MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan Senior National Olympiad
2023 Azerbaijan Senior NMO
2023 Azerbaijan Senior NMO
Part of
Azerbaijan Senior National Olympiad
Subcontests
(5)
5
1
Hide problems
touch points adn equality of BM, M is midpoint
The incircle of the acute-angled triangle
A
B
C
ABC
A
BC
is tangent to the sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
at points
C
1
,
A
1
,
B
1
,
C_1, A_1, B_1,
C
1
,
A
1
,
B
1
,
respectively, and
I
I
I
is the incenter. Let the midpoint of side
B
C
BC
BC
be
M
.
M.
M
.
Let
J
J
J
be the foot of the altitude drawn from
M
M
M
to
C
1
B
1
.
C_1B_1.
C
1
B
1
.
The tangent drawn from
B
B
B
to the circumcircle of
△
B
I
C
\triangle BIC
△
B
I
C
intersects
I
J
IJ
I
J
at
X
.
X.
X
.
If the circumcircle of
△
A
X
I
\triangle AXI
△
A
X
I
intersects
A
B
AB
A
B
at
Y
,
Y,
Y
,
prove that
B
Y
=
B
M
.
BY = BM.
B
Y
=
BM
.
4
1
Hide problems
Griphook tries to find the magic code
To open the magic chest, one needs to say a magic code of length
n
n
n
consisting of digits
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9.
Each time Griphook tells the chest a code it thinks up, the chest's talkative guardian responds by saying the number of digits in that code that match the magic code. (For example, if the magic code is
0423
0423
0423
and Griphook says
3442
,
3442,
3442
,
the chest's talkative guard will say
1
1
1
). Prove that there exists a number
k
k
k
such that for any natural number
n
≥
k
,
n \geq k,
n
≥
k
,
Griphook can find the magic code by checking at most
4
n
−
2023
4n-2023
4
n
−
2023
times, regardless of what the magic code of the box is.
3
1
Hide problems
all polynomials depending on m
Let
m
m
m
be a positive integer. Find polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that
(
x
−
m
)
P
(
x
+
2023
)
=
x
P
(
x
)
(x-m)P(x+2023) = xP(x)
(
x
−
m
)
P
(
x
+
2023
)
=
x
P
(
x
)
is satisfied for all real numbers
x
.
x.
x
.
2
1
Hide problems
square root involving equation with case work
Find all the integer solutions of the equation:
x
+
y
=
x
+
2023
\sqrt{x} + \sqrt{y} = \sqrt{x+2023}
x
+
y
=
x
+
2023
1
1
Hide problems
n = a*b , numbers of the form a^b
The teacher calculates and writes on the board all the numbers
a
b
a^b
a
b
that satisfy the condition
n
=
a
×
b
n = a\times b
n
=
a
×
b
for the natural number
n
.
n.
n
.
Here
a
a
a
and
b
b
b
are natural numbers. Is there a natural number
n
n
n
such that each of the numbers
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.