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Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1979 Poland - Second Round
1979 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
DL^2 + KC^2 = AB^2 , semicircle externally on a rectangle
On the side
D
C
‾
\overline{DC}
D
C
of the rectangle
A
B
C
D
ABCD
A
BC
D
in which
A
B
A
D
=
2
\frac{AB}{AD} = \sqrt{2}
A
D
A
B
=
2
a semicircle is built externally. Any point
M
M
M
of the semicircle is connected by segments with
A
A
A
and
B
B
B
to obtain points
K
K
K
and
L
L
L
on
D
C
‾
\overline{DC}
D
C
, respectively. Prove that
D
L
2
+
K
C
2
=
A
B
2
DL^2 + KC^2 = AB^2
D
L
2
+
K
C
2
=
A
B
2
.
5
1
Hide problems
1 among 9 consecutive is coprime to each of the other nine.
Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.
4
1
Hide problems
lines after symmetry of plane wrt line k
Let
S
k
S_k
S
k
be the symmetry of the plane with respect to the line
k
k
k
. Prove that equality holds for every lines
a
,
b
,
c
a, b, c
a
,
b
,
c
contained in one plane
S
a
S
b
S
c
S
a
S
b
S
c
S
b
S
c
S
a
S
b
S
c
S
a
=
S
b
S
c
S
a
S
b
S
c
S
a
S
a
S
b
S
c
S
a
S
b
S
c
S_aS_bS_cS_aS_bS_cS_bS_cS_aS_bS_cS_a = S_bS_cS_aS_bS_cS_aS_aS_bS_cS_aS_bS_c
S
a
S
b
S
c
S
a
S
b
S
c
S
b
S
c
S
a
S
b
S
c
S
a
=
S
b
S
c
S
a
S
b
S
c
S
a
S
a
S
b
S
c
S
a
S
b
S
c
3
1
Hide problems
area of orthogonal projection of a square on a plane
In space there is a line
k
k
k
and a cube with a vertex
M
M
M
and edges
M
A
‾
\overline{MA}
M
A
,
M
B
‾
\overline{MB}
MB
,
M
C
‾
\overline{MC}
MC
, of length
1
1
1
. Prove that the length of the orthogonal projection of edge
M
A
MA
M
A
on the line
k
k
k
is equal to the area of the orthogonal projection of a square with sides
M
B
MB
MB
and
M
C
MC
MC
onto a plane perpendicular to the line
k
k
k
.[hide=original wording]W przestrzeni dana jest prosta
k
k
k
oraz sześcian o wierzchołku
M
M
M
i krawędziach
M
A
‾
\overline{MA}
M
A
,
M
B
‾
\overline{MB}
MB
,
M
C
‾
\overline{MC}
MC
, długości 1. Udowodnić, że długość rzutu prostokątnego krawędzi
M
A
MA
M
A
na prostą
k
k
k
jest równa polu rzutu prostokątnego kwadratu o bokach
M
B
MB
MB
i
M
C
MC
MC
na płaszczyznę prostopadłą do prostej
k
k
k
.
2
1
Hide problems
a^3 + b^3 + c^3 + 3abc \geq a^2(b + c) + b^2(a + c) + c^2(a + b)
Prove that if
a
,
b
,
c
a, b, c
a
,
b
,
c
are non-negative numbers, then
a
3
+
b
3
+
c
3
+
3
a
b
c
≥
a
2
(
b
+
c
)
+
b
2
(
a
+
c
)
+
c
2
(
a
+
b
)
.
a^3 + b^3 + c^3 + 3abc \geq a^2(b + c) + b^2(a + c) + c^2(a + b).
a
3
+
b
3
+
c
3
+
3
ab
c
≥
a
2
(
b
+
c
)
+
b
2
(
a
+
c
)
+
c
2
(
a
+
b
)
.
1
1
Hide problems
shortest time to among two point on edge of a circular pool.
Given are the points
A
A
A
and
B
B
B
on the edge of a circular pool. The athlete has to get from point
A
A
A
to point
B
B
B
by walking along the edge of the pool or swimming in the pool; he can change the way he moves many times. How should an athlete move to get from point A to B in the shortest time, given that he moves twice as slowly in water as on land?