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Problems
Contests
National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2009 Kyrgyzstan National Olympiad
2009 Kyrgyzstan National Olympiad
Part of
Kyrgyzstan National Olympiad
Subcontests
(9)
6
1
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Divisibility of a,b(KgNM2009)
Find all natural
a
,
b
a,b
a
,
b
such that
a
(
a
+
b
)
+
1
∣
(
a
+
b
)
(
b
+
1
)
−
1
\left. {a(a + b) + 1} \right|(a + b)(b + 1) - 1
a
(
a
+
b
)
+
1
∣
(
a
+
b
)
(
b
+
1
)
−
1
.
4
1
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Equation(KgNM2009)
Find all real
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
x
+
y
2
=
y
3
x + {y^2} = {y^3}
x
+
y
2
=
y
3
y
+
x
2
=
x
3
y + {x^2} = {x^3}
y
+
x
2
=
x
3
5
1
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Divisibility(KgNM2009)
Prove for all natural
n
n
n
that
40
n
⋅
n
!
∣
(
5
n
)
!
\left. {{{40}^n} \cdot n!} \right|(5n)!
40
n
⋅
n
!
∣
(
5
n
)!
1
1
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Estimating Area
a
,
b
,
c
a,b,c
a
,
b
,
c
are sides of triangle
A
B
C
ABC
A
BC
. For any choosen triple from (a \plus{} 1,b,c),(a,b \plus{} 1,c),(a,b,c \plus{} 1) there exist a triangle which sides are choosen triple. Find all possible values of area which triangle
A
B
C
ABC
A
BC
can take.
2
1
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Rational numbers
x
x
x
and
y
y
y
are real numbers.
A
)
A)
A
)
If it is known that x \plus{} y and x \plus{} y^2 are rational numbers, can we conclude that
x
x
x
and
y
y
y
are also rational numbers.
B
)
B)
B
)
If it is known that x \plus{} y , x \plus{} y^2 and x \plus{} y^3 are rational numbers, can we conclude that
x
x
x
and
y
y
y
are also rational numbers.
3
1
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If f(x^2 +x +3) +2*f(x^2 - 3x + 5) = 6x^2 - 10x +17, find f(2009)
For function
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
given that
f
(
x
2
+
x
+
3
)
+
2
⋅
f
(
x
2
−
3
x
+
5
)
=
6
x
2
−
10
x
+
17
f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17
f
(
x
2
+
x
+
3
)
+
2
⋅
f
(
x
2
−
3
x
+
5
)
=
6
x
2
−
10
x
+
17
, calculate
f
(
2009
)
f(2009)
f
(
2009
)
.
8
1
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Fun Eq
Does there exist a function
f
:
N
→
N
f: {\Bbb N} \to {\Bbb N}
f
:
N
→
N
such that f(f(n \minus{} 1)) \equal{} f(n \plus{} 1) \minus{} f(n) for all
n
>
2
n > 2
n
>
2
.
7
1
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Easy one
Does a^2 \plus{} b^2 \plus{} c^2 \leqslant 2(ab \plus{} bc \plus{} ca) hold for every
a
,
b
,
c
a,b,c
a
,
b
,
c
if it is known that a^4 \plus{} b^4 \plus{} c^4 \leqslant 2(a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 ).
9
1
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Old inequality
For any positive
a
1
,
a
2
,
.
.
.
,
a
n
a_1 ,a_2 ,...,a_n
a
1
,
a
2
,
...
,
a
n
prove that \frac {{a_1 }} {{a_2 \plus{} a_3 }} \plus{} \frac {{a_2 }} {{a_3 \plus{} a_4 }} \plus{} ... \plus{} \frac {{a_n }} {{a_1 \plus{} a_2 }} > \frac {n} {4} holds.