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Problems
Contests
International Contests
International Zhautykov Olympiad
2006 International Zhautykov Olympiad
2006 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
3
2
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Good board filled with 0s and 1s
Let
m
≥
n
≥
4
m\geq n\geq 4
m
≥
n
≥
4
be two integers. We call a
m
×
n
m\times n
m
×
n
board filled with 0's or 1's good if 1) not all the numbers on the board are 0 or 1; 2) the sum of all the numbers in
3
×
3
3\times 3
3
×
3
sub-boards is the same; 3) the sum of all the numbers in
4
×
4
4\times 4
4
×
4
sub-boards is the same. Find all
m
,
n
m,n
m
,
n
such that there exists a good
m
×
n
m\times n
m
×
n
board.
Convex hexagon with AD=BC+EF, BE=AF+CD, CF=DE+AB
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon such that AD \equal{} BC \plus{} EF, BE \equal{} AF \plus{} CD, CF \equal{} DE \plus{} AB. Prove that: \frac {AB}{DE} \equal{} \frac {CD}{AF} \equal{} \frac {EF}{BC}.
2
2
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Classical triangle geometry
Let
A
B
C
ABC
A
BC
be a triangle and
K
K
K
and
L
L
L
be two points on
(
A
B
)
(AB)
(
A
B
)
,
(
A
C
)
(AC)
(
A
C
)
such that BK \equal{} CL and let P \equal{} CK\cap BL. Let the parallel through
P
P
P
to the interior angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersect
A
C
AC
A
C
in
M
M
M
. Prove that CM \equal{} AB.
Real numbers with a+b+c+d=0
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be real numbers with sum 0. Prove the inequality: (ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd)^2 \plus{} 12\geq 6(abc \plus{} abd \plus{} acd \plus{} bcd).
1
2
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\phi(n)
Solve in positive integers the equation n \equal{} \varphi(n) \plus{} 402 , where
φ
(
n
)
\varphi(n)
φ
(
n
)
is the number of positive integers less than
n
n
n
having no common prime factors with
n
n
n
.
Pile with 100 stones
In a pile you have 100 stones. A partition of the pile in
k
k
k
piles is good if: 1) the small piles have different numbers of stones; 2) for any partition of one of the small piles in 2 smaller piles, among the k \plus{} 1 piles you get 2 with the same number of stones (any pile has at least 1 stone). Find the maximum and minimal values of
k
k
k
for which this is possible.