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International Contests
Tuymaada Olympiad
2003 Tuymaada Olympiad
2003 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(4)
2
2
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A quadrilateral with two equal sides. One angle to find.
In a quadrilateral
A
B
C
D
ABCD
A
BC
D
sides
A
B
AB
A
B
and
C
D
CD
C
D
are equal,
∠
A
=
15
0
∘
,
\angle A=150^\circ,
∠
A
=
15
0
∘
,
∠
B
=
4
4
∘
,
\angle B=44^\circ,
∠
B
=
4
4
∘
,
∠
C
=
7
2
∘
.
\angle C=72^\circ.
∠
C
=
7
2
∘
.
Perpendicular bisector of the segment
A
D
AD
A
D
meets the side
B
C
BC
BC
at point
P
.
P.
P
.
Find
∠
A
P
D
.
\angle APD.
∠
A
P
D
.
Proposed by F. Bakharev
Positive integers of the form 2x² - 3y² and 10xy - x² - y².
Which number is bigger : the number of positive integers not exceeding 1000000 that can be represented by the form
2
x
2
−
3
y
2
2x^{2}-3y^{2}
2
x
2
−
3
y
2
with integral
x
x
x
and
y
y
y
or that of positive integers not exceeding 1000000 that can be represented by the form
10
x
y
−
x
2
−
y
2
10xy-x^{2}-y^{2}
10
x
y
−
x
2
−
y
2
with integral
x
x
x
and
y
?
y?
y
?
Proposed by A. Golovanov
1
2
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A 2003x2004 rectangle, unit squares and rhombi.
A
2003
×
2004
2003\times 2004
2003
×
2004
rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices?Proposed by A. Golovanov
(\sum 1/sin(a_i))(\sum 1/cos(a_i)) <= 2*(\sum 1/sin(2a_i))².
Prove that for every
α
1
,
α
2
,
…
,
α
n
\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}
α
1
,
α
2
,
…
,
α
n
in the interval
(
0
,
π
/
2
)
(0,\pi/2)
(
0
,
π
/2
)
(
1
sin
α
1
+
1
sin
α
2
+
…
+
1
sin
α
n
)
(
1
cos
α
1
+
1
cos
α
2
+
…
+
1
cos
α
n
)
≤
\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq
(
sin
α
1
1
+
sin
α
2
1
+
…
+
sin
α
n
1
)
(
cos
α
1
1
+
cos
α
2
1
+
…
+
cos
α
n
1
)
≤
≤
2
(
1
sin
2
α
1
+
1
sin
2
α
2
+
…
+
1
sin
2
α
n
)
2
.
\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.
≤
2
(
sin
2
α
1
1
+
sin
2
α
2
1
+
…
+
sin
2
α
n
1
)
2
.
Proposed by A. Khrabrov
3
2
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Finite alphabet ==> finite set of words of finite lengths.
Alphabet
A
A
A
contains
n
n
n
letters.
S
S
S
is a set of words of finite length composed of letters of
A
A
A
. It is known that every infinite sequence of letters of
A
A
A
begins with one and only one word of
S
S
S
. Prove that the set
S
S
S
is finite.Proposed by F. Bakharev
Harmonic convex quadriteral and angles
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
we have
A
B
⋅
C
D
=
B
C
⋅
D
A
AB\cdot CD=BC\cdot DA
A
B
⋅
C
D
=
BC
⋅
D
A
and
2
∠
A
+
∠
C
=
18
0
∘
2\angle A+\angle C=180^\circ
2∠
A
+
∠
C
=
18
0
∘
. Point
P
P
P
lies on the circumcircle of triangle
A
B
D
ABD
A
B
D
and is the midpoint of the arc
B
D
BD
B
D
not containing
A
A
A
. It is known that the point
P
P
P
lies inside the quadrilateral
A
B
C
D
ABCD
A
BC
D
. Prove that
∠
B
C
A
=
∠
D
C
P
\angle BCA=\angle DCP
∠
BC
A
=
∠
D
CP
Proposed by S. Berlov
4
2
Hide problems
functional equation with continous function
Find all continuous functions
f
(
x
)
f(x)
f
(
x
)
defined for all
x
>
0
x>0
x
>
0
such that for every
x
x
x
,
y
>
0
y > 0
y
>
0
f
(
x
+
1
x
)
+
f
(
y
+
1
y
)
=
f
(
x
+
1
y
)
+
f
(
y
+
1
x
)
.
f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) .
f
(
x
+
x
1
)
+
f
(
y
+
y
1
)
=
f
(
x
+
y
1
)
+
f
(
y
+
x
1
)
.
Proposed by F. Petrov
Set of prime divisors of a_{n+1}=f(a_n), f polynomial.
Given are polynomial
f
(
x
)
f(x)
f
(
x
)
with non-negative integral coefficients and positive integer
a
.
a.
a
.
The sequence
{
a
n
}
\{a_{n}\}
{
a
n
}
is defined by
a
1
=
a
,
a_{1}=a,
a
1
=
a
,
a
n
+
1
=
f
(
a
n
)
.
a_{n+1}=f(a_{n}).
a
n
+
1
=
f
(
a
n
)
.
It is known that the set of primes dividing at least one of the terms of this sequence is finite. Prove that
f
(
x
)
=
c
x
k
f(x)=cx^{k}
f
(
x
)
=
c
x
k
for some non-negative integral
c
c
c
and
k
.
k.
k
.
Proposed by F. Petrov[hide="For those of you who liked this problem."] Check [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62259]this thread out.