MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1980 Poland - Second Round
1980 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
a PA^2 + b PB^2 + c PC^2 is fixed if P lies on incircle of (ABC)
Prove that if the point
P
P
P
runs through a circle inscribed in the triangle
A
B
C
ABC
A
BC
, then the value of the expression
a
⋅
P
A
2
+
b
⋅
P
B
2
+
c
⋅
P
C
2
a \cdot PA^2 + b \cdot PB^2 + c \cdot PC^2
a
⋅
P
A
2
+
b
⋅
P
B
2
+
c
⋅
P
C
2
is constant (
a
,
b
,
c
a, b, c
a
,
b
,
c
are the lengths of the sides opposite the vertices
A
,
B
,
C
A, B, C
A
,
B
,
C
, respectively).
5
1
Hide problems
print the terms of the sequence
We print the terms of the sequence
(
n
1
,
n
2
,
…
,
n
k
)
(n_1, n_2, \ldots, n_k)
(
n
1
,
n
2
,
…
,
n
k
)
, where
n
1
=
1000
n_1 = 1000
n
1
=
1000
, and
n
j
n_j
n
j
for
j
>
1
j > 1
j
>
1
is an integer selected randomly from the range
[
0
,
n
j
−
1
−
1
]
[0, n_{j-1 } - 1]
[
0
,
n
j
−
1
−
1
]
(each number in this range is equally likely to be selected). We stop printing when the selected number is zero, i.e.
n
k
−
1
n_{k-1}
n
k
−
1
,
n
k
=
0
n_k = 0
n
k
=
0
, The length
k
k
k
of the sequence
(
n
1
,
n
2
,
…
,
n
k
)
(n_1, n_2, \ldots, n_k)
(
n
1
,
n
2
,
…
,
n
k
)
is a random variable. Prove that the expected value of this random variable is greater than 7.
4
1
Hide problems
ax^3 - ax^2 + 9bx - b has 3 positive roots
Prove that if
a
a
a
and
b
b
b
are real numbers and the polynomial
a
x
3
−
a
x
2
+
9
b
x
−
b
ax^3 - ax^2 + 9bx - b
a
x
3
−
a
x
2
+
9
b
x
−
b
has three positive roots, then they are equal.
3
1
Hide problems
3d locus with a sphere
There is a sphere
K
K
K
in space and points
A
,
B
A, B
A
,
B
outside the sphere such that the segment
A
B
AB
A
B
intersects the interior of the sphere. Prove that the set of points
P
P
P
for which the segments
A
P
AP
A
P
and
B
P
BP
BP
are tangent to the sphere
K
K
K
is contained in a certain plane.
2
1
Hide problems
x_1x_2x_3...x_n <= x_1^2 /2+ x_2^4 / 4 + x_3^8 /8 + ...+1/2^n
Prove that for any real numbers
x
1
,
x
2
,
x
3
,
…
,
x
n
x_1, x_2, x_3, \ldots, x_n
x
1
,
x
2
,
x
3
,
…
,
x
n
the inequality is true
x
1
x
2
x
3
…
x
n
≤
x
1
2
2
+
x
2
4
4
+
x
3
8
8
+
…
+
x
n
2
n
2
n
+
1
2
n
x_1x_2x_3\ldots x_n \leq \frac{x_1^2}{2} + \frac{x_2^4}{4} + \frac{x_3^8}{8} + \ldots + \frac{x_n^{2^ n}}{2^n} + \frac{1}{2^n}
x
1
x
2
x
3
…
x
n
≤
2
x
1
2
+
4
x
2
4
+
8
x
3
8
+
…
+
2
n
x
n
2
n
+
2
n
1
1
1
Hide problems
2player game with vectors
Students
A
A
A
and
B
B
B
play according to the following rules: student
A
A
A
selects a vector
a
1
→
\overrightarrow{a_1}
a
1
of length 1 in the plane, then student
B
B
B
gives the number
s
1
s_1
s
1
, equal to
1
1
1
or
−
-
−
1; then the student
A
A
A
chooses a vector
a
1
→
\overrightarrow{a_1}
a
1
of length
1
1
1
, and in turn the student
B
B
B
gives a number
s
2
s_2
s
2
equal to
1
1
1
or
−
1
-1
−
1
etc.
B
B
B
wins if for a certain
n
n
n
vector
∑
j
=
1
n
ε
j
a
j
→
\sum_{j=1}^n \varepsilon_j \overrightarrow{a_j}
∑
j
=
1
n
ε
j
a
j
has a length greater than the number
R
R
R
determined before the start of the game. Prove that student
B
B
B
can achieve a win in no more than
R
2
+
1
R^2 + 1
R
2
+
1
steps regardless of partner
A
A
A
's actions.