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Problems
Contests
National and Regional Contests
Russia Contests
Saint Petersburg Mathematical Olympiad
2014 Saint Petersburg Mathematical Olympiad
2014 Saint Petersburg Mathematical Olympiad
Part of
Saint Petersburg Mathematical Olympiad
Subcontests
(7)
7
3
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Another geometry
I
I
I
- incenter ,
M
M
M
- midpoint of arc
B
A
C
BAC
B
A
C
of circumcircle,
A
L
AL
A
L
- angle bisector of triangle
A
B
C
ABC
A
BC
.
M
I
MI
M
I
intersect circumcircle in
K
K
K
. Circumcircle of
A
K
L
AKL
A
K
L
intersect
B
C
BC
BC
at
L
L
L
and
P
P
P
. Prove that
∠
A
I
P
=
90
\angle AIP=90
∠
A
I
P
=
90
Cities and roads
Some cities in country are connected with oneway road. It is known that every closed cyclic route, that don`t break traffic laws, consists of even roads. Prove that king of city can place military bases in some cities such that there are not roads between these cities, but for every city without base we can go from city with base by no more than
1
1
1
road.I think it should be one more condition, like there is cycle that connect all cities
Numbers in the table
Natural
a
,
b
,
c
a,b,c
a
,
b
,
c
are pairwise prime. There is infinite table with one integer number in every cell. Sum of numbers in every
a
×
a
a \times a
a
×
a
, every
b
×
b
b \times b
b
×
b
, every
c
×
c
c \times c
c
×
c
squares is even. Is it true, that every number in table must be even?
6
2
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Childs in cells
In the
n
×
n
n \times n
n
×
n
table in every cell there is one child. Every child looks in neigbour cell. So every child sees ear or back of the head of neighbour. What is minimal number children, that see ear ?
Some geometry
Points
A
,
B
A,B
A
,
B
are on circle
ω
\omega
ω
. Points
C
C
C
and
D
D
D
are moved on the arc
A
B
AB
A
B
, such that
C
D
CD
C
D
has constant length.
I
1
,
I
2
I_1,I_2
I
1
,
I
2
- incenters of
A
B
C
ABC
A
BC
and
A
B
D
ABD
A
B
D
. Prove that line
I
1
I
2
I_1I_2
I
1
I
2
is tangent to some fixed circle.
5
3
Hide problems
Set of natural
M
M
M
is infinite set of natural numbers. If
a
,
b
,
a
≠
b
a,b, a\neq b
a
,
b
,
a
=
b
are in
M
M
M
, then
a
b
+
2
a^b+2
a
b
+
2
or
a
b
−
2
a^b-2
a
b
−
2
( or both) are in
M
M
M
. Prove that there is composite number in
M
M
M
Some geometry
Incircle
ω
\omega
ω
of
A
B
C
ABC
A
BC
touch
A
C
AC
A
C
at
B
1
B_1
B
1
. Point
E
,
F
E,F
E
,
F
on the
ω
\omega
ω
such that
∠
A
E
B
1
=
∠
B
1
F
C
=
90
\angle AEB_1=\angle B_1FC=90
∠
A
E
B
1
=
∠
B
1
FC
=
90
. Tangents to
ω
\omega
ω
at
E
,
F
E,F
E
,
F
intersects in
D
D
D
, and
B
B
B
and
D
D
D
are on different sides for line
A
C
AC
A
C
.
M
M
M
- midpoint of
A
C
AC
A
C
. Prove, that
A
E
,
C
F
,
D
M
AE,CF,DM
A
E
,
CF
,
D
M
intersects at one point.
Napkin on the plane
On a cellular plane with a cell side equal to
1
1
1
, arbitrarily
100
×
100
100 \times 100
100
×
100
napkin is thrown. It covers some nodes (the node lying on the border of a napkin, is also considered covered). What is the smallest number of lines (going not necessarily along grid lines) you can certainly cover all these nodes?
4
3
Hide problems
Another geometry
Points
B
1
,
C
1
B_1,C_1
B
1
,
C
1
are on
A
C
AC
A
C
and
A
B
AB
A
B
and
B
1
C
1
∥
B
C
B_1C_1 \parallel BC
B
1
C
1
∥
BC
. Circumcircle of
A
B
B
1
ABB_1
A
B
B
1
intersect
C
C
1
CC_1
C
C
1
at
L
L
L
. Circumcircle
C
L
B
1
CLB_1
C
L
B
1
is tangent to
A
L
AL
A
L
. Prove
A
L
≤
A
C
+
A
C
1
2
AL \leq \frac{AC+AC_1}{2}
A
L
≤
2
A
C
+
A
C
1
Interesting inequality
a
1
≥
a
2
≥
.
.
.
≥
a
100
n
>
0
a_1\geq a_2\geq... \geq a_{100n}>0
a
1
≥
a
2
≥
...
≥
a
100
n
>
0
If we take from
(
a
1
,
a
2
,
.
.
.
,
a
100
n
)
(a_1,a_2,...,a_{100n})
(
a
1
,
a
2
,
...
,
a
100
n
)
some
2
n
+
1
2n+1
2
n
+
1
numbers
b
1
≥
b
2
≥
.
.
.
≥
b
2
n
+
1
b_1\geq b_2 \geq ... \geq b_{2n+1}
b
1
≥
b
2
≥
...
≥
b
2
n
+
1
then
b
1
+
.
.
.
+
b
n
>
b
n
+
1
+
.
.
.
b
2
n
+
1
b_1+...+b_n > b_{n+1}+...b_{2n+1}
b
1
+
...
+
b
n
>
b
n
+
1
+
...
b
2
n
+
1
Prove, that
(
n
+
1
)
(
a
1
+
.
.
.
+
a
n
)
>
a
n
+
1
+
a
n
+
2
+
.
.
.
+
a
100
n
(n+1)(a_1+...+a_n)>a_{n+1}+a_{n+2}+...+a_{100n}
(
n
+
1
)
(
a
1
+
...
+
a
n
)
>
a
n
+
1
+
a
n
+
2
+
...
+
a
100
n
Venerable numbers
We call a natural number venerable if the sum of all its divisors, including
1
1
1
, but not including the number itself, is
1
1
1
less than this number. Find all the venerable numbers, some exact degree of which is also venerable.
3
3
Hide problems
Some geometry
D
D
D
is inner point of triangle
A
B
C
ABC
A
BC
.
E
E
E
is on
B
D
BD
B
D
and
C
E
=
B
D
CE=BD
CE
=
B
D
.
∠
A
B
D
=
∠
E
C
D
=
10
,
∠
B
A
D
=
40
,
∠
C
E
D
=
60
\angle ABD=\angle ECD=10,\angle BAD=40,\angle CED=60
∠
A
B
D
=
∠
EC
D
=
10
,
∠
B
A
D
=
40
,
∠
CE
D
=
60
Prove, that
A
B
>
A
C
AB>AC
A
B
>
A
C
Coloring numbers
N
N
N
in natural. There are natural numbers from
N
3
N^3
N
3
to
N
3
+
N
N^3+N
N
3
+
N
on the board.
a
a
a
numbers was colored in red,
b
b
b
numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that
b
∣
a
b|a
b
∣
a
Deputies in commisions
100
100
100
deputies formed
450
450
450
commissions. Each two commissions has no more than three common deputies, and every
5
5
5
- no more than one. Prove that, that there are
4
4
4
commissions that has exactly one common deputy each.
2
3
Hide problems
Destroying the roads
There are cities in country, and some cities are connected by roads. Not more than
100
100
100
roads go from every city. Set of roads is called as ideal if all roads in set have not common ends, and we can not add one more road in set without breaking this rule. Every day minister destroy one ideal set of roads. Prove, that he need not more than
199
199
199
days to destroy all roads in country.
Points on lines
There are
40
40
40
points on the two parallel lines. We divide it to pairs, such that line segments, that connects point in pair, do not intersect each other ( endpoint from one segment cannot lies on another segment). Prove, that number of ways to do it is less than
3
39
3^{39}
3
39
Some geometry
All angles of
A
B
C
ABC
A
BC
are in
(
30
,
90
)
(30,90)
(
30
,
90
)
. Circumcenter of
A
B
C
ABC
A
BC
is
O
O
O
and circumradius is
R
R
R
. Point
K
K
K
is projection of
O
O
O
to angle bisector of
∠
B
\angle B
∠
B
, point
M
M
M
is midpoint
A
C
AC
A
C
. It is known, that
2
K
M
=
R
2KM=R
2
K
M
=
R
. Find
∠
B
\angle B
∠
B
1
2
Hide problems
Building function
Let
f
(
x
)
f(x)
f
(
x
)
is such function, that
f
(
x
)
=
1
f(x)=1
f
(
x
)
=
1
for integer
x
x
x
and
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
for non integer
x
x
x
. Build such function using only variable
x
x
x
, integer numbers, and operations
+
,
−
,
∗
,
/
,
[
.
]
+,-,*,/,[.]
+
,
−
,
∗
,
/
,
[
.
]
(plus, minus, multiply,divide and integer part)
Algebra with trinomial
f
(
x
)
f(x)
f
(
x
)
is square polynomial and
a
≠
b
a \neq b
a
=
b
such that
f
(
a
)
=
b
,
f
(
b
)
=
a
f(a)=b,f(b)=a
f
(
a
)
=
b
,
f
(
b
)
=
a
. Prove that there is not other pair
(
c
,
d
)
(c,d)
(
c
,
d
)
that
f
(
c
)
=
d
,
f
(
d
)
=
c
f(c)=d,f(d)=c
f
(
c
)
=
d
,
f
(
d
)
=
c