MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan IZHO TST
2020 Azerbaijan IZHO TST
2020 Azerbaijan IZHO TST
Part of
Azerbaijan IZHO TST
Subcontests
(6)
1
1
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combinatorics problem
Let
F
F
F
be the set of all
n
−
t
u
p
l
e
s
n-tuples
n
−
t
u
pl
es
(
A
1
,
A
2
,
…
,
A
n
)
(A_1,A_2,…,A_n)
(
A
1
,
A
2
,
…
,
A
n
)
such that each
A
i
A_i
A
i
is a subset of
1
,
2
,
…
,
2019
{1,2,…,2019}
1
,
2
,
…
,
2019
. Let
∣
A
∣
\mid{A}\mid
∣
A
∣
denote the number of elements o the set
A
A
A
. Find
∑
(
A
1
,
…
,
A
n
)
∈
F
∣
A
1
∪
A
2
∪
.
.
.
∪
A
n
∣
\sum_{(A_1,…,A_n)\in{F}}^{}\mid{A_1\cup{A_2}\cup...\cup{A_n}}\mid
∑
(
A
1
,
…
,
A
n
)
∈
F
∣
A
1
∪
A
2
∪
...
∪
A
n
∣
2
1
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Geometry Problem
Consider two circles
k
1
,
k
2
k_1,k_2
k
1
,
k
2
touching at point
T
T
T
. A line touches
k
2
k_2
k
2
at point
X
X
X
and intersects
k
1
k_1
k
1
at points
A
,
B
A,B
A
,
B
where
B
B
B
lies between
A
A
A
and
X
X
X
.Let
S
S
S
be the second intersection point of
k
1
k_1
k
1
with
X
T
XT
XT
. On the arc \overarc{TS} not containing
A
A
A
and
B
B
B
, a point
C
C
C
is choosen. Let
C
Y
CY
C
Y
be the tangent line to
k
2
k_2
k
2
with
Y
∈
k
2
Y\in{k_2}
Y
∈
k
2
, such that the segment
C
Y
CY
C
Y
doesn't intersect the segment
S
T
ST
ST
.If
I
=
X
Y
∩
S
C
I=XY\cap{SC}
I
=
X
Y
∩
SC
, prove that :
(
a
)
(a)
(
a
)
the points
C
,
T
,
Y
,
I
C,T,Y,I
C
,
T
,
Y
,
I
are concyclic.
(
b
)
(b)
(
b
)
I
I
I
is the
A
−
e
x
c
e
n
t
e
r
A-excenter
A
−
e
x
ce
n
t
er
of
△
A
B
C
\triangle ABC
△
A
BC
3
1
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Function problem
Find all functions
u
:
R
→
R
u:R\rightarrow{R}
u
:
R
→
R
for which there exists a strictly monotonic function
f
:
R
→
R
f:R\rightarrow{R}
f
:
R
→
R
such that
f
(
x
+
y
)
=
f
(
x
)
u
(
y
)
+
f
(
y
)
f(x+y)=f(x)u(y)+f(y)
f
(
x
+
y
)
=
f
(
x
)
u
(
y
)
+
f
(
y
)
for all
x
,
y
∈
R
x,y\in{\mathbb{R}}
x
,
y
∈
R
4
1
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Hard NT Problem
Consider an odd prime number
p
p
p
and
p
p
p
consecutive positive integers
m
1
,
m
2
,
…
,
m
p
m_1,m_2,…,m_p
m
1
,
m
2
,
…
,
m
p
. Choose a permutation
σ
\sigma
σ
of
1
,
2
,
…
,
p
1,2,…,p
1
,
2
,
…
,
p
. Show that there exist two different numbers
k
,
l
∈
(
1
,
2
,
…
,
p
)
k,l\in{(1,2,…,p)}
k
,
l
∈
(
1
,
2
,
…
,
p
)
such that
p
∣
m
k
.
m
σ
(
k
)
−
m
l
.
m
σ
(
l
)
p\mid{m_k.m_{\sigma(k)}-m_l.m_{\sigma(l)}}
p
∣
m
k
.
m
σ
(
k
)
−
m
l
.
m
σ
(
l
)
6
1
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Sequence problem
Define a sequence
a
n
n
≥
1
{{a_n}}_{n\ge1}
a
n
n
≥
1
such that
a
1
=
1
a_1=1
a
1
=
1
,
a
2
=
2
a_2=2
a
2
=
2
and
a
n
+
1
a_{n+1}
a
n
+
1
is the smallest positive integer
m
m
m
such that
m
m
m
hasn't yet occurred in the sequence and also
g
c
d
(
m
,
a
n
)
≠
1
gcd(m,a_n)\neq{1}
g
c
d
(
m
,
a
n
)
=
1
. Show that all positive integers occur in the sequence.
5
1
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Hard one ...
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers such that
x
4
+
y
4
+
z
4
=
1
x^4+y^4+z^4=1
x
4
+
y
4
+
z
4
=
1
. Determine with proof the minimum value of
x
3
1
−
x
8
+
y
3
1
−
y
8
+
z
3
1
−
z
8
\frac{x^3}{1-x^8}+\frac{y^3}{1-y^8}+\frac{z^3}{1-z^8}
1
−
x
8
x
3
+
1
−
y
8
y
3
+
1
−
z
8
z
3