MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan EGMO TST
2021 Azerbaijan EGMO TST
2021 Azerbaijan EGMO TST
Part of
Azerbaijan EGMO TST
Subcontests
(3)
2
2
Hide problems
symmetric lines with respect to angle bisector
Let
ω
\omega
ω
be a circle with center
O
,
O,
O
,
and let
A
A
A
be a point with tangents
A
P
AP
A
P
and
A
Q
AQ
A
Q
to the circle. Denote by
K
K
K
the intersection point of
A
O
AO
A
O
and
P
Q
.
PQ.
PQ
.
l
1
l_1
l
1
and
l
2
l_2
l
2
are two lines passing through
A
A
A
that intersect
ω
.
\omega.
ω
.
Call
B
B
B
the intersection point of
l
1
l_1
l
1
with
ω
,
\omega,
ω
,
which is located nearer to
A
A
A
on
l
1
.
l_1.
l
1
.
Call
C
C
C
the intersection point of
l
2
l_2
l
2
with
ω
,
\omega,
ω
,
which is located further to
A
A
A
on
l
2
.
l_2.
l
2
.
Prove that
∠
P
A
B
=
∠
Q
A
C
\angle PAB = \angle QAC
∠
P
A
B
=
∠
Q
A
C
if and only if the points
B
,
K
,
C
B, K, C
B
,
K
,
C
are on line.
constructing unbounded sequence form non-decreasing, unbounded sequence
Given a non-decreasing unbounded sequence
a
n
,
a_n,
a
n
,
construct a new sequence
b
n
b_n
b
n
as follows
b
n
=
a
2
−
a
1
a
2
+
a
3
−
a
2
a
3
+
.
.
.
+
a
n
−
a
n
−
1
a
n
b_n = \frac{a_2 - a_1}{a_2} + \frac{a_3 - a_2}{a_3} + ... + \frac{a_n - a_{n-1}}{a_n}
b
n
=
a
2
a
2
−
a
1
+
a
3
a
3
−
a
2
+
...
+
a
n
a
n
−
a
n
−
1
Prove that
b
n
b_n
b
n
is also unbounded.
1
2
Hide problems
two numbers replaced with their product
Let
n
n
n
be an even positive integer. There are
n
n
n
real numbers written on the blackboard. In every step, we choose two numbers, erase them, and replace each of them with their product. Show that for any initial
n
n
n
-tuple it is possible to obtain
n
n
n
equal numbers on the blackboard after a finite number of steps.
Mathematics
p is a prime number, k is a positive integer Find all (p, k):
k
!
=
(
p
3
−
1
)
(
p
3
−
p
)
(
p
3
−
p
2
)
k!=(p^3-1)(p^3-p)(p^3-p^2)
k
!
=
(
p
3
−
1
)
(
p
3
−
p
)
(
p
3
−
p
2
)
4
1
Hide problems
tangent line to 3 circles of center A, B, C and all pass through orthocenter
Let
A
B
C
ABC
A
BC
be an acute, non isosceles with
I
I
I
is its incenter. Denote
D
,
E
D, E
D
,
E
as tangent points of
(
I
)
(I)
(
I
)
on
A
B
,
A
C
AB,AC
A
B
,
A
C
, respectively. The median segments respect to vertex
A
A
A
of triangles
A
B
E
ABE
A
BE
and
A
C
D
ACD
A
C
D
meet
(
I
)
(I)
(
I
)
at
P
,
Q
,
P,Q,
P
,
Q
,
respectively. Take points
M
,
N
M, N
M
,
N
on the line
D
E
DE
D
E
such that
A
M
⊥
B
E
AM \perp BE
A
M
⊥
BE
and
A
N
⊥
C
D
AN \perp C D
A
N
⊥
C
D
respectively. a) Prove that
A
A
A
lies on the radical axis of
(
M
I
P
)
(MIP)
(
M
I
P
)
and
(
N
I
Q
)
(NIQ)
(
N
I
Q
)
. b) Suppose that the orthocenter
H
H
H
of triangle
A
B
C
ABC
A
BC
lies on
(
I
)
(I)
(
I
)
. Prove that there exists a line which is tangent to three circles of center
A
,
B
,
C
A, B, C
A
,
B
,
C
and all pass through
H
H
H
.