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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1992 Poland - Second Round
1992 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
3
1
Hide problems
ABC is isosceles if |BP| x|CQ| x|AR| = |PC| x |QA| x|RB|
Through the center of gravity of the acute-angled triangle
A
B
C
ABC
A
BC
, lines are drawn perpendicular to the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
, intersecting them at the points
P
P
P
,
Q
Q
Q
,
R
R
R
, respectively. Prove that if
∣
B
P
∣
⋅
∣
C
Q
∣
⋅
∣
A
R
∣
=
∣
P
C
∣
⋅
∣
Q
A
∣
⋅
∣
R
B
∣
|BP|\cdot |CQ| \cdot |AR| = |PC| \cdot |QA| \cdot |RB|
∣
BP
∣
⋅
∣
CQ
∣
⋅
∣
A
R
∣
=
∣
PC
∣
⋅
∣
Q
A
∣
⋅
∣
RB
∣
, then the triangle
A
B
C
ABC
A
BC
is isosceles.Note: According to Ceva's theorem, the assumed equality of products is equivalent to the fact that the lines
A
P
AP
A
P
,
B
Q
BQ
BQ
,
C
R
CR
CR
have a common point.
5
1
Hide problems
max vol of sphere in tetrahedra all heights <=1
Determine the upper limit of the volume of spheres contained in tetrahedra of all heights not longer than
1
1
1
.
4
1
Hide problems
SE=SF wanted , 3 circles ext. tangent
The circles
k
1
k_1
k
1
,
k
2
k_2
k
2
,
k
3
k_3
k
3
are externally tangent:
k
1
k_1
k
1
to
k
2
k_2
k
2
at point
A
A
A
,
k
2
k_2
k
2
to
k
3
k_3
k
3
at point
B
B
B
,
k
3
k_3
k
3
to
k
4
k_4
k
4
at point
C
C
C
,
k
4
k_4
k
4
to
k
1
k_1
k
1
at point
D
D
D
. The lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect at the point
S
S
S
. A line
p
p
p
is drawn through point
S
S
S
, tangent to
k
4
k_4
k
4
at point
F
F
F
. Prove that
∣
S
E
∣
=
∣
S
F
∣
|SE|=|SF|
∣
SE
∣
=
∣
SF
∣
.
6
1
Hide problems
x_{n+1} = (x_n+2)/(x_n+1) ,y _{n+1}=(y_n^2+2)/2y_n
The sequences
(
x
n
)
(x_n)
(
x
n
)
and
(
y
n
)
(y_n)
(
y
n
)
are defined as follows: x_{n+1} = \frac{x_n+2}{x_n+1}, y_{n+1}=\frac{y_n^2+2}{2y_n} \text{ for } n= 0,1,2,\ldots. Prove that for every integer
n
≥
0
n\geq 0
n
≥
0
the equality
y
n
=
x
2
n
−
1
y_n = x_{2^n-1}
y
n
=
x
2
n
−
1
holds.
2
1
Hide problems
sum a_i x_i <= sum b_i x_i.
Given a natural number
n
≥
2
n \geq 2
n
≥
2
. Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots , a_n
a
1
,
a
2
,
…
,
a
n
,
b
1
,
b
2
,
…
,
b
n
b_1, b_2, \ldots , b_n
b
1
,
b
2
,
…
,
b
n
be real numbers. Prove that the following conditions are equivalent:- For any real numbers
x
1
≤
x
2
≤
…
≤
x
n
x_1 \leq x_2 \leq \ldots \leq x_n
x
1
≤
x
2
≤
…
≤
x
n
holds the inequality
∑
i
=
1
n
a
i
x
i
≤
∑
i
=
1
n
b
i
x
i
.
\sum_{i=1}^n a_i x_i \leq \sum_{i=1}^n b_i x_i.
i
=
1
∑
n
a
i
x
i
≤
i
=
1
∑
n
b
i
x
i
.
- For every natural number
k
∈
{
1
,
2
,
…
,
n
−
1
}
k\in \{1,2,\ldots, n-1\}
k
∈
{
1
,
2
,
…
,
n
−
1
}
holds the inequality
∑
i
=
1
k
a
i
≥
∑
i
=
1
k
b
i
,
and
∑
i
=
1
n
a
i
=
∑
i
=
1
n
b
i
.
\sum_{i=1}^k a_i \geq \sum_{i=1}^k b_i, \ \ \text{ and } \\ \ \sum_{i=1}^n a_i = \sum_{i=1 }^n b_i.
i
=
1
∑
k
a
i
≥
i
=
1
∑
k
b
i
,
and
i
=
1
∑
n
a
i
=
i
=
1
∑
n
b
i
.
1
1
Hide problems
perimeter of the polygon with lattice vertices is an even number.
Every vertex of a polygon has both integer coordinates; the length of each side of this polygon is a natural number. Prove that the perimeter of the polygon is an even number.