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Problems
Contests
National and Regional Contests
Russia Contests
239 Open Math Olympiad
2024 239 Open Mathematical Olympiad
2024 239 Open Mathematical Olympiad
Part of
239 Open Math Olympiad
Subcontests
(8)
6
1
Hide problems
Magic trick again
Let
X
X
X
denotes the set of integers from
1
1
1
to
239
239
239
. A magician with an assistant perform a trick. The magician leaves the hall and the spectator writes a sequence of
10
10
10
elements on the board from the set
X
X
X
. The magician’s assistant looks at them and adds
k
k
k
more elements from
X
X
X
to the existing sequence. After that the spectator replaces three of these
k
+
10
k+10
k
+
10
numbers by random elements of
X
X
X
(it is permitted to change them by themselves, that is to not change anything at all, for example). The magician enters and looks at the resulting row of
k
+
10
k+10
k
+
10
numbers and without error names the original
10
10
10
numbers written by the spectator. Find the minimal possible
k
k
k
for which the trick is possible.
8
2
Hide problems
Combo geo with circles
There are
2
n
2n
2
n
points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into
n
n
n
pairs and cover each pair of points with a circle containing no other points.
Weird binary sequence
Let
x
1
,
x
2
,
…
x_1, x_2, \ldots
x
1
,
x
2
,
…
be a sequence of
0
,
1
0,1
0
,
1
, such that it satisfies the following three conditions: 1)
x
2
=
x
100
=
1
x_2=x_{100}=1
x
2
=
x
100
=
1
,
x
i
=
0
x_i=0
x
i
=
0
for
1
≤
i
≤
100
1 \leq i \leq 100
1
≤
i
≤
100
and
i
≠
2
,
100
i \neq 2,100
i
=
2
,
100
; 2)
x
2
n
−
1
=
x
n
−
50
+
1
,
x
2
n
=
x
n
−
50
x_{2n-1}=x_{n-50}+1, x_{2n}=x_{n-50}
x
2
n
−
1
=
x
n
−
50
+
1
,
x
2
n
=
x
n
−
50
for
51
≤
n
≤
100
51 \leq n \leq 100
51
≤
n
≤
100
; 3)
x
2
n
=
x
n
−
50
,
x
2
n
−
1
=
x
n
−
50
+
x
n
−
100
x_{2n}=x_{n-50}, x_{2n-1}=x_{n-50}+x_{n-100}
x
2
n
=
x
n
−
50
,
x
2
n
−
1
=
x
n
−
50
+
x
n
−
100
for
n
>
100
n>100
n
>
100
. Show that the sequence is periodic.
7
2
Hide problems
Weird sequence NT
Let
n
>
3
n>3
n
>
3
be a positive integer satisfying
2
n
+
1
=
3
p
2^n+1=3p
2
n
+
1
=
3
p
, where
p
p
p
is a prime. Let
s
0
=
2
n
−
2
+
1
3
s_0=\frac{2^{n-2}+1}{3}
s
0
=
3
2
n
−
2
+
1
and
s
i
=
s
i
−
1
2
−
2
s_i=s_{i-1}^2-2
s
i
=
s
i
−
1
2
−
2
for
i
>
0
i>0
i
>
0
. Show that
p
∣
2
s
n
−
2
−
3
p \mid 2s_{n-2}-3
p
∣
2
s
n
−
2
−
3
.
Sets and sums
Prove that there exists a positive integer
k
>
100
k>100
k
>
100
, such that for any set
A
A
A
of
k
k
k
positive reals, there exists a subset
B
B
B
of
100
100
100
numbers, so that none of the sums of at least two numbers in
B
B
B
is in the set
A
A
A
.
5
2
Hide problems
Circumscribed quadrilateral and lengths
A quadrilateral
A
B
C
D
ABCD
A
BC
D
has an incircle
Γ
\Gamma
Γ
. The points
X
,
Y
X, Y
X
,
Y
are chosen so that
A
X
−
C
X
=
A
B
−
B
C
AX-CX=AB-BC
A
X
−
CX
=
A
B
−
BC
,
B
X
−
D
X
=
B
C
−
C
D
BX-DX=BC-CD
BX
−
D
X
=
BC
−
C
D
,
C
Y
−
A
Y
=
A
D
−
D
C
CY-AY=AD-DC
C
Y
−
A
Y
=
A
D
−
D
C
and
D
Y
−
B
Y
=
A
B
−
A
D
DY-BY=AB-AD
D
Y
−
B
Y
=
A
B
−
A
D
. Given that the center of
Γ
\Gamma
Γ
lies on
X
Y
XY
X
Y
, show that
A
C
,
B
D
,
X
Y
AC, BD, XY
A
C
,
B
D
,
X
Y
are concurrent.
3-variable inequality with a weird condition
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be reals such that
a
2
(
c
2
−
2
b
−
1
)
+
b
2
(
a
2
−
2
c
−
1
)
+
c
2
(
b
2
−
2
a
−
1
)
=
0.
a^2(c^2-2b-1)+b^2(a^2-2c-1)+c^2(b^2-2a-1)=0.
a
2
(
c
2
−
2
b
−
1
)
+
b
2
(
a
2
−
2
c
−
1
)
+
c
2
(
b
2
−
2
a
−
1
)
=
0.
Show that
3
(
a
2
+
b
2
+
c
2
)
+
4
(
a
+
b
+
c
)
+
3
≥
6
a
b
c
.
3(a^2+b^2+c^2)+4(a+b+c)+3 \geq 6abc.
3
(
a
2
+
b
2
+
c
2
)
+
4
(
a
+
b
+
c
)
+
3
≥
6
ab
c
.
4
2
Hide problems
Good sets covering a table
Let
n
n
n
be a positive integer greater than
1
1
1
and let us call an arbitrary set of cells in a
n
×
n
n\times n
n
×
n
square
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
if they are the intersection cells of several rows and several columns, such that none of those cells lie on the main diagonal. What is the minimum number of pairwise disjoint
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
sets required to cover the entire table without the main diagonal?
Incircle geo with angle conditions
Let
I
I
I
be the incenter of a triangle
A
B
C
ABC
A
BC
. The points
X
,
Y
X, Y
X
,
Y
lie on the prolongations of the lines
I
B
,
I
C
IB, IC
I
B
,
I
C
after
I
I
I
so that
∠
I
A
X
=
∠
I
B
A
\angle IAX=\angle IBA
∠
I
A
X
=
∠
I
B
A
and
∠
I
A
Y
=
∠
I
C
A
\angle IAY=\angle ICA
∠
I
A
Y
=
∠
I
C
A
. Show that the line through the midpoints of
I
A
IA
I
A
and
X
Y
XY
X
Y
passes through the circumcenter of
A
B
C
ABC
A
BC
.
3
2
Hide problems
Two products of distances
a) (version for grades 10-11)Let
P
P
P
be a point lying in the interior of a triangle. Show that the product of the distances from
P
P
P
to the sides of the triangle is at least
8
8
8
times less than the product of the distances from
P
P
P
to the tangents to the circumcircle at the vertices of the triangle.b) (version for grades 8-9)Is it true that for any triangle there exists a point
P
P
P
for which equality in the inequality from a) holds?
Digits on a circle
There are
169
169
169
non-zero digits written around a circle. Prove that they can be split into
14
14
14
non-empty blocks of consecutive digits so that among the
14
14
14
natural numbers formed by the digits in those blocks, at least
13
13
13
of them are divisible by
13
13
13
(the digits in each block are read in clockwise direction).
2
2
Hide problems
Rich knight adds coins
A rich knight has a chest and a lot of coins, so every day he puts into the chest some quantity of coins - among the numbers
1
,
2
,
…
,
100
1, 2, \ldots, 100
1
,
2
,
…
,
100
. If there exist two days on which he added equal quantities of coins (say,
k
k
k
coins) and he has added in total at most
100
k
100k
100
k
coins on the days between these two days, he stops putting coins into the chest. Prove that this will necessarily happen eventually.
Intersecting segments
There are
2
n
2n
2
n
points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose
n
n
n
drawn segments having no common ends?
1
2
Hide problems
Continuous function
Let
f
:
R
≥
0
→
R
≥
0
f:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}
f
:
R
≥
0
→
R
≥
0
be a continuous function such that
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
and
f
(
x
)
+
f
(
f
(
x
)
)
+
f
(
f
(
f
(
x
)
)
)
=
3
x
f(x)+f(f(x))+f(f(f(x)))=3x
f
(
x
)
+
f
(
f
(
x
))
+
f
(
f
(
f
(
x
)))
=
3
x
for all
x
>
0
x>0
x
>
0
. Show that
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
for all
x
>
0
x>0
x
>
0
.
Linked rows and columns
We will say that two sets of distinct numbers are
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
l
i
n
k
e
d
<
/
s
p
a
n
>
<span class='latex-italic'>linked</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
l
ink
e
d
<
/
s
p
an
>
to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a
100
×
200
100 \times 200
100
×
200
rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?