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Problems
Contests
National and Regional Contests
Russia Contests
Junior Tuymaada Olympiad
2003 Junior Tuymaada Olympiad
2003 Junior Tuymaada Olympiad
Part of
Junior Tuymaada Olympiad
Subcontests
(7)
8
1
Hide problems
2 rooms, everyone has even number of acquaintances in his room
A few people came to the party. Prove that they can be placed in two rooms so that each of them has in their own room an even number of acquaintances. (One of the rooms can be left empty.)
7
1
Hide problems
right angle wanted, tangents, angle bisector related
Through the point
K
K
K
lying outside the circle
ω
\omega
ω
, the tangents are drawn
K
B
KB
K
B
and
K
D
KD
KD
to this circle (
B
B
B
and
D
D
D
are tangency points) and a line intersecting a circle at points
A
A
A
and
C
C
C
. The bisector of angle
A
B
C
ABC
A
BC
intersects the segment
A
C
AC
A
C
at the point
E
E
E
and circle
ω
\omega
ω
at
F
F
F
. Prove that
∠
F
D
E
=
9
0
∘
\angle FDE = 90^\circ
∠
F
D
E
=
9
0
∘
.
6
1
Hide problems
1-100 numbers on a circle, pair of good numbers include only smaller
On a circle, numbers from
1
1
1
to
100
100
100
are arranged in some order. We call a pair of numbers good if these two numbers do not stand side by side, and at least on one of the two arcs into which they break a circle, all the numbers are less than each of them. What can be the total number of good pairs?
5
1
Hide problems
x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})
Prove that for any real
x
x
x
and
y
y
y
the inequality
x
2
1
+
2
y
2
+
y
2
1
+
2
x
2
≥
x
y
(
x
+
y
+
2
)
x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})
x
2
1
+
2
y
2
+
y
2
1
+
2
x
2
≥
x
y
(
x
+
y
+
2
)
.
4
1
Hide problems
1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 , sum is <= n ^ {2 ^ n}
The natural numbers
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\dots
…
,
a
n
a_n
a
n
satisfy the condition
1
/
a
1
+
1
/
a
2
+
…
+
1
/
a
n
=
1
1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1
1/
a
1
+
1/
a
2
+
…
+
1/
a
n
=
1
. Prove that all these numbers do not exceed
n
2
n
n ^ {2 ^ n}
n
2
n
3
1
Hide problems
equal angles wanted, incenter, circumcenter, excenter related
In the acute triangle
A
B
C
ABC
A
BC
, the point
I
I
I
is the center of the inscribed the circle, the point
O
O
O
is the center of the circumscribed circle and the point
I
a
I_a
I
a
is the center the excircle tangent to the side
B
C
BC
BC
and the extensions of the sides
A
B
AB
A
B
and
A
C
AC
A
C
. Point
A
′
A'
A
′
is symmetric to vertex
A
A
A
with respect to the line
B
C
BC
BC
. Prove that
∠
I
O
I
a
=
∠
I
A
′
I
a
\angle IOI_a = \angle IA'I_a
∠
I
O
I
a
=
∠
I
A
′
I
a
.
2
1
Hide problems
3x+1 and 6x-2 are perfect squares while 6x^2 -1 is prime
Find all natural
x
x
x
for which
3
x
+
1
3x+1
3
x
+
1
and
6
x
−
2
6x-2
6
x
−
2
are perfect squares, and the number
6
x
2
−
1
6x^2-1
6
x
2
−
1
is prime.