MathDB
Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
2022 All-Russian Olympiad
2022 All-Russian Olympiad
Part of
All-Russian Olympiad
Subcontests
(8)
4
1
Hide problems
Cutting squares
There are
18
18
18
children in the class. Parents decided to give children from this class a cake. To do this, they first learned from each child the area of the piece he wants to get. After that, they showed a square-shaped cake, the area of which is exactly equal to the sum of
18
18
18
named numbers. However, when they saw the cake, the children wanted their pieces to be squares too. The parents cut the cake with lines parallel to the sides of the cake (cuts do not have to start or end on the side of the cake). For what maximum k the parents are guaranteed to cut out
k
k
k
square pieces from the cake, which you can give to
k
k
k
children so that each of them gets what they want?
7
2
Hide problems
max 5k+10 edges in each group of k vertices
There are
998
998
998
cities in a country. Some pairs of cities are connected by two-way flights. According to the law, between any pair cities should be no more than one flight. Another law requires that for any group of cities there will be no more than
5
k
+
10
5k+10
5
k
+
10
flights connecting two cities from this group, where
k
k
k
is the number number of cities in the group. Prove that several new flights can be introduced so that laws still hold and the total number of flights in the country is equal to
5000
5000
5000
.
Parallelogram geo
Point
E
E
E
is marked on side
B
C
BC
BC
of parallelogram
A
B
C
D
ABCD
A
BC
D
, and on the side
A
D
AD
A
D
- point
F
F
F
so that the circumscribed circle of
A
B
E
ABE
A
BE
is tangent to line segment
C
F
CF
CF
. Prove that the circumcircle of triangle
C
D
F
CDF
C
D
F
is tangent to line
A
E
AE
A
E
.
5
1
Hide problems
Product of numbers
There are
11
11
11
integers (not necessarily distinct) written on the board. Can it turn out that the product of any five of them is greater than the product of the other six?
6
3
Hide problems
Interesting quadratic
What is the smallest natural number
a
a
a
for which there are numbers
b
b
b
and
c
c
c
such that the quadratic trinomial
a
x
2
+
b
x
+
c
ax^2 + bx + c
a
x
2
+
b
x
+
c
has two different positive roots not exceeding
1
1000
\frac {1}{1000}
1000
1
?
Binary strings
Given is a natural number
n
>
5
n > 5
n
>
5
. On a circular strip of paper is written a sequence of zeros and ones. For each sequence
w
w
w
of
n
n
n
zeros and ones we count the number of ways to cut out a fragment from the strip on which is written
w
w
w
. It turned out that the largest number
M
M
M
is achieved for the sequence
1100...0
11 00...0
1100...0
(
n
−
2
n-2
n
−
2
zeros) and the smallest - for the sequence
00...011
00...011
00...011
(
n
−
2
n-2
n
−
2
zeros). Prove that there is another sequence of
n
n
n
zeros and ones that occurs exactly
M
M
M
times.
3D combo geo
Given is natural number
n
n
n
. Sasha claims that for any
n
n
n
rays in space, no two of which have a common point, he will be able to mark on these rays
k
k
k
points lying on one sphere. What is the largest
k
k
k
for which his statement is true?
8
3
Hide problems
Symmetrical circle of the incircle wrt A
A circle
ω
\omega
ω
is inscribed in triangle
A
B
C
ABC
A
BC
, tangent to the side
B
C
BC
BC
at point
K
K
K
. Circle
ω
′
\omega'
ω
′
is symmetrical to the circle
ω
\omega
ω
with respect to point
A
A
A
. The point
A
0
A_0
A
0
is chosen so that the segments
B
A
0
BA_0
B
A
0
and
C
A
0
CA_0
C
A
0
touch
ω
′
\omega'
ω
′
. Let
M
M
M
be the midpoint of side
B
C
BC
BC
. Prove that the line
A
M
AM
A
M
bisects the segment
K
A
0
KA_0
K
A
0
.
Delete a digit to obtain a square
For a natural number
N
N
N
, consider all distinct perfect squares that can be obtained from
N
N
N
by deleting one digit from its decimal representation. Prove that the number of such squares is bounded by some value that doesn't depend on
N
N
N
.
geometry
From each vertex of triangle
A
B
C
ABC
A
BC
we draw two rays, red and blue, symmetric about the angle bisector of the corresponding angle. The circumcircles of triangles formed by the intersection of rays of the same color. Prove that if the circumcircle of triangle
A
B
C
ABC
A
BC
touches one of these circles then it also touches to the other one.
2
2
Hide problems
Cool geo with many equal segments
On side
B
C
BC
BC
of an acute triangle
A
B
C
ABC
A
BC
are marked points
D
D
D
and
E
E
E
so that
B
D
=
C
E
BD = CE
B
D
=
CE
. On the arc
D
E
DE
D
E
of the circumscribed circle of triangle
A
D
E
ADE
A
D
E
that does not contain the point
A
A
A
, there are points
P
P
P
and
Q
Q
Q
such that
A
B
=
P
C
AB = PC
A
B
=
PC
and
A
C
=
B
Q
AC = BQ
A
C
=
BQ
. Prove that
A
P
=
A
Q
AP=AQ
A
P
=
A
Q
.
Intersting problem
In the coordinate plane,the graps of functions
y
=
s
i
n
x
y=sin x
y
=
s
in
x
and
y
=
t
a
n
x
y=tan x
y
=
t
an
x
are drawn, along with the coordinate axes. Using compass and ruler, construct a line tangent to the graph of sine at a point above the axis,
O
x
Ox
O
x
, as well at a point below that axis (the line can also meet the graph at several other points)
1
1
Hide problems
Equal main divisors imply a=b
We call the
m
a
i
n
main
main
d
i
v
i
s
o
r
s
divisors
d
i
v
i
sors
of a composite number
n
n
n
the two largest of its natural divisors other than
n
n
n
. Composite numbers
a
a
a
and
b
b
b
are such that the main divisors of
a
a
a
and
b
b
b
coincide. Prove that
a
=
b
a=b
a
=
b
.
3
2
Hide problems
Numbers in a row
200
200
200
natural numbers are written in a row. For any two adjacent numbers of the row, the right one is either
9
9
9
times greater than the left one,
2
2
2
times smaller than the left one. Can the sum of all these 200 numbers be equal to
2
4
2022
24^{2022}
2
4
2022
?
center of (XYZ) lies on a fixed circle
An acute-angled triangle
A
B
C
ABC
A
BC
is fixed on a plane with largest side
B
C
BC
BC
. Let
P
Q
PQ
PQ
be an arbitrary diameter of its circumscribed circle, and the point
P
P
P
lies on the smaller arc
A
B
AB
A
B
, and the point
Q
Q
Q
is on the smaller arc
A
C
AC
A
C
. Points
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
are feet of perpendiculars dropped from point
P
P
P
to the line
A
B
AB
A
B
, from point
Q
Q
Q
to the line
A
C
AC
A
C
and from point
A
A
A
to line
P
Q
PQ
PQ
. Prove that the center of the circumscribed circle of triangle
X
Y
Z
XYZ
X
Y
Z
lies on a fixed circle.