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Problems
Contests
National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
2013 Saint Petersburg Mathematical Olympiad
2013 Saint Petersburg Mathematical Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(7)
7
3
Hide problems
St.Peterburg, P7 Grade 11, 2013
In the language of wolves has two letters
F
F
F
and
P
P
P
, any finite sequence which forms a word. А word
Y
Y
Y
is called 'subpart' of word
X
X
X
if Y is obtained from X by deleting some letters (for example, the word
F
F
P
F
FFPF
FFPF
has 8 'subpart's: F, P, FF, FP, PF, FFP, FPF, FFF). Determine
n
n
n
such that the
n
n
n
is the greatest number of 'subpart's can have n-letter word language of wolves. F. Petrov, V. Volkov
Divisors of naturals
Let
a
1
,
a
2
a_1,a_2
a
1
,
a
2
- two naturals, and
1
<
b
1
<
a
1
,
1
<
b
2
<
a
2
1<b_1<a_1,1<b_2<a_2
1
<
b
1
<
a
1
,
1
<
b
2
<
a
2
and
b
1
∣
a
1
,
b
2
∣
a
2
b_1|a_1,b_2|a_2
b
1
∣
a
1
,
b
2
∣
a
2
. Prove that
a
1
b
1
+
a
2
b
2
−
1
a_1b_1+a_2b_2-1
a
1
b
1
+
a
2
b
2
−
1
is not divided by
a
1
a
2
a_1a_2
a
1
a
2
St.Peterburg, P7 Grade 9, 2013
Given is a natural number
a
a
a
with
54
54
54
digits, each digit equal to
0
0
0
or
1
1
1
. Prove the remainder of
a
a
a
when divide by
33
⋅
34
⋯
39
33\cdot 34\cdots 39
33
⋅
34
⋯
39
is larger than
100000
100000
100000
. (It's mean:
a
≡
r
(
m
o
d
33
⋅
34
⋯
39
)
a \equiv r \pmod{33\cdot 34\cdots 39 }
a
≡
r
(
mod
33
⋅
34
⋯
39
)
with
0
<
r
<
33
⋅
34
⋯
39
0<r<33\cdot 34\cdots 39
0
<
r
<
33
⋅
34
⋯
39
then prove that
r
>
100000
r>100000
r
>
100000
) M. Antipov
2
3
Hide problems
St.Peterburg, P2 Grade 10, 2013
in a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
,
M
,
N
M,N
M
,
N
are midpoints of
B
C
,
A
D
BC,AD
BC
,
A
D
respectively. If
A
M
=
B
N
AM=BN
A
M
=
BN
and
D
M
=
C
N
DM=CN
D
M
=
CN
then prove that
A
C
=
B
D
AC=BD
A
C
=
B
D
.S. Berlov
St.Peterburg, P2 Grade 11, 2013
At the faculty of mathematics study
40
40
40
boys and
10
10
10
girls. Every girl acquaintance with all boys, who older than her, or tall(higher) than her. Prove that there exist two boys such that the sets of acquainted-girls of the boys are same.
St.Peterburg, P2 Grade 9, 2013
if
a
2
+
b
2
+
c
2
+
d
2
=
1
a^2+b^2+c^2+d^2=1
a
2
+
b
2
+
c
2
+
d
2
=
1
prove that
(
1
−
a
)
(
1
−
b
)
≥
c
d
.
(1-a)(1-b)\ge cd.
(
1
−
a
)
(
1
−
b
)
≥
c
d
.
A. Khrabrov
3
3
Hide problems
St.Peterburg, P3 Grade 11, 2013
Let
M
M
M
and
N
N
N
are midpoint of edges
A
B
AB
A
B
and
C
D
CD
C
D
of the tetrahedron
A
B
C
D
ABCD
A
BC
D
,
A
N
=
D
M
AN=DM
A
N
=
D
M
and
C
M
=
B
N
CM=BN
CM
=
BN
. Prove that
A
C
=
B
D
AC=BD
A
C
=
B
D
. S. Berlov
St.Peterburg, P3 Grade 10, 2013
On a circle there are some black and white points (there are at least
12
12
12
points). Each point has
10
10
10
neighbors (
5
5
5
left and
5
5
5
right neighboring points),
5
5
5
being black and
5
5
5
white. Prove that the number of points on the circle is divisible by
4
4
4
.
Bisectors and parallels
A
B
C
ABC
A
BC
is triangle.
l
1
l_1
l
1
- line passes through
A
A
A
and parallel to
B
C
BC
BC
,
l
2
l_2
l
2
- line passes through
C
C
C
and parallel to
A
B
AB
A
B
. Bisector of
∠
B
\angle B
∠
B
intersect
l
1
l_1
l
1
and
l
2
l_2
l
2
at
X
,
Y
X,Y
X
,
Y
.
X
Y
=
A
C
XY=AC
X
Y
=
A
C
. What value can take
∠
A
−
∠
C
\angle A- \angle C
∠
A
−
∠
C
?
6
2
Hide problems
St.Peterburg, P6 Grade 11, 2013
Let
(
I
b
)
(I_b)
(
I
b
)
,
(
I
c
)
(I_c)
(
I
c
)
are excircles of a triangle
A
B
C
ABC
A
BC
. Given a circle
ω
\omega
ω
passes through
A
A
A
and externally tangents to the circles
(
I
b
)
(I_b)
(
I
b
)
and
(
I
c
)
(I_c)
(
I
c
)
such that it intersects with
B
C
BC
BC
at points
M
M
M
,
N
N
N
. Prove that
∠
B
A
M
=
∠
C
A
N
\angle BAM=\angle CAN
∠
B
A
M
=
∠
C
A
N
. A. Smirnov
Soldiers color grass
There are
85
85
85
soldiers with different heigth and age. Every day commander chooses random soldier and send him and also all soldiers that are taller and older than this soldier, or all soldiers that are lower and younger than this soldier to color grass. Prove that after
10
10
10
days we can find two soldiers, that color grass at same days.
1
2
Hide problems
St.Peterburg, P1 Grade 11, 2013
Find the minimum positive noninteger root of
sin
x
=
sin
⌊
x
⌋
\sin x=\sin \lfloor x \rfloor
sin
x
=
sin
⌊
x
⌋
. F. Petrov
Interesting number
Call number
A
A
A
as interesting if
A
A
A
is divided by every number that can be received from
A
A
A
by crossing some last digits. Find maximum interesting number with different digits.
4
3
Hide problems
St.Peterburg, P4 Grade 11, 2013
Find all pairs
(
p
,
q
)
(p,q)
(
p
,
q
)
of prime numbers, such that
2
p
−
1
2p-1
2
p
−
1
,
2
q
−
1
2q-1
2
q
−
1
,
2
p
q
−
1
2pq-1
2
pq
−
1
are perfect square. F. Petrov, A. Smirnov
Roots on the board
There are
100
100
100
numbers from
(
0
,
1
)
(0,1)
(
0
,
1
)
on the board. On every move we replace two numbers
a
,
b
a,b
a
,
b
with roots of
x
2
−
a
x
+
b
=
0
x^2-ax+b=0
x
2
−
a
x
+
b
=
0
(if it has two roots). Prove that process is not endless.
Cents in glasses
There are
100
100
100
glasses, with
101
,
102
,
.
.
.
,
200
101,102,...,200
101
,
102
,
...
,
200
cents.Two players play next game. In every move they can take some cents from one glass, but after move should be different number of cents in every glass. Who will win with right strategy?
5
2
Hide problems
St.Peterburg, P5 Grade 11, 2013
Let
x
1
x_1
x
1
, ... ,
x
n
+
1
∈
[
0
,
1
]
x_{n+1} \in [0,1]
x
n
+
1
∈
[
0
,
1
]
and
x
1
=
x
n
+
1
x_1=x_{n+1}
x
1
=
x
n
+
1
. Prove that
∏
i
=
1
n
(
1
−
x
i
x
i
+
1
+
x
i
2
)
≥
1.
\prod_{i=1}^{n} (1-x_ix_{i+1}+x_i^2)\ge 1.
i
=
1
∏
n
(
1
−
x
i
x
i
+
1
+
x
i
2
)
≥
1.
A. Khrabrov, F. Petrov
St.Peterburg, P5 Grade 10, 2013
Given quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
B
=
B
C
=
C
D
AB=BC=CD
A
B
=
BC
=
C
D
. Let
A
C
∩
B
D
=
O
AC\cap BD=O
A
C
∩
B
D
=
O
,
X
,
Y
X,Y
X
,
Y
are symmetry points of
O
O
O
respect to midpoints of
B
C
BC
BC
,
A
D
AD
A
D
, and
Z
Z
Z
is intersection point of lines, which perpendicular bisects of
A
C
AC
A
C
,
B
D
BD
B
D
. Prove that
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
are collinear.