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Contests
National and Regional Contests
Russia Contests
239 Open Math Olympiad
2011 239 Open Mathematical Olympiad
2011 239 Open Mathematical Olympiad
Part of
239 Open Math Olympiad
Subcontests
(7)
6
1
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Rotating regular polygons
Some regular polygons are inscribed in a circle. Fedir turned some of them, so all polygons have a common vertice. Prove that the number of vertices did not increase.
3
1
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a+b+c+d=4, prove sum_{cyc}\frac{a}{a^3 + 4} < 4/5
Positive reals
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
satisfy
a
+
b
+
c
+
d
=
4
a+b+c+d=4
a
+
b
+
c
+
d
=
4
. Prove that
∑
c
y
c
a
a
3
+
4
≤
4
5
\sum_{cyc}\frac{a}{a^3 + 4} \le \frac{4}{5}
∑
cyc
a
3
+
4
a
≤
5
4
7
1
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(ab+bc+ca+1)(a+b)(b+c)(c+a) \ge 2abc(a+b+c+1)^2
Prove for positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
that
(
a
b
+
b
c
+
c
a
+
1
)
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≥
2
a
b
c
(
a
+
b
+
c
+
1
)
2
(ab+bc+ca+1)(a+b)(b+c)(c+a) \ge 2abc(a+b+c+1)^2
(
ab
+
b
c
+
c
a
+
1
)
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
≥
2
ab
c
(
a
+
b
+
c
+
1
)
2
5
2
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exist 1000 consecutive numbers, none divides by sum of its digits
Prove that there exist 1000 consecutive numbers such that none of them is divisible by its sum of the digits
20 blue points on circle, 1123 contain 10 red
There are 20 blue points on the circle and some red inside so no three are collinear. It turned out that there exists
1123
1123
1123
triangles with blue vertices having 10 red points inside. Prove that all triangles have 10 red points inside
4
2
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acute ABCD, BAD similar to BKL, then APK= LBC
In convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, where
A
B
=
A
D
AB=AD
A
B
=
A
D
, on
B
D
BD
B
D
point
K
K
K
is chosen. On
K
C
KC
K
C
point
L
L
L
is such that
△
B
A
D
∼
△
B
K
L
\bigtriangleup BAD \sim \bigtriangleup BKL
△
B
A
D
∼
△
B
K
L
. Line parallel to
D
L
DL
D
L
and passes through
K
K
K
, intersect
C
D
CD
C
D
at
P
P
P
. Prove that
∠
A
P
K
=
∠
L
B
C
\angle APK = \angle LBC
∠
A
P
K
=
∠
L
BC
.@below edited
Rombus ABCD, \angle AQO=\angle PBC
Rombus ABCD with acute angle
B
B
B
is given.
O
O
O
is a circumcenter of
A
B
C
ABC
A
BC
. Point
P
P
P
lies on line
O
C
OC
OC
beyond
C
C
C
.
P
D
PD
P
D
intersect the line that goes through
O
O
O
and parallel to
A
B
AB
A
B
at
Q
Q
Q
. Prove that
∠
A
Q
O
=
∠
P
B
C
\angle AQO=\angle PBC
∠
A
QO
=
∠
PBC
.
2
1
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for each pair 50 people that know one in a pair
There are
100
100
100
people in the group. Is it possible that for each pair of people exist at least
50
50
50
others, so every in that group knows exactly one person from the pair?
1
2
Hide problems
a+b=b(a-c), c+1 square of prime, then a+b of ab square
Positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy that
a
+
b
=
b
(
a
−
c
)
a+b=b(a-c)
a
+
b
=
b
(
a
−
c
)
and c+1 is a square of a prime. Prove that
a
+
b
a+b
a
+
b
or
a
b
ab
ab
is a square.
2AP=BC, X,Y symmetric to P, find C
In the acute triangle
A
B
C
ABC
A
BC
on
A
C
AC
A
C
point
P
P
P
is chosen such that
2
A
P
=
B
C
2AP=BC
2
A
P
=
BC
. Points
X
X
X
and
Y
Y
Y
are symmetric to
P
P
P
wrt
A
A
A
and
C
C
C
respectively. It turned out that
B
X
=
B
Y
BX=BY
BX
=
B
Y
. Find angle
C
C
C
.