MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1980 Polish MO Finals
1980 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
3
1
Hide problems
fair coin is to be flipped 100 times
Let
k
k
k
be an integer in the interval
[
1
,
99
]
[1,99]
[
1
,
99
]
. A fair coin is to be flipped
100
100
100
times. Let
ε
j
=
{
1
,
if the j-th flip is a head
2
,
f the j-th flip is a tail
\varepsilon_j =\begin{cases} 1, \text{if the j-th flip is a head} \\ 2, \text{f the j-th flip is a tail}\end{cases}
ε
j
=
{
1
,
if the j-th flip is a head
2
,
f the j-th flip is a tail
Let
M
k
M_k
M
k
denote the probability that there exists a number
i
i
i
such that
k
+
ε
1
+
.
.
.
+
ε
i
=
100
k+\varepsilon_1 +...+\varepsilon_i = 100
k
+
ε
1
+
...
+
ε
i
=
100
. How to choose
k
k
k
so as to maximize the probability
M
k
M_k
M
k
?
4
1
Hide problems
every 3 var polymial = equal 2 3 var. polynomials
Show that for every polynomial
W
W
W
in three variables there exist polynomials
U
U
U
and
V
V
V
such that:
W
(
x
,
y
,
z
)
=
U
(
x
,
y
,
z
)
+
V
(
x
,
y
,
z
)
,
W(x,y,z) = U(x,y,z)+V(x,y,z),
W
(
x
,
y
,
z
)
=
U
(
x
,
y
,
z
)
+
V
(
x
,
y
,
z
)
,
U
(
x
,
y
,
z
)
=
U
(
y
,
x
,
z
)
,
U(x,y,z) = U(y,x,z),
U
(
x
,
y
,
z
)
=
U
(
y
,
x
,
z
)
,
V
(
x
,
y
,
z
)
=
−
V
(
x
,
z
,
y
)
.
V(x,y,z) = -V(x,z,y).
V
(
x
,
y
,
z
)
=
−
V
(
x
,
z
,
y
)
.
6
1
Hide problems
\sum_{s=n}^{2n} 2^{2n-s}{s \choose n}= 2^{2n}
Prove that for every natural number
n
n
n
we have
∑
s
=
n
2
n
2
2
n
−
s
(
s
n
)
=
2
2
n
.
\sum_{s=n}^{2n} 2^{2n-s}{s \choose n}= 2^{2n}.
s
=
n
∑
2
n
2
2
n
−
s
(
n
s
)
=
2
2
n
.
5
1
Hide problems
regular tetrahedron criterion by areas
In a tetrahedron, the six triangles determined by an edge of the tetrahedron and the midpoint of the opposite edge all have equal area. Prove that the tetrahedron is regular.
2
1
Hide problems
a^2 +b^2 +c^2 = 3abc NT
Prove that for every
n
n
n
there exists a solution of the equation
a
2
+
b
2
+
c
2
=
3
a
b
c
a^2 +b^2 +c^2 = 3abc
a
2
+
b
2
+
c
2
=
3
ab
c
in natural numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
greater than
n
n
n
.
1
1
Hide problems
area of cyclic octagon
Compute the area of an octagon inscribed in a circle, whose four sides have length
1
1
1
and the other four sides have length
2
2
2
.