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Problems
Contests
National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
2015 Saint Petersburg Mathematical Olympiad
2015 Saint Petersburg Mathematical Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(7)
3
3
Hide problems
Weights sets
There are weights with mass
1
,
3
,
5
,
.
.
.
.
,
2
i
+
1
,
.
.
.
1,3,5,....,2i+1,...
1
,
3
,
5
,
....
,
2
i
+
1
,
...
Let
A
(
n
)
A(n)
A
(
n
)
-is number of different sets with total mass equal
n
n
n
( For example
A
(
9
)
=
2
A(9)=2
A
(
9
)
=
2
, because we have two sets
9
=
9
=
1
+
3
+
5
9=9=1+3+5
9
=
9
=
1
+
3
+
5
). Prove that
A
(
n
)
≤
A
(
n
+
1
)
A(n) \leq A(n+1)
A
(
n
)
≤
A
(
n
+
1
)
for
n
>
1
n>1
n
>
1
Inequality with bisectors
A
B
C
D
ABCD
A
BC
D
- convex quadrilateral. Bisectors of angles
A
A
A
and
D
D
D
intersect in
K
K
K
, Bisectors of angles
B
B
B
and
C
C
C
intersect in
L
L
L
. Prove
2
K
L
≥
∣
A
B
−
B
C
+
C
D
−
D
A
∣
2KL \geq |AB-BC+CD-DA|
2
K
L
≥
∣
A
B
−
BC
+
C
D
−
D
A
∣
Colored table
All cells of
2015
×
2015
2015 \times 2015
2015
×
2015
table colored in one of
4
4
4
colors. We count number of ways to place one tetris T-block in table. Prove that T-block has cell of all
4
4
4
colors in less than
51
%
51\%
51%
of total number of ways.
2
3
Hide problems
Interesting primes
a
,
b
>
1
a,b>1
a
,
b
>
1
- are naturals, and
a
2
+
b
,
a
+
b
2
a^2+b,a+b^2
a
2
+
b
,
a
+
b
2
are primes. Prove
(
a
b
+
1
,
a
+
b
)
=
1
(ab+1,a+b)=1
(
ab
+
1
,
a
+
b
)
=
1
Chessboard coloring
The beaver is chess piece that move to
2
2
2
cells by horizontal or vertical. Every cell of
100
×
100
100 \times 100
100
×
100
chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need?
Trapezoid geometry
A
B
=
C
D
,
A
D
∥
B
C
AB=CD,AD \parallel BC
A
B
=
C
D
,
A
D
∥
BC
and
A
D
>
B
C
AD>BC
A
D
>
BC
.
Ω
\Omega
Ω
is circumcircle of
A
B
C
D
ABCD
A
BC
D
. Point
E
E
E
is on
Ω
\Omega
Ω
such that
B
E
⊥
A
D
BE \perp AD
BE
⊥
A
D
. Prove that
A
E
+
B
C
>
D
E
AE+BC>DE
A
E
+
BC
>
D
E
5
2
Hide problems
Squares, rectangles and areas
Square with side 100 was cut by 99 horizontal and 99 vertical lines into 10000 rectangles (not necessarily with integer sides). How many rectangles in this square with area not exceeding 1 at least can be?
Pentagon angles
A
B
C
D
E
ABCDE
A
BC
D
E
is convex pentagon.
∠
B
C
A
=
∠
B
E
A
=
∠
B
D
A
2
,
∠
B
D
C
=
∠
E
D
A
\angle BCA=\angle BEA = \frac{\angle BDA}{2}, \angle BDC =\angle EDA
∠
BC
A
=
∠
BE
A
=
2
∠
B
D
A
,
∠
B
D
C
=
∠
E
D
A
. Prove, that
∠
D
E
B
=
∠
D
A
C
\angle DEB=\angle DAC
∠
D
EB
=
∠
D
A
C
4
3
Hide problems
Parallelogram geometry
A
B
C
D
ABCD
A
BC
D
is convex quadrilateral. Circumcircle of
A
B
C
ABC
A
BC
intersect
A
D
AD
A
D
and
D
C
DC
D
C
at points
P
P
P
and
Q
Q
Q
. Circumcircle of
A
D
C
ADC
A
D
C
intersect
A
B
AB
A
B
and
B
C
BC
BC
at points
S
S
S
and
R
R
R
. Prove that if
P
Q
R
S
PQRS
PQRS
is parallelogram then
A
B
C
D
ABCD
A
BC
D
is parallelogram
Largest 'Olympic' number not exceeding 2015
A positive integer
n
n
n
is called Olympic, if there exists a quadratic trinomial with integer coeffecients
f
(
x
)
f(x)
f
(
x
)
satisfying
f
(
f
(
n
)
)
=
0
f(f(\sqrt{n}))=0
f
(
f
(
n
))
=
0
. Determine, with proof, the largest Olympic number not exceeding
2015
2015
2015
.A. Khrabrov
Nice inequality: prove 4x+y+z>=2
Positive numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfy the condition
x
y
+
y
z
+
z
x
+
2
x
y
z
=
1.
xy + yz + zx + 2xyz = 1.
x
y
+
yz
+
z
x
+
2
x
yz
=
1.
Prove that
4
x
+
y
+
z
≥
2.
4x + y + z \ge 2.
4
x
+
y
+
z
≥
2.
A. Khrabrov
1
3
Hide problems
Child camps
There is child camp with some rooms. Call room as
4
−
4-
4
−
room, if
4
4
4
children live here. Not less then half of all rooms are
4
−
4-
4
−
rooms , other rooms are
3
−
3-
3
−
rooms. Not less than
2
/
3
2/3
2/3
girls live in
3
−
3-
3
−
rooms. Prove that not less than
35
%
35\%
35%
of all children are boys.
find the value
x
,
y
x,y
x
,
y
are real numbers such that
x
2
+
y
2
=
1
,
20
x
3
−
15
x
=
3
x^2+y^2=1 , 20x^3-15x=3
x
2
+
y
2
=
1
,
20
x
3
−
15
x
=
3
Find the value of
∣
20
y
3
−
15
y
∣
|20y^3-15y|
∣20
y
3
−
15
y
∣
.(K. Tyshchuk)
Quadratic with f(f(sqrt2))=0
Is there a quadratic trinomial
f
(
x
)
f(x)
f
(
x
)
with integer coefficients such that
f
(
f
(
2
)
)
=
0
f(f(\sqrt{2}))=0
f
(
f
(
2
))
=
0
? A. Khrabrov
7
2
Hide problems
ST passes through midpoint of arc ABC
Let
B
L
BL
B
L
be angle bisector of acute triangle
A
B
C
ABC
A
BC
.Point
K
K
K
choosen on
B
L
BL
B
L
such that
∡
A
K
C
−
∡
A
B
C
=
90
º
\measuredangle AKC-\measuredangle ABC=90º
∡
A
K
C
−
∡
A
BC
=
90º
.point
S
S
S
lies on the extention of
B
L
BL
B
L
from
L
L
L
such that
∡
A
S
C
=
90
º
\measuredangle ASC=90º
∡
A
SC
=
90º
.Point
T
T
T
is diametrically opposite the point
K
K
K
on the circumcircle of
△
A
K
C
\triangle AKC
△
A
K
C
.Prove that
S
T
ST
ST
passes through midpoint of arc
A
B
C
ABC
A
BC
.(S. Berlov) :trampoline: my 100th post :trampoline:
Colored segment with numbers
There is convex
n
−
n-
n
−
gon. We color all its sides and also diagonals, that goes out from one vertex. So we have
2
n
−
3
2n-3
2
n
−
3
colored segments. We write positive numbers on colored segments. In one move we can take quadrilateral
A
B
C
D
ABCD
A
BC
D
such, that
A
C
AC
A
C
and all sides are colored, then remove
A
C
AC
A
C
and color
B
D
BD
B
D
with number
x
z
+
y
t
w
\frac{xz+yt}{w}
w
x
z
+
y
t
, where
x
,
y
,
z
,
t
,
w
x,y,z,t,w
x
,
y
,
z
,
t
,
w
- numbers on
A
B
,
B
C
,
C
D
,
D
A
,
A
C
AB,BC,CD,DA,AC
A
B
,
BC
,
C
D
,
D
A
,
A
C
. After some moves we found that all colored segments are same that was at beginning. Prove, that they have same number that was at beginning.
6
3
Hide problems
All numbers occur exactly once
A sequence of integers is defined as follows:
a
1
=
1
,
a
2
=
2
,
a
3
=
3
a_1=1,a_2=2,a_3=3
a
1
=
1
,
a
2
=
2
,
a
3
=
3
and for
n
>
3
n>3
n
>
3
,
a
n
=
The
smallest
integer
not
occurring
earlier,
which
is
relatively
prime
to
a
n
−
1
but
not
relatively
prime
to
a
n
−
2
.
a_n=\textsf{The smallest integer not occurring earlier, which is relatively prime to }a_{n-1}\textsf{ but not relatively prime to }a_{n-2}.
a
n
=
The smallest integer not occurring earlier, which is relatively prime to
a
n
−
1
but not relatively prime to
a
n
−
2
.
Prove that every natural number occurs exactly once in this sequence.M. Ivanov
Bunches of roads
In country there are some cities, some pairs of cities are connected with roads.From every city go out exactly
100
100
100
roads. We call
10
10
10
roads, that go out from same city, as bunch. Prove, that we can split all roads in several bunches.
Removing Edges for Connected Subgraph
There are
1
0
2015
10^{2015}
1
0
2015
planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of
2015
2015
2015
travel companies. The Emperor would like to close
k
k
k
of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of
k
k
k
for which this is always possible. (D. Karpov)