MathDB
Problems
Contests
International Contests
International Zhautykov Olympiad
2008 International Zhautykov Olympiad
2008 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
2
2
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Good polynomials
A polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable
x
x
x
) with integer coefficients.For example,the polynomials x^3 \minus{} 1 and 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3 are good. a)Is the polynomial P(x) \equal{} 3x \plus{} 3x^7 good? b)Is the polynomial P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008} good? Justify your answers.
Nice,but easy
Let
A
1
A
2
A_1A_2
A
1
A
2
be the external tangent line to the nonintersecting cirlces
ω
1
(
O
1
)
\omega_1(O_1)
ω
1
(
O
1
)
and
ω
2
(
O
2
)
\omega_2(O_2)
ω
2
(
O
2
)
,
A
1
∈
ω
1
A_1\in\omega_1
A
1
∈
ω
1
,
A
2
∈
ω
2
A_2\in\omega_2
A
2
∈
ω
2
.Points
K
K
K
is the midpoint of
A
1
A
2
A_1A_2
A
1
A
2
.And
K
B
1
KB_1
K
B
1
and
K
B
2
KB_2
K
B
2
are tangent lines to
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
,respectvely(
B
1
≠
A
1
B_1\neq A_1
B
1
=
A
1
,
B
2
≠
A
2
B_2\neq A_2
B
2
=
A
2
).Lines
A
1
B
1
A_1B_1
A
1
B
1
and
A
2
B
2
A_2B_2
A
2
B
2
meet in point
L
L
L
,and lines
K
L
KL
K
L
and
O
1
O
2
O_1O_2
O
1
O
2
meet in point
P
P
P
. Prove that points
B
1
,
B
2
,
P
B_1,B_2,P
B
1
,
B
2
,
P
and
L
L
L
are concyclic.
1
2
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Nice geometry problem
Points
K
,
L
,
M
,
N
K,L,M,N
K
,
L
,
M
,
N
are repectively the midpoints of sides
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
in a convex quadrliateral
A
B
C
D
ABCD
A
BC
D
.Line
K
M
KM
K
M
meets dioganals
A
C
AC
A
C
and
B
D
BD
B
D
at points
P
P
P
and
Q
Q
Q
,respectively.Line
L
N
LN
L
N
meets dioganals
A
C
AC
A
C
and
B
D
BD
B
D
at points
R
R
R
and
S
S
S
,respectively. Prove that if AP\cdot PC\equal{}BQ\cdot QD,then AR\cdot RC\equal{}BS\cdot SD.
The sum of digits
For each positive integer
n
n
n
,denote by
S
(
n
)
S(n)
S
(
n
)
the sum of all digits in decimal representation of
n
n
n
. Find all positive integers
n
n
n
,such that n\equal{}2S(n)^3\plus{}8.
3
2
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Hard combinatorics problem
Let A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N} , 1\leq a_i\leq i \plus{} 1 for each i \equal{} 1,2\dots,8\}.A subset
X
⊂
A
X\subset A
X
⊂
A
is called sparse if for each two distinct elements
(
a
1
,
…
,
a
8
)
(a_1,\dots,a_8)
(
a
1
,
…
,
a
8
)
,
(
b
1
,
…
,
b
8
)
∈
X
(b_1,\dots,b_8)\in X
(
b
1
,
…
,
b
8
)
∈
X
,there exist at least three indices
i
i
i
,such that
a
i
≠
b
i
a_i\neq b_i
a
i
=
b
i
. Find the maximal possible number of elements in a sparse subset of set
A
A
A
.
Three variables, cyclic
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers for which abc \equal{} 1. Prove that \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}.