MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1987 Bulgaria National Olympiad
1987 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 6
1
Hide problems
iterated triangle mapping
Let
Δ
\Delta
Δ
be the set of all triangles inscribed in a given circle, with angles whose measures are integer numbers of degrees different than
4
5
∘
,
9
0
∘
45^\circ,90^\circ
4
5
∘
,
9
0
∘
and
13
5
∘
135^\circ
13
5
∘
. For each triangle
T
∈
Δ
T\in\Delta
T
∈
Δ
,
f
(
T
)
f(T)
f
(
T
)
denotes the triangle with vertices at the second intersection points of the altitudes of
T
T
T
with the circle.(a) Prove that there exists a natural number
n
n
n
such that for every triangle
T
∈
Δ
T\in\Delta
T
∈
Δ
, among the triangles
T
,
f
(
T
)
,
…
,
f
n
(
T
)
T,f(T),\ldots,f^n(T)
T
,
f
(
T
)
,
…
,
f
n
(
T
)
(where
f
0
(
T
)
=
T
f^0(T)=T
f
0
(
T
)
=
T
and
f
k
(
T
)
=
f
(
f
k
−
1
(
T
)
)
f^k(T)=f(f^{k-1}(T))
f
k
(
T
)
=
f
(
f
k
−
1
(
T
))
) at least two are equal. (b) Find the smallest
n
n
n
with the property from (a).
Problem 5
1
Hide problems
point on median, projection to sides and perpendiculars
Let
E
E
E
be a point on the median
A
D
AD
A
D
of a triangle
A
B
C
ABC
A
BC
, and
F
F
F
be the projection of
E
E
E
onto
B
C
BC
BC
. From a point
M
M
M
on
E
F
EF
EF
the perpendiculars
M
N
MN
MN
to
A
C
AC
A
C
and
M
P
MP
MP
to
A
B
AB
A
B
are drawn. Prove that if the points
N
,
E
,
P
N,E,P
N
,
E
,
P
lie on a line, then
M
M
M
lies on the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
.
Problem 4
1
Hide problems
perfect square recurrence
The sequence
(
x
n
)
n
∈
N
(x_n)_{n\in\mathbb N}
(
x
n
)
n
∈
N
is defined by
x
1
=
x
2
=
1
x_1=x_2=1
x
1
=
x
2
=
1
,
x
n
+
2
=
14
x
n
+
1
−
x
n
−
4
x_{n+2}=14x_{n+1}-x_n-4
x
n
+
2
=
14
x
n
+
1
−
x
n
−
4
for each
n
∈
N
n\in\mathbb N
n
∈
N
. Prove that all terms of this sequence are perfect squares.
Problem 3
1
Hide problems
largest ball in pyramid given side equations
Let
M
A
B
C
D
MABCD
M
A
BC
D
be a pyramid with the square
A
B
C
D
ABCD
A
BC
D
as the base, in which
M
A
=
M
D
MA=MD
M
A
=
M
D
,
M
A
2
+
A
B
2
=
M
B
2
MA^2+AB^2=MB^2
M
A
2
+
A
B
2
=
M
B
2
and the area of
△
A
D
M
\triangle ADM
△
A
D
M
is equal to
1
1
1
. Determine the radius of the largest ball that is contained in the given pyramid.
Problem 2
1
Hide problems
double rotation, rational ratio of angles
Let there be given a polygon
P
P
P
which is mapped onto itself by two rotations:
ρ
1
\rho_1
ρ
1
with center
O
1
O_1
O
1
and angle
ω
1
\omega_1
ω
1
, and
ρ
2
\rho_2
ρ
2
with center
O
2
O_2
O
2
and angle
ω
2
(
0
<
ω
i
<
2
π
)
\omega_2~(0<\omega_i<2\pi)
ω
2
(
0
<
ω
i
<
2
π
)
. Show that the ratio
ω
1
ω
2
\frac{\omega_1}{\omega_2}
ω
2
ω
1
is rational.
Problem 1
1
Hide problems
bounding root of polynomial
Let
f
(
x
)
=
x
n
+
a
1
x
n
−
1
+
…
+
a
n
(
n
≥
3
)
f(x)=x^n+a_1x^{n-1}+\ldots+a_n~(n\ge3)
f
(
x
)
=
x
n
+
a
1
x
n
−
1
+
…
+
a
n
(
n
≥
3
)
be a polynomial with real coefficients and
n
n
n
real roots, such that
a
n
−
1
a
n
>
n
+
1
\frac{a_{n-1}}{a_n}>n+1
a
n
a
n
−
1
>
n
+
1
. Prove that if
a
n
−
2
=
0
a_{n-2}=0
a
n
−
2
=
0
, then at least one root of
f
(
x
)
f(x)
f
(
x
)
lies in the open interval
(
−
1
2
,
1
n
+
1
)
\left(-\frac12,\frac1{n+1}\right)
(
−
2
1
,
n
+
1
1
)
.