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Problems
Contests
International Contests
International Zhautykov Olympiad
2007 International Zhautykov Olympiad
2007 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
3
2
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infinity of positive integers satisfying...
Show that there are an infinity of positive integers
n
n
n
such that
2
n
+
3
n
2^{n}+3^{n}
2
n
+
3
n
is divisible by
n
2
n^{2}
n
2
.
super nice 6-th problem, an interesting result.
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon and it`s diagonals have one common point
M
M
M
. It is known that the circumcenters of triangles
M
A
B
,
M
B
C
,
M
C
D
,
M
D
E
,
M
E
F
,
M
F
A
MAB,MBC,MCD,MDE,MEF,MFA
M
A
B
,
MBC
,
MC
D
,
M
D
E
,
MEF
,
MF
A
lie on a circle. Show that the quadrilaterals
A
B
D
E
,
B
C
E
F
,
C
D
F
A
ABDE,BCEF,CDFA
A
B
D
E
,
BCEF
,
C
D
F
A
have equal areas.
2
2
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convex quadrilateral suggesting a trigonometric solution
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, with
∠
B
A
C
=
∠
D
A
C
\angle BAC=\angle DAC
∠
B
A
C
=
∠
D
A
C
and
M
M
M
a point inside such that
∠
M
B
A
=
∠
M
C
D
\angle MBA=\angle MCD
∠
MB
A
=
∠
MC
D
and
∠
M
B
C
=
∠
M
D
C
\angle MBC=\angle MDC
∠
MBC
=
∠
M
D
C
. Show that the angle
∠
A
D
C
\angle ADC
∠
A
D
C
is equal to
∠
B
M
C
\angle BMC
∠
BMC
or
∠
A
M
B
\angle AMB
∠
A
MB
.
a partition in three non-empty subsets...
The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets
(
A
∪
B
∪
C
)
(A\cup B\cup C)
(
A
∪
B
∪
C
)
. a) show that there exists a triangle of side-lengths
a
,
b
,
c
a,b,c
a
,
b
,
c
, such that
a
∈
A
,
b
∈
B
,
c
∈
C
a\in A, b\in B, c\in C
a
∈
A
,
b
∈
B
,
c
∈
C
. b) does it always happen that there exists a right triangle with the above property ?
1
2
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easy n x n table problem
There are given
111
111
111
coins and a
n
×
n
n\times n
n
×
n
table divided into unit cells. This coins are placed inside the unit cells (one unit cell may contain one coin, many coins, or may be empty), such that the difference between the number of coins from two neighbouring cells (that have a common edge) is
1
1
1
. Find the maximal
n
n
n
for this to be possible.
functional ecuation, a bit strange, but not difficult
Does there exist a function
f
:
R
→
R
f: \mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
such that
f
(
x
+
f
(
y
)
)
=
f
(
x
)
+
sin
y
f(x+f(y))=f(x)+\sin y
f
(
x
+
f
(
y
))
=
f
(
x
)
+
sin
y
, for all reals
x
,
y
x,y
x
,
y
?