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Problems
Contests
National and Regional Contests
Russia Contests
Junior Tuymaada Olympiad
2007 Junior Tuymaada Olympiad
2007 Junior Tuymaada Olympiad
Part of
Junior Tuymaada Olympiad
Subcontests
(5)
7
1
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concyclic points wanted, parallels and circumcircle given
On the
A
B
AB
A
B
side of the triangle
A
B
C
ABC
A
BC
, points
X
X
X
and
Y
Y
Y
are chosen, on the side of
A
C
AC
A
C
is a point of
Z
Z
Z
, and on the side of
B
C
BC
BC
is a point of
T
T
T
. Wherein
X
Z
∥
B
C
XZ \parallel BC
XZ
∥
BC
,
Y
T
∥
A
C
YT \parallel AC
Y
T
∥
A
C
. Line
T
Z
TZ
TZ
intersects the circumscribed circle of triangle
A
B
C
ABC
A
BC
at points
D
D
D
and
E
E
E
. Prove that points
X
X
X
,
Y
Y
Y
,
D
D
D
and
E
E
E
lie on the same circle.
6
1
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1-round chess tournament involves 10 players from 2 countries
One-round chess tournament involves
10
10
10
players from two countries. For a victory, one point is given, for a draw - half a point, for defeat - zero. All players scored a different number of points. Prove that one of the chess players scored in meetings with his countrymen less points, than meeting with players from another country.
4
1
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circumcircle of OIH passes through vertex A, prove an angle is 60^o
An acute-angle non-isosceles triangle
A
B
C
ABC
A
BC
is given. The point
H
H
H
is its orthocenter, the points
O
O
O
and
I
I
I
are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle
O
I
H
OIH
O
I
H
passes through the vertex
A
A
A
. Prove that one of the angles of the triangle is
6
0
∘
60^\circ
6
0
∘
.
3
1
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dividing a 600x600 square in 4 types of cells
A square
600
×
600
600 \times 600
600
×
600
divided into figures of
4
4
4
cells of the forms in the figure: In the figures of the first two types in shaded cells The number
2
k
2 ^ k
2
k
is written, where
k
k
k
is the number of the column in which this cell. Prove that the sum of all the numbers written is divisible by
9
9
9
.
2
1
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a permutation of coefficients of trinomials , f(x) \geq g(x) for all real x?
Two quadratic trinomials
f
(
x
)
f (x)
f
(
x
)
and
g
(
x
)
g (x)
g
(
x
)
differ from each other only by a permutation of coefficients. Could it be that
f
(
x
)
≥
g
(
x
)
f (x) \geq g (x)
f
(
x
)
≥
g
(
x
)
for all real
x
x
x
?