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Problems
Contests
National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
2010 Saint Petersburg Mathematical Olympiad
2010 Saint Petersburg Mathematical Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(7)
7
3
Hide problems
Yummy segments
600
600
600
integer numbers from
[
1
,
1000
]
[1,1000]
[
1
,
1000
]
colored in red. Natural segment
[
n
,
k
]
[n,k]
[
n
,
k
]
is called yummy if for every natural
t
t
t
from
[
1
,
k
−
n
]
[1,k-n]
[
1
,
k
−
n
]
there are two red numbers
a
,
b
a,b
a
,
b
from
[
n
,
k
]
[n,k]
[
n
,
k
]
and
b
−
a
=
t
b-a=t
b
−
a
=
t
. Prove that there is yummy segment with
[
a
,
b
]
[a,b]
[
a
,
b
]
with
b
−
a
≥
199
b-a \geq 199
b
−
a
≥
199
Coloring chess square
200
×
200
200 \times 200
200
×
200
square is colored in chess order. In one move we can take every
2
×
3
2 \times 3
2
×
3
rectangle and change color of all its cells. Can we make all cells of square in same color ?
Some geometry
Incircle of
A
B
C
ABC
A
BC
tangent
A
B
,
A
C
,
B
C
AB,AC,BC
A
B
,
A
C
,
BC
in
C
1
,
B
1
,
A
1
C_1,B_1,A_1
C
1
,
B
1
,
A
1
.
A
A
1
AA_1
A
A
1
intersect incircle in
E
E
E
.
N
N
N
is midpoint
B
1
A
1
B_1A_1
B
1
A
1
.
M
M
M
is symmetric to
N
N
N
relatively
A
A
1
AA_1
A
A
1
. Prove that
∠
E
M
C
=
90
\angle EMC= 90
∠
EMC
=
90
6
3
Hide problems
Inequality
For positive numbers is true that
a
b
+
a
c
+
b
c
=
a
+
b
+
c
ab+ac+bc=a+b+c
ab
+
a
c
+
b
c
=
a
+
b
+
c
Prove
a
+
b
+
c
+
1
≥
4
a
b
c
a+b+c+1 \geq 4abc
a
+
b
+
c
+
1
≥
4
ab
c
Extract primes
Natural number
N
N
N
is given. Let
p
N
p_N
p
N
- biggest prime, that
≤
N
\leq N
≤
N
. On every move we replace
N
N
N
by
N
−
p
N
N-p_N
N
−
p
N
. We repeat this until we get
0
0
0
or
1
1
1
. Prove that exists such number
N
N
N
, that we need exactly
1000
1000
1000
turns to make
0
0
0
Inequality
For positive is true
3
a
b
c
≥
a
+
b
+
c
\frac{3}{abc} \geq a+b+c
ab
c
3
≥
a
+
b
+
c
Prove
1
a
+
1
b
+
1
c
≥
a
+
b
+
c
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq a+b+c
a
1
+
b
1
+
c
1
≥
a
+
b
+
c
5
2
Hide problems
Pyramid geometry
S
A
B
C
D
SABCD
S
A
BC
D
is quadrangular pyramid. Lateral faces are acute triangles with orthocenters lying in one plane.
A
B
C
D
ABCD
A
BC
D
is base of pyramid and
A
C
AC
A
C
and
B
D
BD
B
D
intersects at
P
P
P
, where
S
P
SP
SP
is height of pyramid. Prove that
A
C
⊥
B
D
AC \perp BD
A
C
⊥
B
D
Roads and cities
There are
2010
2010
2010
cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Ministry destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly
11
11
11
different cities?
4
3
Hide problems
Roads and cities
There are
2010
2010
2010
cities in country, and
3
3
3
roads go from every city. President and Prime Minister play next game. They sell roads by turn to one of
3
3
3
companies( one road is one turn). President will win, if three roads from some city are sold to different companies. Who will win?
Big degree
A
A
A
-is
20
20
20
-digit number. We write
101
101
101
numbers
A
A
A
then erase last
11
11
11
digits. Prove that this
2009
2009
2009
-digit number can not be degree of
2
2
2
Substract primes
Natural number
N
N
N
is given. Let
p
N
p_N
p
N
- biggest prime, that
≤
N
\leq N
≤
N
. On every move we replace
N
N
N
by
N
−
p
N
N-p_N
N
−
p
N
. We repeat this until we get
0
0
0
or
1
1
1
. If we get
1
1
1
then
N
N
N
is called as good, else is bad. For example,
95
95
95
is good because we get
95
→
6
→
1
95 \to 6 \to 1
95
→
6
→
1
. Prove that among numbers from
1
1
1
to
1000000
1000000
1000000
there are between one quarter and half good numbers
3
3
Hide problems
Irrational root
a
a
a
is irrational , but
a
a
a
and
a
3
−
6
a
a^3-6a
a
3
−
6
a
are roots of square polynomial with integer coefficients.Find
a
a
a
Cities and roads.
There are
2009
2009
2009
cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Minisrty destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly
75
75
75
different cities?
Some geometry
M
,
N
M,N
M
,
N
are midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
for convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Points
X
X
X
and
Y
Y
Y
are on
A
D
AD
A
D
and
B
C
BC
BC
and
X
D
=
3
A
X
,
Y
C
=
3
B
Y
XD=3AX,YC=3BY
X
D
=
3
A
X
,
Y
C
=
3
B
Y
.
∠
M
X
A
=
∠
M
Y
B
=
90
\angle MXA=\angle MYB = 90
∠
MX
A
=
∠
M
Y
B
=
90
. Prove that
∠
X
M
N
=
∠
A
B
C
\angle XMN=\angle ABC
∠
XMN
=
∠
A
BC
2
2
Hide problems
Geometry with midpoints
A
B
C
ABC
A
BC
is triangle with
A
B
=
B
C
AB=BC
A
B
=
BC
.
X
,
Y
X,Y
X
,
Y
are midpoints of
A
C
AC
A
C
and
A
B
AB
A
B
.
Z
Z
Z
is base of perpendicular from
B
B
B
to
C
Y
CY
C
Y
. Prove, that circumcenter of
X
Y
Z
XYZ
X
Y
Z
lies on
A
C
AC
A
C
Divisors of big numbers.
There are
10
10
10
consecutive 30-digit numbers. We write the biggest divisor for every number ( divisor is not equal number). Prove that some written numbers ends with same digit.
1
3
Hide problems
King in chess
Chess king is standing in some square of chessboard. Every sunday it is moved to one square by diagonal, and every another day it is moved to one square by horisontal or vertical. What maximal numbers of moves can be made ?
System of equations
Solve in positives
x
y
=
z
,
y
z
=
x
,
z
x
=
y
x^y=z,y^z=x,z^x=y
x
y
=
z
,
y
z
=
x
,
z
x
=
y
Some polynomials
f
(
x
)
f(x)
f
(
x
)
is square trinomial. Is it always possible to find polynomial
g
(
x
)
g(x)
g
(
x
)
with fourth degree, such that
f
(
g
(
x
)
)
=
0
f(g(x))=0
f
(
g
(
x
))
=
0
has not roots?