MathDB
Problems
Contests
National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2016 Kyrgyzstan National Olympiad
2016 Kyrgyzstan National Olympiad
Part of
Kyrgyzstan National Olympiad
Subcontests
(6)
6
1
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three pairwise tangent circles
Given three pairwise tangent equal circles
Ω
i
(
i
=
1
,
2
,
3
)
\Omega_i (i=1,2,3)
Ω
i
(
i
=
1
,
2
,
3
)
with radius
r
r
r
. The circle
Γ
\Gamma
Γ
touches the three circles internally (circumscribed about 3 circles).The three equal circles
ω
i
(
i
=
1
,
2
,
3
)
\omega_i (i=1,2,3)
ω
i
(
i
=
1
,
2
,
3
)
with radius
x
x
x
touches
Ω
i
\Omega_i
Ω
i
and
Ω
i
+
1
\Omega_{i+1}
Ω
i
+
1
externally (
Ω
4
=
Ω
1
\Omega_4= \Omega_1
Ω
4
=
Ω
1
) and touches
Γ
\Gamma
Γ
internally.Find
x
x
x
in terms of
r
r
r
5
1
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polynomials
Given two monic polynomials
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
with degrees 2016.
P
(
x
)
=
Q
(
x
)
P(x)=Q(x)
P
(
x
)
=
Q
(
x
)
has no real root. Prove that P(x)=Q(x+1) has at least one real root.
4
1
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combinatorics problem
Aibek wrote 6 letters to 6 different person.In how many ways can he send the letters to them,such that no person gets his letter.
3
1
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find the area,
Given a
△
A
B
C
\triangle ABC
△
A
BC
with sides
a
,
b
,
c
.
a,b,c.
a
,
b
,
c
.
Three tangents are drawn to the incircle of
△
A
B
C
\triangle ABC
△
A
BC
parallel to the sides of
△
A
B
C
\triangle ABC
△
A
BC
.These tangents cut three new little triangles.Three little incircles are drawn into new little triangles.Find the sum of the area of these 4 incircles.
2
1
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nice combinatorics problem
The number
N
N
N
consists only
2
′
s
2's
2
′
s
and
1
′
s
1's
1
′
s
in its decimal representation.We know that,after deleting digits from N,we can get any number consisting
9999
9999
9999
-
1
′
s
1's
1
′
s
and
o
n
e
one
o
n
e
-
2
′
s
2's
2
′
s
in its decimal representation.Find the least number of digits in the decimal representation of N
1
1
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a+b+c=0,then?
If
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
,then find the value of
(
a
b
−
c
+
b
c
−
a
+
c
a
−
b
)
(
b
−
c
a
+
c
−
a
b
+
a
−
b
c
)
(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})
(
b
−
c
a
+
c
−
a
b
+
a
−
b
c
)
(
a
b
−
c
+
b
c
−
a
+
c
a
−
b
)