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Contests
National and Regional Contests
Russia Contests
Junior Tuymaada Olympiad
2004 Junior Tuymaada Olympiad
2004 Junior Tuymaada Olympiad
Part of
Junior Tuymaada Olympiad
Subcontests
(5)
6
1
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pos. integer with integer sum of the reciprocals of all its natural divisors
We call a positive integer good if the sum of the reciprocals of all its natural divisors are integers. Prove that if
m
m
m
is a good number, and
p
>
m
p> m
p
>
m
is a prime number, then
p
m
pm
p
m
is not good.
4
1
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every natural belonging to A or B, at least one of: k +17 \in A , k-31 \in B
Given the disjoint finite sets of natural numbers
A
A
A
and
B
B
B
, consisting of
n
n
n
and
m
m
m
elements, respectively. It is known that every natural number belonging to
A
A
A
or
B
B
B
satisfies at least one of the conditions
k
+
17
∈
A
k + 17 \in A
k
+
17
∈
A
,
k
−
31
∈
B
k-31 \in B
k
−
31
∈
B
. Prove that
17
n
=
31
m
17n = 31m
17
n
=
31
m
3
1
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collinearity wanted, 2 points inside arc are chosen, perpendiculars related
Point
O
O
O
is the center of the circumscribed circle of an acute triangle
A
b
c
Abc
A
b
c
. A certain circle passes through the points
B
B
B
and
C
C
C
and intersects sides
A
B
AB
A
B
and
A
C
AC
A
C
of a triangle. On its arc lying inside the triangle, points
D
D
D
and
E
E
E
are chosen so that the segments
B
D
BD
B
D
and
C
E
CE
CE
pass through the point
O
O
O
. Perpendicular
D
D
1
DD_1
D
D
1
to
A
B
AB
A
B
side and perpendicular
E
E
1
EE_1
E
E
1
to
A
C
AC
A
C
side intersect at
M
M
M
. Prove that the points
A
A
A
,
M
M
M
and
O
O
O
lie on the same straight line.
2
1
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n nunbers in a circle such that noone exceed 60% of sum of 2 neighbors
For which natural
n
≥
3
n \geq 3
n
≥
3
numbers from 1 to
n
n
n
can be arranged by a circle so that each number does not exceed
60
60
60
% of the sum of its two neighbors?
1
1
Hide problems
starting with rational, r -> \sqrt {r + 1} -> \sqrt {r + 1} -> ..., irrational
A positive rational number is written on the blackboard. Every minute Vasya replaces the number
r
r
r
written on the board with
r
+
1
\sqrt {r + 1}
r
+
1
. Prove that someday he will get an irrational number.