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Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad Regional Round
2023 All-Russian Olympiad Regional Round
2023 All-Russian Olympiad Regional Round
Part of
All-Russian Olympiad Regional Round
Subcontests
(14)
11.10
1
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Find a coloring of a connected graph satisfying weird condition
Given is a simple connected graph with
2
n
2n
2
n
vertices. Prove that its vertices can be colored with two colors so that if there are
k
k
k
edges connecting vertices with different colors and
m
m
m
edges connecting vertices with the same color, then
k
−
m
≥
n
k-m \geq n
k
−
m
≥
n
.
11.9
1
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Weird but easy inequality with absolute values
If
a
,
b
,
c
a, b, c
a
,
b
,
c
are non-zero reals, prove that
∣
b
a
−
b
c
∣
+
∣
c
a
−
c
b
∣
+
∣
b
c
+
1
∣
>
1
|\frac{b} {a}-\frac{b} {c}|+|\frac{c} {a}-\frac{c}{b}|+|bc+1|>1
∣
a
b
−
c
b
∣
+
∣
a
c
−
b
c
∣
+
∣
b
c
+
1∣
>
1
.
11.8
1
Hide problems
Geometry with angle bisector and perpendicular bisectors
Given is a triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
. Points
D
,
E
D, E
D
,
E
are chosen on the angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
such that
E
A
=
E
B
,
D
B
=
D
C
EA=EB, DB=DC
E
A
=
EB
,
D
B
=
D
C
. If
P
,
Q
P, Q
P
,
Q
are the circumcenters of
(
A
O
E
)
,
(
C
O
D
)
(AOE), (COD)
(
A
OE
)
,
(
CO
D
)
, prove that either the line
P
Q
PQ
PQ
coincides with
A
C
AC
A
C
or
P
Q
C
A
PQCA
PQC
A
is cyclic.
10.10
1
Hide problems
Another easy inequality with square roots
Prove that for all positive reals
x
,
y
,
z
x, y, z
x
,
y
,
z
, the inequality
(
x
−
y
)
3
x
2
+
y
2
+
(
y
−
z
)
3
y
2
+
z
2
+
(
z
−
x
)
3
z
2
+
x
2
≥
0
(x-y)\sqrt{3x^2+y^2}+(y-z)\sqrt{3y^2+z^2}+(z-x)\sqrt{3z^2+x^2} \geq 0
(
x
−
y
)
3
x
2
+
y
2
+
(
y
−
z
)
3
y
2
+
z
2
+
(
z
−
x
)
3
z
2
+
x
2
≥
0
is satisfied.
10.8
1
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Equal segments in a parallelogram
The bisector of
∠
B
A
D
\angle BAD
∠
B
A
D
of a parallelogram
A
B
C
D
ABCD
A
BC
D
meets
B
C
BC
BC
at
K
K
K
. The point
L
L
L
lies on
A
B
AB
A
B
such that
A
L
=
C
K
AL=CK
A
L
=
C
K
. The lines
A
K
AK
A
K
and
C
L
CL
C
L
meet at
M
M
M
. Let
(
A
L
M
)
(ALM)
(
A
L
M
)
meet
A
D
AD
A
D
after
D
D
D
at
N
N
N
. Prove that
∠
C
N
L
=
9
0
o
\angle CNL=90^{o}
∠
CN
L
=
9
0
o
9.10
1
Hide problems
Switching operations in a cube
A
100
×
100
×
100
100 \times 100 \times 100
100
×
100
×
100
cube is divided into a million unit cubes and in each small cube there is a light bulb. Three faces
100
×
100
100 \times 100
100
×
100
of the large cube having a common vertex are painted: one in red, one in blue and the other in green. Call a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
o
l
u
m
n
<
/
s
p
a
n
>
<span class='latex-italic'>column</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
co
l
u
mn
<
/
s
p
an
>
a set of
100
100
100
cubes forming a block
1
×
1
×
100
1 \times 1 \times 100
1
×
1
×
100
. Each of the
30000
30 000
30000
columns have one painted end cell, on which there is a switch. After pressing a switch, the states of all light bulbs of this column are changed. Petya pressed several switches, getting a situation with exactly
k
k
k
lamps on. Prove that Vasya can press several switches so that all lamps are off, but by using no more than
k
100
\frac {k} {100}
100
k
switches on the red face.
9.9
1
Hide problems
Easy inequality with square roots
Find the largest real
m
m
m
, such that for all positive real
a
,
b
,
c
a, b, c
a
,
b
,
c
with sum
1
1
1
, the inequality
a
b
a
b
+
c
+
b
c
b
c
+
a
+
c
a
c
a
+
b
≥
m
\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m
ab
+
c
ab
+
b
c
+
a
b
c
+
c
a
+
b
c
a
≥
m
is satisfied.
9.6
1
Hide problems
Easy but nice NT with lcm
Does there exist a positive integer
m
m
m
, such that if
S
n
S_n
S
n
denotes the lcm of
1
,
2
,
…
,
n
1,2, \ldots, n
1
,
2
,
…
,
n
, then
S
m
+
1
=
4
S
m
S_{m+1}=4S_m
S
m
+
1
=
4
S
m
?
11.5
1
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BY, CX, AH are concurrent
Given is a triangle
A
B
C
ABC
A
BC
with altitude
A
H
AH
A
H
and median
A
M
AM
A
M
. The line
O
H
OH
O
H
meets
A
M
AM
A
M
at
D
D
D
. Let
A
B
∩
C
D
=
E
,
A
C
∩
B
D
=
F
AB \cap CD=E, AC \cap BD=F
A
B
∩
C
D
=
E
,
A
C
∩
B
D
=
F
. If
E
H
EH
E
H
and
F
H
FH
F
H
meet
(
A
B
C
)
(ABC)
(
A
BC
)
at
X
,
Y
X, Y
X
,
Y
, prove that
B
Y
,
C
X
,
A
H
BY, CX, AH
B
Y
,
CX
,
A
H
are concurrent.
11.4
1
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Process with pairs
We write pairs of integers on a blackboard. Initially, the pair
(
1
,
2
)
(1,2)
(
1
,
2
)
is written. On a move, if
(
a
,
b
)
(a, b)
(
a
,
b
)
is on the blackboard, we can add
(
−
a
,
−
b
)
(-a, -b)
(
−
a
,
−
b
)
or
(
−
b
,
a
+
b
)
(-b, a+b)
(
−
b
,
a
+
b
)
. In addition, if
(
a
,
b
)
(a, b)
(
a
,
b
)
and
(
c
,
d
)
(c, d)
(
c
,
d
)
are written on the blackboard, we can add
(
a
+
c
,
b
+
d
)
(a+c, b+d)
(
a
+
c
,
b
+
d
)
. Can we reach
(
2022
,
2023
)
(2022, 2023)
(
2022
,
2023
)
?
10.5
1
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Interesting orthocenter configuration
In a triangle
A
B
C
ABC
A
BC
, let
B
D
BD
B
D
be its altitude and let
H
H
H
be its orthocenter. The perpendicular bisector of of
H
D
HD
HD
meets
(
B
C
D
)
(BCD)
(
BC
D
)
at
P
,
Q
P, Q
P
,
Q
. Prove that
∠
A
P
B
+
∠
A
Q
B
=
18
0
o
\angle APB+\angle AQB=180^{o}
∠
A
PB
+
∠
A
QB
=
18
0
o
10.3
1
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Every 30 sets intersect, then all intersect
Given are
50
50
50
distinct sets of positive integers, each of size
30
30
30
, such that every
30
30
30
of them have a common element. Prove that all of them have a common element.
9.5
1
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Tangent circles in a quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral such that the circles with diameters
A
B
AB
A
B
and
C
D
CD
C
D
touch at
S
S
S
. If
M
,
N
M, N
M
,
N
are the midpoints of
A
B
,
C
D
AB, CD
A
B
,
C
D
, prove that the perpendicular through
M
M
M
to
M
N
MN
MN
meets
C
S
CS
CS
on the circumcircle of
A
B
C
D
ABCD
A
BC
D
.
9.3
1
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Grid with rooks
Given is a positive integer
n
n
n
. There are
2
n
2n
2
n
mutually non-attacking rooks placed on a grid
2
n
×
2
n
2n \times 2n
2
n
×
2
n
. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?