MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1979 Bulgaria National Olympiad
1979 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 6
1
Hide problems
partition [2n] into k disjoint subsets, existence of even numbers in subset
The set
M
=
{
1
,
2
,
…
,
2
n
}
(
n
≥
2
)
M=\{1,2,\ldots,2n\}~(n\ge2)
M
=
{
1
,
2
,
…
,
2
n
}
(
n
≥
2
)
is partitioned into
k
k
k
nonintersecting subsets
M
1
,
M
2
,
…
,
M
k
M_1,M_2,\ldots,M_k
M
1
,
M
2
,
…
,
M
k
, where
k
3
+
1
≤
n
k^3+1\le n
k
3
+
1
≤
n
. Prove that there exist
k
+
1
k+1
k
+
1
even numbers
2
j
1
,
2
j
2
,
…
,
2
j
k
+
1
2j_1,2j_2,\ldots,2j_{k+1}
2
j
1
,
2
j
2
,
…
,
2
j
k
+
1
in
M
M
M
that are in one and the same subset
M
j
M_j
M
j
(
1
≤
j
≤
k
)
(1\le j\le k)
(
1
≤
j
≤
k
)
such that the numbers
2
j
1
−
1
,
2
j
2
−
1
,
…
,
2
j
k
+
1
−
1
2j_1-1,2j_2-1,\ldots,2j_{k+1}-1
2
j
1
−
1
,
2
j
2
−
1
,
…
,
2
j
k
+
1
−
1
are also in one and the same subset
M
r
M_r
M
r
(
1
≤
r
≤
k
)
(1\le r\le k)
(
1
≤
r
≤
k
)
.
Problem 5
1
Hide problems
triangles similar in pentagon, three vertices make equilateral triangle
A convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
satisfies
A
B
=
B
C
=
C
A
AB=BC=CA
A
B
=
BC
=
C
A
and
C
D
=
D
E
=
E
C
CD=DE=EC
C
D
=
D
E
=
EC
. Let
S
S
S
be the center of the equilateral triangle
A
B
C
ABC
A
BC
and
M
M
M
and
N
N
N
be the midpoints of
B
D
BD
B
D
and
A
E
AE
A
E
, respectively. Prove that the triangles
S
M
E
SME
SME
and
S
N
D
SND
SN
D
are similar.
Problem 3
1
Hide problems
dividing triangle sides, counting triangles without parallel sides
Each side of a triangle
A
B
C
ABC
A
BC
has been divided into
n
+
1
n+1
n
+
1
equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of
△
A
B
C
\triangle ABC
△
A
BC
.
Problem 2
1
Hide problems
lines concurrent in tetrahedron
Points
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
are taken on respective edges
A
C
AC
A
C
,
A
B
AB
A
B
,
B
D
BD
B
D
, and
C
D
CD
C
D
of a tetrahedron
A
B
C
D
ABCD
A
BC
D
so that
P
R
PR
PR
and
Q
S
QS
QS
intersect at point
N
N
N
and
P
S
PS
PS
and
Q
R
QR
QR
intersect at point
M
M
M
. The line
M
N
MN
MN
meets the plane
A
B
C
ABC
A
BC
at point
L
L
L
. Prove that the lines
A
L
AL
A
L
,
B
P
BP
BP
, and
C
Q
CQ
CQ
are concurrent.
Problem 4
1
Hide problems
larger root of quadratic, parameter involved
For each real number
k
k
k
, denote by
f
(
k
)
f(k)
f
(
k
)
the larger of the two roots of the quadratic equation
(
k
2
+
1
)
x
2
+
10
k
x
−
6
(
9
k
2
+
1
)
=
0.
(k^2+1)x^2+10kx-6(9k^2+1)=0.
(
k
2
+
1
)
x
2
+
10
k
x
−
6
(
9
k
2
+
1
)
=
0.
Show that the function
f
(
k
)
f(k)
f
(
k
)
attains a minimum and maximum and evaluate these two values.
Problem 1
1
Hide problems
x^2 + 5 = y^3
Show that there are no integers
x
x
x
and
y
y
y
satisfying
x
2
+
5
=
y
3
x^2 + 5 = y^3
x
2
+
5
=
y
3
.Daniel Harrer