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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1995 Bulgaria National Olympiad
1995 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
5
1
Hide problems
exists g(g(a)) = f(a) if f(m) = 1, f(m+n) = m+1 and f(i) = i+1 for all other i
Let
A
=
{
1
,
2
,
.
.
.
,
m
+
n
}
A = \{1,2,...,m + n\}
A
=
{
1
,
2
,
...
,
m
+
n
}
, where
m
,
n
m,n
m
,
n
are positive integers, and let the function f :
A
→
A
A \to A
A
→
A
be defined by:
f
(
m
)
=
1
f(m) = 1
f
(
m
)
=
1
,
f
(
m
+
n
)
=
m
+
1
f(m+n) = m+1
f
(
m
+
n
)
=
m
+
1
and
f
(
i
)
=
i
+
1
f(i) = i+1
f
(
i
)
=
i
+
1
for all the other
i
i
i
. (a) Prove that if
m
m
m
and
n
n
n
are odd, then there exists a function
g
:
A
→
A
g : A \to A
g
:
A
→
A
such that
g
(
g
(
a
)
)
=
f
(
a
)
g(g(a)) = f(a)
g
(
g
(
a
))
=
f
(
a
)
for all
a
∈
A
a \in A
a
∈
A
. (b) Prove that if
m
m
m
is even, then there is a function
g
:
A
→
A
g : A\to A
g
:
A
→
A
such that
g
(
g
(
a
)
)
=
f
(
a
)
g(g(a))=f(a)
g
(
g
(
a
))
=
f
(
a
)
for all
a
∈
A
a \in A
a
∈
A
is and only if
n
=
m
n = m
n
=
m
.
4
1
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midpoints wanted, 4 equal inradii in an equilateral triangle
Points
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
are selected on the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
respectively of an equilateral triangle
A
B
C
ABC
A
BC
in such a way that the inradii of the triangles
C
1
A
B
1
C_1AB_1
C
1
A
B
1
,
A
1
B
C
1
A_1BC_1
A
1
B
C
1
,
B
1
C
A
1
B_1CA_1
B
1
C
A
1
and
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
are equal. Prove that
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
are the midpoints of the corresponding sides.
3
1
Hide problems
2 player game with a heap of n stones
Two players
A
A
A
and
B
B
B
take stones one after the other from a heap with
n
≥
2
n \ge 2
n
≥
2
stones.
A
A
A
begins the game and takes at least one stone, but no more than
n
−
1
n -1
n
−
1
stones. Thereafter, a player on turn takes at least one, but no more than the other player has taken before him. The player who takes the last stone wins. Who of the players has a winning strategy?
1
1
Hide problems
n>1 divides a^{25} - a for evey integer a
Find the number of integers
n
>
1
n > 1
n
>
1
which divide
a
25
−
a
a^{25} - a
a
25
−
a
for every integer
a
a
a
.
6
1
Hide problems
Problem in 1995 Bulgaria MO (Number Theory)
Suppose that
x
x
x
and
y
y
y
are different real numbers such that
x
n
−
y
n
x
−
y
\frac{x^n-y^n}{x-y}
x
−
y
x
n
−
y
n
is an integer for some four consecutive positive integers
n
n
n
. Prove that
x
n
−
y
n
x
−
y
\frac{x^n-y^n}{x-y}
x
−
y
x
n
−
y
n
is an integer for all positive integers n.
2
1
Hide problems
tangent to the circumcircle
Let triangle ABC has semiperimeter
p
p
p
. E,F are located on AB such that CE\equal{}CF\equal{}p. Prove that the C-excircle of triangle ABC touches the circumcircle (EFC).