MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1980 Bulgaria National Olympiad
1980 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
Problem 6
1
Hide problems
pentagonal pyramid is regular if edges,angles are equal
Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular.
Problem 5
1
Hide problems
choose 6 integers, 2 consecutive, out of [49]
Prove that the number of ways of choosing
6
6
6
among the first
49
49
49
positive integers, at least two of which are consecutive, is equal to
(
49
6
)
−
(
44
6
)
\binom{49}6-\binom{44}6
(
6
49
)
−
(
6
44
)
.
Problem 3
1
Hide problems
nonagonal pyramid, coloring edges and diagonals
Each diagonal of the base and each lateral edge of a
9
9
9
-gonal pyramid is colored either green or red. Show that there must exist a triangle with the vertices at vertices of the pyramid having all three sides of the same color.
Problem 1
1
Hide problems
decimal digits, unique sequence existence
Show that there exists a unique sequence of decimal digits
p
0
=
5
,
p
1
,
p
2
,
…
p_0=5,p_1,p_2,\ldots
p
0
=
5
,
p
1
,
p
2
,
…
such that, for any
k
k
k
, the square of any positive integer ending with
p
k
p
k
−
1
⋯
p
0
‾
\overline{p_kp_{k-1}\cdots p_0}
p
k
p
k
−
1
⋯
p
0
ends with the same digits.
Problem 2
1
Hide problems
S>4Q in tetrahedron, Q is cross-sectional area
(a) Prove that the area of a given convex quadrilateral is at least twice the area of an arbitrary convex quadrilateral inscribed in it whose sides are parallel to the diagonals of the original one. (b) A tetrahedron with surface area
S
S
S
is intersected by a plane perpendicular to two opposite edges. If the area of the cross-section is
Q
Q
Q
, prove that
S
>
4
Q
S>4Q
S
>
4
Q
.
Problem 4
1
Hide problems
Might be well-known...
Let
a
a
a
,
b
b
b
, and
c
c
c
be non-negative reals. Prove that
a
3
+
b
3
+
c
3
+
6
a
b
c
≥
(
a
+
b
+
c
)
3
4
a^3+b^3+c^3+6abc\ge \frac{(a+b+c)^3}{4}
a
3
+
b
3
+
c
3
+
6
ab
c
≥
4
(
a
+
b
+
c
)
3
.