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National and Regional Contests
Ukraine Contests
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Kyiv Mathematical Festival
2011 Kyiv Mathematical Festival
2011 Kyiv Mathematical Festival
Part of
Kyiv Mathematical Festival
Subcontests
(5)
2
2
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represent 2011... 2011 as a product of some number and sum of its digits
Is it possible to represent number
2011...2011
2011... 2011
2011...2011
, where number
2011
2011
2011
is written
20112011
20112011
20112011
times, as a product of some number and sum of its digits?
max of (a -b^2)(b - a^2), where 0 \<= a,b <= 1
Find maximum of the expression
(
a
−
b
2
)
(
b
−
a
2
)
(a -b^2)(b - a^2)
(
a
−
b
2
)
(
b
−
a
2
)
, where
0
≤
a
,
b
≤
1
0 \le a,b \le 1
0
≤
a
,
b
≤
1
.
4
1
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erasing 2 of those numbers x,y of 1,2,..., n and write 2x - y instead
There are
n
≥
2
n \ge 2
n
≥
2
numbers on the blackboard:
1
,
2
,
.
.
.
,
n
1, 2,..., n
1
,
2
,
...
,
n
. It is permitted to erase two of those numbers
x
,
y
x,y
x
,
y
and write
2
x
−
y
2x - y
2
x
−
y
instead. Find all values of
n
n
n
such that it is possible to leave number
0
0
0
on the blackboard after
n
−
1
n - 1
n
−
1
such procedures.
5
3
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7 pupils with 20 candies, 5 candies of 4 different kinds, each max1 of each kind
7
7
7
pupils has been given
20
20
20
candies,
5
5
5
candies of
4
4
4
different kinds, so that each pupil has no more then one candy of each kind. Prove that there are two pupils that have three or more pairs of candies of the same kind.
3 segments of length 1 and a circle of radius less than \sqrt3 / 3 ...
Pete claims that he can draw
3
3
3
segments of length
1
1
1
and a circle of radius less than
3
/
3
\sqrt3 / 3
3
/3
on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of
3
3
3
segments. Is Pete right?
4 segments of length 1 and a circle of radius less than \sqrt3 /3 ...
Pete claims that he can draw
4
4
4
segments of length
1
1
1
and a circle of radius less than
3
/
3
\sqrt3 /3
3
/3
on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of
4
4
4
segments. Is Pete right?
3
2
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cutting a quadrilateral into 2 isosceles triangles in 2 different ways
Quadrilateral can be cut into two isosceles triangles in two different ways. a) Can this quadrilateral be nonconvex? b) If given quadrilateral is convex, is it necessarily a rhomb?
square on side of a right triangle, concurrency from Kyiv
A
B
C
ABC
A
BC
is right triangle with right angle near vertex
B
,
M
B, M
B
,
M
is the midpoint of
A
C
AC
A
C
. The square
B
K
L
M
BKLM
B
K
L
M
is built on
B
M
BM
BM
, such that segments
M
L
ML
M
L
and
B
C
BC
BC
intersect. Segment
A
L
AL
A
L
intersects
B
C
BC
BC
in point
E
E
E
. Prove that lines
A
B
,
C
L
AB,CL
A
B
,
C
L
and
K
E
KE
K
E
intersect in one point.
1
2
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equation with gcd and lcm of two numbers
Solve the equation
m
n
=
mn =
mn
=
(gcd(
m
,
n
m,n
m
,
n
))
2
^2
2
+ lcm(
m
,
n
m, n
m
,
n
) in positive integers, where gcd(
m
,
n
m, n
m
,
n
) – greatest common divisor of
m
,
n
m,n
m
,
n
, and lcm(
m
,
n
m, n
m
,
n
) – least common multiple of
m
,
n
m,n
m
,
n
.
exponential equation with gcm and lcm of 2 numbers
Solve the equation
m
g
c
d
(
m
,
n
)
=
n
l
c
m
(
m
,
n
)
m^{gcd(m,n)} = n^{lcm(m,n)}
m
g
c
d
(
m
,
n
)
=
n
l
c
m
(
m
,
n
)
in positive integers, where gcd(
m
,
n
m, n
m
,
n
) – greatest common divisor of
m
,
n
m,n
m
,
n
, and lcm(
m
,
n
m, n
m
,
n
) – least common multiple of
m
,
n
m,n
m
,
n
.