MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1984 Poland - Second Round
1984 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
x_1=c,\; x_{n+1} = cx_n + \sqrt{(c^2-1)(x_n^2-1)}
The sequence
(
x
n
)
(x_n)
(
x
n
)
is defined by formulas x_1=c,\; x_{n+1} = cx_n + \sqrt{(c^2-1)(x_n^2-1)} \text{ for } n=1,2,\ldots Prove that if
c
c
c
is a natural number, then all numbers
x
n
x_n
x
n
are natural.
5
1
Hide problems
lower bound of areas of convex hexagons whose vertices are lattice points
Calculate the lower bound of the areas of convex hexagons whose vertices all have integer coordinates.
4
1
Hide problems
3n participants in the Mathematical Olympiad competition
There are
3
n
3n
3
n
participants in the Mathematical Olympiad competition. They are assigned seats in three rows, with
n
n
n
seats in each, and are admitted into the hall one at a time, after which they immediately take their seats. Calculate the probability that until the last competitor takes his seat, at any moment for each two rows the difference in the number of players sitting in them is no greater than 1.
3
1
Hide problems
(x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) }
The given sequences are
(
x
1
,
x
2
,
…
,
x
n
)
(x_1, x_2, \ldots, x_n)
(
x
1
,
x
2
,
…
,
x
n
)
,
(
y
1
,
y
2
,
…
,
y
n
)
(y_1, y_2, \ldots, y_n)
(
y
1
,
y
2
,
…
,
y
n
)
with positive terms. Prove that there exists a permutation
p
p
p
of the set
{
1
,
2
,
…
,
n
}
\{1, 2, \ldots, n\}
{
1
,
2
,
…
,
n
}
such that for every real
t
t
t
the sequence
(
x
p
(
1
)
+
t
y
p
(
1
)
,
x
p
(
2
)
+
t
y
p
(
2
)
,
…
,
x
p
(
n
)
+
t
y
p
(
n
)
)
(x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) })
(
x
p
(
1
)
+
t
y
p
(
1
)
,
x
p
(
2
)
+
t
y
p
(
2
)
,
…
,
x
p
(
n
)
+
t
y
p
(
n
)
)
has the following property: there is a number
k
k
k
such that
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
and all non-zero terms of the sequence with indices less than
k
k
k
are of the same sign and all non-zero terms of the sequence with indices not less than
k
k
k
are the same sign.
2
1
Hide problems
similar isosceles triangles on sides of a triangle, # or colinear
We construct similar isosceles triangles on the sides of the triangle
A
B
C
ABC
A
BC
: triangle
A
P
B
APB
A
PB
outside the triangle
A
B
C
ABC
A
BC
(
A
P
=
P
B
AP = PB
A
P
=
PB
), triangle
C
Q
A
CQA
CQ
A
outside the triangle
A
B
C
ABC
A
BC
(
C
Q
=
Q
A
CQ = QA
CQ
=
Q
A
), triangle
C
R
B
CRB
CRB
inside the triangle
A
B
C
ABC
A
BC
(
C
R
=
R
B
CR = RB
CR
=
RB
). Prove that
A
P
R
Q
APRQ
A
PRQ
is a parallelogram or that the points
A
,
P
,
R
,
Q
A, P, R, Q
A
,
P
,
R
,
Q
lie on a straight line.
1
1
Hide problems
\sqrt{x} + \sqrt{y} = n , NT
For a given natural number
n
n
n
, find the number of solutions to the equation
x
+
y
=
n
\sqrt{x} + \sqrt{y} = n
x
+
y
=
n
in natural numbers
x
,
y
x, y
x
,
y
.