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Problems
Contests
National and Regional Contests
Ukraine Contests
Other Ukrainian Contests
Kyiv Mathematical Festival
2010 Kyiv Mathematical Festival
2010 Kyiv Mathematical Festival
Part of
Kyiv Mathematical Festival
Subcontests
(5)
5
1
Hide problems
table filled with primes and products of primes
1) Cells of
8
×
8
8 \times 8
8
×
8
table contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer
a
a
a
from the table there exists integer written in the same row or in the same column such that it is not relatively prime with
a
a
a
. Find maximum possible number of prime integers in the table.2) Cells of
2
n
×
2
n
2n \times 2n
2
n
×
2
n
table,
n
≥
2
,
n \ge 2,
n
≥
2
,
contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer
a
a
a
from the table there exist integers written in the same row and in the same column such that they are not relatively prime with
a
a
a
. Find maximum possible number of prime integers in the table.
4
1
Hide problems
two games
1) The numbers
1
,
2
,
3
,
…
,
2010
1,2,3,\ldots,2010
1
,
2
,
3
,
…
,
2010
are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers
a
a
a
and
b
b
b
with
a
b
−
a
−
b
ab-a-b
ab
−
a
−
b
while the second player replaces them with
a
b
+
a
+
b
.
ab+a+b.
ab
+
a
+
b
.
The game ends when a single number remains on the blackboard. If this number is smaller than
1
⋅
2
⋅
3
⋅
…
⋅
2010
1\cdot2\cdot3\cdot\ldots\cdot2010
1
⋅
2
⋅
3
⋅
…
⋅
2010
then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy?2) The numbers
1
,
2
,
3
,
…
,
2010
1,2,3,\ldots,2010
1
,
2
,
3
,
…
,
2010
are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers
a
a
a
and
b
b
b
with
a
b
−
a
−
b
+
2
ab-a-b+2
ab
−
a
−
b
+
2
while the second player replaces them with
a
b
+
a
+
b
.
ab+a+b.
ab
+
a
+
b
.
The game ends when a single number remains on the blackboard. If this number is smaller than
1
⋅
2
⋅
3
⋅
…
⋅
2010
1\cdot2\cdot3\cdot\ldots\cdot2010
1
⋅
2
⋅
3
⋅
…
⋅
2010
then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy?
3
1
Hide problems
angle bisectors and radii
Let
O
O
O
be the circumcenter and
I
I
I
be the incenter of triangle
A
B
C
.
ABC.
A
BC
.
Prove that if
A
I
⊥
O
B
AI\perp OB
A
I
⊥
OB
and
B
I
⊥
O
C
BI\perp OC
B
I
⊥
OC
then
C
I
∥
O
A
CI\parallel OA
C
I
∥
O
A
.
2
1
Hide problems
sums of sums of digits
Denote by
S
(
n
)
S(n)
S
(
n
)
the sum of digits of integer
n
.
n.
n
.
Find 1)
S
(
3
)
+
S
(
6
)
+
S
(
9
)
+
…
+
S
(
300
)
;
S(3)+S(6)+S(9)+\ldots+S(300);
S
(
3
)
+
S
(
6
)
+
S
(
9
)
+
…
+
S
(
300
)
;
2)
S
(
3
)
+
S
(
6
)
+
S
(
9
)
+
…
+
S
(
3000
)
.
S(3)+S(6)+S(9)+\ldots+S(3000).
S
(
3
)
+
S
(
6
)
+
S
(
9
)
+
…
+
S
(
3000
)
.
1
1
Hide problems
Alice guesses an integer
Bob has picked positive integer
1
<
N
<
100
1<N<100
1
<
N
<
100
. Alice tells him some integer, and Bob replies with the remainder of division of this integer by
N
N
N
. What is the smallest number of integers which Alice should tell Bob to determine
N
N
N
for sure?