
,
n
≥
1
}
\{a_n, n\ge 1\}
{
a
n
,
n
≥
1
}
is such that
a
n
≤
a
n
+
1
≤
a
n
+
5
a_n\le a_{n+1}\le a_n+5
a
n
≤
a
n
+
1
≤
a
n
+
5
and
a
n
a_n
a
n
is divisible by
n
n
n
for all
n
≥
1
n \ge 1
n
≥
1
. What are the possible values of
a
1
a_1
a
1
?
inequality of sequence by recursion
a) Suppose that a sequence of numbers
x
1
,
x
2
,
x
3
,
.
.
.
x_1,x_2,x_3,...
x
1
,
x
2
,
x
3
,
...
satisfies the inequality
x
n
−
2
x
n
+
1
+
x
n
+
2
≤
0
x_n-2x_{n+1}+x_{n+2} \le 0
x
n
−
2
x
n
+
1
+
x
n
+
2
≤
0
for any
n
n
n
. Moreover
x
o
=
1
,
x
20
=
9
,
x
200
=
6
x_o=1,x_{20}=9,x_{200}=6
x
o
=
1
,
x
20
=
9
,
x
200
=
6
. What is the maximal value of
x
2009
x_{2009}
x
2009
can be? b) Suppose that a sequence of numbers
x
1
,
x
2
,
x
3
,
.
.
.
x_1,x_2,x_3,...
x
1
,
x
2
,
x
3
,
...
satisfies the inequality
2
x
n
−
3
x
n
+
1
+
x
n
+
2
≤
0
2x_n-3x_{n+1}+x_{n+2} \le 0
2
x
n
−
3
x
n
+
1
+
x
n
+
2
≤
0
for any
n
n
n
. Moreover
x
o
=
1
,
x
1
=
2
,
x
3
=
1
x_o=1,x_1=2,x_3=1
x
o
=
1
,
x
1
=
2
,
x
3
=
1
. Can
x
2009
x_{2009}
x
2009
be greater then
0
,
678
0,678
0
,
678
?
6
1
Hide problems
inequalitiy with subsets of of (-1,1)
Let
{
a
1
,
.
.
.
,
a
n
}
⊂
{
−
1
,
1
}
\{a_1,...,a_n\}\subset \{-1,1\}
{
a
1
,
...
,
a
n
}
⊂
{
−
1
,
1
}
and
a
>
0
a>0
a
>
0
. Denote by
X
X
X
and
Y
Y
Y
the number of collections
{
ε
1
,
.
.
.
,
ε
n
}
⊂
{
−
1
,
1
}
\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}
{
ε
1
,
...
,
ε
n
}
⊂
{
−
1
,
1
}
, such that
m
a
x
1
≤
k
≤
n
(
ε
1
a
1
+
.
.
.
+
ε
k
a
k
)
>
α
max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha
ma
x
1
≤
k
≤
n
(
ε
1
a
1
+
...
+
ε
k
a
k
)
>
α
and
ε
1
a
1
+
.
.
.
+
ε
n
a
n
>
a
\varepsilon_1a_1+...+\varepsilon_na_n>a
ε
1
a
1
+
...
+
ε
n
a
n
>
a
respectively. Prove that
X
≤
2
Y
X\le 2Y
X
≤
2
Y
.
3
2
Hide problems
coloring plane with 2009 colors under conditions of distances
Let
A
B
AB
A
B
be a segment of a plane. Is it possible to paint the plane in
2009
2009
2009
colors in such a way that both of the following conditions are satisfied? 1) Every two points of the same color can be connected by a polygonal line. 2) For any point
C
C
C
of
A
B
AB
A
B
, every
n
∈
N
n \in N
n
∈
N
and every
k
∈
{
1
,
2
,
3
,
.
.
.
,
2009
}
k\in \{1,2,3,...,2009\}
k
∈
{
1
,
2
,
3
,
...
,
2009
}
, there exists a point
D
D
D
, painted in
k
k
k
-th color such that the length of
C
D
CD
C
D
is less than
0
,
0...01
0,0...01
0
,
0...01
, where all the zeros after the decimal point are exactly
n
n
n
.
sum of least distances of n points in a equilateral triangle of side 1
Points
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
are selected from the equilateral triangle with a side that is equal to
1
1
1
. Denote by
d
k
d_k
d
k
the least distance from
A
k
A_k
A
k
to all other selected points. Prove that
d
1
2
+
.
.
.
+
d
n
2
≤
3
,
5
d_1^2+...+d_n^2 \le 3,5
d
1
2
+
...
+
d
n
2
≤
3
,
5
.
4
1
Hide problems
2 convex polygons inside a square of side 1, perimeter question
Two convex polygons can be placed into a square with the side
1
1
1
without intersection. Prove that at least one polygon has the perimeter that is less than or equal to
3
,
5
3,5
3
,
5
.
2
1
Hide problems
x+y+z ? x^2+y^2+z^2 ? x^3+y^3+z^3 , in positive
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive numebrs such that
x
+
y
+
z
≤
x
3
+
y
3
+
z
3
x+y+z\le x^3+y^3+z^3
x
+
y
+
z
≤
x
3
+
y
3
+
z
3
. Is it true that a)
x
2
+
y
2
+
z
2
≤
x
3
+
y
3
+
z
3
x^2+y^2+z^2 \le x^3+y^3+z^3
x
2
+
y
2
+
z
2
≤
x
3
+
y
3
+
z
3
? b)
x
+
y
+
z
≤
x
2
+
y
2
+
z
2
x+y+z\le x^2+y^2+z^2
x
+
y
+
z
≤
x
2
+
y
2
+
z
2
?
1
2
Hide problems
last digit of sum of divisors of (3*2009)^{((2* 2009)^{2009}-1)}
Let
X
X
X
be the sum of all divisors of the number
(
3
⋅
2009
)
(
(
2
⋅
2009
)
2009
−
1
)
(3\cdot 2009)^{((2\cdot 2009)^{2009}-1)}
(
3
⋅
2009
)
((
2
⋅
2009
)
2009
−
1
)
. Find the last digit of
X
X
X
.
cos(x-pi/4)+tgx)^3=54 sin^2x, trigonometric equation
Solve the equation
(
2
c
o
s
(
x
−
π
4
)
+
t
g
x
)
3
=
54
s
i
n
2
x
\big(2cos(x-\frac{\pi}{4})+tgx\big)^3=54 sin^2x
(
2
cos
(
x
−
4
π
)
+
t
gx
)
3
=
54
s
i
n
2
x
,
x
∈
[
0
,
π
2
)
x\in \big[0,\frac{\pi}{2}\big)
x
∈
[
0
,
2
π
)