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National and Regional Contests
Russia Contests
Assara - South Russian Girl's MO
2024 Assara - South Russian Girl's MO
2024 Assara - South Russian Girl's MO
Part of
Assara - South Russian Girl's MO
Subcontests
(8)
8
2
Hide problems
Chosing any 10 (9) we can add 1 such a their sum is divisible by 10
Given a set
S
S
S
of
2024
2024
2024
natural numbers satisfying the following condition: if you select any
10
10
10
(different) numbers from
S
S
S
, then you can select another number from
S
S
S
so that the sum of all
11
11
11
selected numbers is divisible by
10
10
10
. Prove that one of the numbers can be thrown out of
S
S
S
so that the resulting set
S
′
S'
S
′
of
2023
2023
2023
numbers satisfies the condition: if you choose any
9
9
9
(different) numbers from
S
′
S'
S
′
, then you can choose another number from
S
′
S'
S
′
so that the sum of all
10
10
10
selected numbers is divisible by
10
10
10
. K.A.Sukhov
Friendly classmates
There are
15
15
15
boys and
15
15
15
girls in the class. The first girl is friends with
4
4
4
boys, the second with
5
5
5
, the third with
6
6
6
, . . . , the
11
11
11
th with
14
14
14
, and each of the other four girls is friends with all the boys. It turned out that there are exactly
3
⋅
2
25
3 \cdot 2^{25}
3
⋅
2
25
ways to split the entire class into pairs, so that each pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends with all the other girls too. G.M.Sharafetdinova
7
2
Hide problems
Chip on a board makes moves
There is a chip in one of the squares on the checkered board. In one move, she can move either
1
1
1
square to the right, or diagonally
1
1
1
to the left and
1
1
1
up, or
1
1
1
to the left and
3
3
3
down (see Fig.). The chip made
n
n
n
moves and returned to the starting square. Prove that a)
n
n
n
is divisible by
2
2
2
, b)
n
n
n
is divisible by
8
8
8
. K.A.Sukhov
n|(a+b)^2,(b+c)^2,(c+a)^2
Find all positive integers
n
n
n
for such the following condition holds: "If
a
a
a
,
b
b
b
and
c
c
c
are positive integers such are all numbers
a
2
+
2
a
b
+
b
2
,
b
2
+
2
b
c
+
c
2
,
c
2
+
2
c
a
+
a
2
a^2+2ab+b^2,\ b^2+2bc+c^2, \ c^2+2ca+a^2
a
2
+
2
ab
+
b
2
,
b
2
+
2
b
c
+
c
2
,
c
2
+
2
c
a
+
a
2
are divisible by
n
n
n
, then
(
a
+
b
+
c
)
2
(a+b+c)^2
(
a
+
b
+
c
)
2
is also divisible by
n
n
n
." G.M.Sharafetdinova
6
2
Hide problems
\angle AEX = 65^\circ in regular hexagon
In the regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
, a point
X
X
X
was marked on the diagonal
A
D
AD
A
D
such that
∠
A
E
X
=
6
5
∘
\angle AEX = 65^\circ
∠
A
EX
=
6
5
∘
. What is the degree measure of the angle
∠
X
C
D
\angle XCD
∠
XC
D
? A.V.Smirnov, I.A.Efremov
XY \perp AD Assara
The points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
are marked on the straight line in this order. Circle
ω
1
\omega_1
ω
1
passes through points
A
A
A
and
C
C
C
, and the circle
ω
2
\omega_2
ω
2
passes through points
B
B
B
and
D
D
D
. On the circle
ω
2
\omega_2
ω
2
, the point
E
E
E
is marked so that
A
B
=
B
E
AB = BE
A
B
=
BE
, and on the circle
ω
1
\omega_1
ω
1
, the point
F
F
F
is marked so that
C
D
=
C
F
CD = CF
C
D
=
CF
. The line
A
E
AE
A
E
intersects the circle
ω
2
\omega_2
ω
2
a second time at point
X
X
X
, and the line
D
F
DF
D
F
intersects the circle
ω
1
\omega_1
ω
1
at point
Y
Y
Y
. Prove that the
X
Y
XY
X
Y
lines and
A
D
AD
A
D
is perpendicular. A.D.Tereshin
5
2
Hide problems
2024^8 | 2024!
Prove that
2024
!
2024!
2024
!
is divisible by a)
202
4
2
2024^2
202
4
2
; b)
202
4
8
2024^8
202
4
8
. (
n
!
=
1
⋅
2
⋅
3
⋅
.
.
.
⋅
n
n!=1\cdot 2 \cdot 3 \cdot ... \cdot n
n
!
=
1
⋅
2
⋅
3
⋅
...
⋅
n
) Z.Smysl
(100!)^99 > (99!)^100 > (100!)^98 .
Prove that
(
100
!
)
99
>
(
99
!
)
100
>
(
100
!
)
98
(100!)^{99} > (99!)^{100} > (100!)^{98}
(
100
!
)
99
>
(
99
!
)
100
>
(
100
!
)
98
. K.A.Sukhov
4
2
Hide problems
Sides are more than diameter of incircle
Is there a described
n
n
n
-gon in which each side is longer than the diameter of the inscribed circle a) at
n
=
4
n = 4
n
=
4
? b) when
n
=
7
n = 7
n
=
7
? c) when
n
=
6
n = 6
n
=
6
? P.A.Kozhevnikov
XY passes from fixed point; parabolas
A parabola
p
p
p
is drawn on the coordinate plane — the graph of the equation
y
=
−
x
2
y =-x^2
y
=
−
x
2
, and a point
A
A
A
is marked that does not lie on the parabola
p
p
p
. All possible parabolas
q
q
q
of the form
y
=
x
2
+
a
x
+
b
y = x^2+ax+b
y
=
x
2
+
a
x
+
b
are drawn through point
A
A
A
, intersecting
p
p
p
at two points
X
X
X
and
Y
Y
Y
. Prove that all possible
X
Y
XY
X
Y
lines pass through a fixed point in the plane. P.A.Kozhevnikov
3
2
Hide problems
Numbers in cells are modulo not more than 2024
In the cells of the
4
×
N
4\times N
4
×
N
table, integers are written, modulo no more than
2024
2024
2024
(i.e. numbers from the set
{
−
2024
,
−
2023
,
…
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
…
,
2024
}
\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}
{
−
2024
,
−
2023
,
…
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
…
,
2024
}
) so that in each of the four lines there are no two equal numbers. At what maximum
N
N
N
could it turn out that in each column the sum of the numbers is equal to
2
2
2
? G.M.Sharafetdinova
Numbers in cells are modulo not more than 2024; seniors version (2 to 23)
In the cells of the
4
×
N
4\times N
4
×
N
table, integers are written, modulo no more than
2024
2024
2024
(i.e. numbers from the set
{
−
2024
,
−
2023
,
…
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
…
,
2024
}
\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}
{
−
2024
,
−
2023
,
…
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
…
,
2024
}
) so that in each of the four lines there are no two equal numbers. At what maximum
N
N
N
could it turn out that in each column the sum of the numbers is equal to
23
23
23
? G.M.Sharafetdinova
2
2
Hide problems
p/a+p/b=1, p|a+b
Let
p
p
p
be a prime number. Positive integers numbers
a
a
a
and
b
b
b
are such
p
a
+
p
b
=
1
\frac{p}{a}+\frac{p}{b}=1
a
p
+
b
p
=
1
and
a
+
b
a+b
a
+
b
is divisible by
p
p
p
. What values can an expression
a
+
b
p
\frac{a+b}{p}
p
a
+
b
take? Yu.A.Karpenko
There is a side that does not more than diameter of incircle in 8-gon
Prove that in any described
8
8
8
-gon there is a side that does not exceed the diameter of the inscribed circle in length. P.A.Kozhevnikov
1
2
Hide problems
cards lies beautifully
There is a set of
50
50
50
cards. Each card on both sides is colored in one of three colors — red, blue or white, and for each card its two sides are colored in different colors. The cards were laid out on the table. The card lies beautifully if at least one of two conditions is met: its upper side — red; its underside is blue. It turned out that exactly
25
25
25
cards are lying beautifully. Then all the cards were turned over. Now some of the cards are lying beautifully on the table. How many of them can there be? K.A.Sukhov
cards lies beautifully; seniors version
There is a set of
2024
2024
2024
cards. Each card on both sides is colored in one of three colors — red, blue or white, and for each card its two sides are colored in different colors. The cards were laid out on the table. The card lies beautifully if at least one of two conditions is met: its upper side — red; its underside is blue. It turned out that exactly
150
150
150
cards are lying beautifully. Then all the cards were turned over. Now some of the cards are lying beautifully on the table. How many of them can there be? K.A.Sukhov