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Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan EGMO TST
2017 Azerbaijan EGMO TST
2017 Azerbaijan EGMO TST
Part of
Azerbaijan EGMO TST
Subcontests
(4)
1
2
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The finite integer set which is closed
M
M
M
is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of
M
.
M.
M
.
How many elements can
M
M
M
have at most?
PB = PA + PC, equilateral , iff form (2009 Indonedia MO Province P2 q4 OSP)
Given an equilateral triangle
A
B
C
ABC
A
BC
and a point
P
P
P
so that the distances
P
P
P
to
A
A
A
and to
C
C
C
are not farther than the distances
P
P
P
to
B
B
B
. Prove that
P
B
=
P
A
+
P
C
PB = PA + PC
PB
=
P
A
+
PC
if and only if
P
P
P
lies on the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
.
3
2
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Day 1 Problem 3
In
△
\bigtriangleup
△
A
B
C
ABC
A
BC
B
L
BL
B
L
is bisector. Arbitrary point
M
M
M
on segment
C
L
CL
C
L
is chosen. Tangent to
⊙
\odot
⊙
(
A
B
C
)
(ABC)
(
A
BC
)
at
B
B
B
intersects
C
A
CA
C
A
at
P
P
P
. Tangents to
⊙
\odot
⊙
B
L
M
BLM
B
L
M
at
B
B
B
and
M
M
M
intersect at point
Q
Q
Q
. Prove that
P
Q
PQ
PQ
∥
\parallel
∥
B
L
BL
B
L
.
Real roots of the equation P(P(x))=0
The degree of the polynomial
P
(
x
)
P(x)
P
(
x
)
is
2017.
2017.
2017.
Prove that the number of distinct real roots of the equation
P
(
P
(
x
)
)
=
0
P(P(x)) = 0
P
(
P
(
x
))
=
0
is not less than the number of distinct real roots of the equation
P
(
x
)
=
0.
P(x) = 0.
P
(
x
)
=
0.
2
2
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Sequence
Let
(
a
n
)
n
≥
0
(a_n)_n\geq 0
(
a
n
)
n
≥
0
and
a
m
+
n
+
a
m
−
n
=
1
2
(
a
2
m
+
a
2
n
)
a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n})
a
m
+
n
+
a
m
−
n
=
2
1
(
a
2
m
+
a
2
n
)
for every
m
≥
n
≥
0.
m\geq n\geq0.
m
≥
n
≥
0.
If
a
1
=
1
,
a_1=1,
a
1
=
1
,
then find the value of
a
2007
.
a_{2007}.
a
2007
.
four numbers two deleted and replaced with a+b and ab
Four numbers are written on the board:
1
,
3
,
6
,
10.
1, 3, 6, 10.
1
,
3
,
6
,
10.
Each time two arbitrary numbers,
a
a
a
and
b
b
b
are deleted, and numbers
a
+
b
a + b
a
+
b
and
a
b
ab
ab
are written in their place. Is it possible to get numbers
2015
,
2016
,
2017
,
2018
2015, 2016, 2017, 2018
2015
,
2016
,
2017
,
2018
on the board after several such operations?
4
2
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Bulgarian National Mathematical Olympiad 2016,problem 1
Find all positive integers
m
m
m
and
n
n
n
such that
(
2
2
n
+
1
)
(
2
2
m
+
1
)
(2^{2^{n}}+1)(2^{2^{m}}+1)
(
2
2
n
+
1
)
(
2
2
m
+
1
)
is divisible by
m
⋅
n
m\cdot n
m
⋅
n
.
find all naturals a,b such that a^{n}+ b^{n} = c^{n+1}
Find all natural numbers a, b such that a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1} where c and n are naturals.