MathDB
Problems
Contests
National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2010 Kyrgyzstan National Olympiad
2010 Kyrgyzstan National Olympiad
Part of
Kyrgyzstan National Olympiad
Subcontests
(8)
6
1
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p*k with 1's (KgNM2010)
Let
p
p
p
- a prime, where
p
>
11
p> 11
p
>
11
. Prove that there exists a number
k
k
k
such that the product
p
⋅
k
p \cdot k
p
⋅
k
can be written in the decimal system with only ones.
2
1
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coprime numbers(KgNM2010)
Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.
1
1
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ineq
Given that
a
,
b
,
c
>
0
a,b,c > 0
a
,
b
,
c
>
0
and
a
+
b
+
c
=
1
a + b + c = 1
a
+
b
+
c
=
1
. Prove that
a
b
a
b
+
c
+
b
c
b
c
+
a
+
c
a
c
a
+
b
⩽
3
2
\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}
ab
+
c
ab
+
b
c
+
a
b
c
+
c
a
+
b
c
a
⩽
2
3
.
5
1
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polynomial equation (KgNM2010)
Let
k
k
k
be a constant number larger than
1
1
1
. Find all polynomials
P
(
x
)
P(x)
P
(
x
)
such that
P
(
x
k
)
=
(
P
(
x
)
)
k
P({x^k}) = {\left( {P(x)} \right)^k}
P
(
x
k
)
=
(
P
(
x
)
)
k
for all real
x
x
x
.
7
1
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divisible triples(KgNM2010)
Find all natural triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
, such that:
a
−
)
a
≤
b
≤
c
a - )\,a \le b \le c
a
−
)
a
≤
b
≤
c
b
−
)
(
a
,
b
,
c
)
=
1
b - )\,(a,b,c) = 1
b
−
)
(
a
,
b
,
c
)
=
1
c
−
)
a
2
b
∣
a
3
+
b
3
+
c
3
,
b
2
c
∣
a
3
+
b
3
+
c
3
,
c
2
a
∣
a
3
+
b
3
+
c
3
c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}
c
−
)
a
2
b
a
3
+
b
3
+
c
3
,
b
2
c
a
3
+
b
3
+
c
3
,
c
2
a
a
3
+
b
3
+
c
3
.
8
1
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none-negative(KgNM2010)
Solve in none-negative integers
x
3
+
7
x
2
+
35
x
+
27
=
y
3
{x^3} + 7{x^2} + 35x + 27 = {y^3}
x
3
+
7
x
2
+
35
x
+
27
=
y
3
.
3
1
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friendship(KgNM2010)
At the meeting, each person is familiar with 22 people. If two persons
A
A
A
and
B
B
B
know each with one another, among the remaining people they do not have a common friend. For each pair individuals
A
A
A
and
B
B
B
are not familiar with each other, there are among the remaining six common acquaintances. How many people were at the meeting?
4
1
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distance product(KgNM2010)
Point
O
O
O
is chosen in a triangle
A
B
C
ABC
A
BC
such that
d
a
,
d
b
,
d
c
{d_a},{d_b},{d_c}
d
a
,
d
b
,
d
c
are distance from point
O
O
O
to sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively. Find position of point
O
O
O
so that product
d
a
⋅
d
b
⋅
d
c
{d_a} \cdot {d_b} \cdot {d_c}
d
a
⋅
d
b
⋅
d
c
becomes maximum.