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International Contests
Tuymaada Olympiad
2009 Tuymaada Olympiad
2009 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(4)
4
4
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3
4
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2
3
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P(x) is a quadratic trinomial
P
(
x
)
P(x)
P
(
x
)
is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence
P
(
1
)
P(1)
P
(
1
)
,
P
(
2
)
P(2)
P
(
2
)
,
P
(
3
)
P(3)
P
(
3
)
,
…
?
\dots?
…
?
Proposed by A. Golovanov
Perpendicular to bases drawn through P meets line BQ at K
M
M
M
is the midpoint of base
B
C
BC
BC
in a trapezoid
A
B
C
D
ABCD
A
BC
D
. A point
P
P
P
is chosen on the base
A
D
AD
A
D
. The line
P
M
PM
PM
meets the line
C
D
CD
C
D
at a point
Q
Q
Q
such that
C
C
C
lies between
Q
Q
Q
and
D
D
D
. The perpendicular to the bases drawn through
P
P
P
meets the line
B
Q
BQ
BQ
at
K
K
K
. Prove that \angle QBC \equal{} \angle KDA. Proposed by S. Berlov
A necklace consists of 100 blue and several red beads
A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace? Proposed by A. Golovanov
1
3
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Naoki and Richard in turn put chips in free squares
All squares of a
20
×
20
20\times 20
20
×
20
table are empty. Misha* and Sasha** in turn put chips in free squares (Misha* begins). The player after whose move there are four chips on the intersection of two rows and two columns wins. Which of the players has a winning strategy? Proposed by A. Golovanov US Name Conversions: Misha*: Naoki Sasha**: Richard
Fractional part of the product of every two of them is 0.5
Three real numbers are given. Fractional part of the product of every two of them is
1
2
1\over 2
2
1
. Prove that these numbers are irrational. Proposed by A. Golovanov
Magician asked a spectator to think of a three-digit number
A magician asked a spectator to think of a three-digit number
a
b
c
‾
\overline{abc}
ab
c
and then to tell him the sum of numbers
a
c
b
‾
\overline{acb}
a
c
b
,
b
a
c
‾
\overline{bac}
ba
c
,
b
c
a
‾
\overline{bca}
b
c
a
,
c
a
b
‾
\overline{cab}
c
ab
, and
c
b
a
‾
\overline{cba}
c
ba
. He claims that when he knows this sum he can determine the original number. Is that so?