MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1968 Poland - Second Round
1968 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
4
1
Hide problems
a^2yz + b^2zx + c^2xy >= 0
Prove that if the numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
, are the lengths of the sides of a triangle and the sum of the numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
is zero, then
a
2
y
z
+
b
2
z
x
+
c
2
x
y
≤
0.
a^2yz + b^2zx + c^2xy \leq 0.
a
2
yz
+
b
2
z
x
+
c
2
x
y
≤
0.
5
1
Hide problems
ratio of volumes of tetrahedrons
The tetrahedrons
A
B
C
D
ABCD
A
BC
D
and
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
are situated so that the midpoints of the segments
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
,
C
C
1
CC_1
C
C
1
,
D
D
1
DD_1
D
D
1
are the centroids of the triangles
B
C
D
BCD
BC
D
,
A
C
D
ACD
A
C
D
,
A
B
D
A B D
A
B
D
and
A
B
C
ABC
A
BC
, respectively. What is the ratio of the volumes of these tetrahedrons?
2
1
Hide problems
construct triangle in circle with gicen orthocenter
Given a circle
k
k
k
and a point inside it
H
H
H
. Inscribe a triangle in the circle such that this point
H
H
H
is the point of intersection of the triangle's altitudes.
6
1
Hide problems
k different lines passing through n points, not all on the same line.
On the plane are chosen
n
≥
3
n \ge 3
n
≥
3
points, not all on the same line. Drawing all lines passing through two of these points one obtains k different lines. Prove that
k
≥
n
k \ge n
k
≥
n
.
3
1
Hide problems
at least 5 persons sit at a round table,
Show that if at least five persons are sitting at a round table, then it is possible to rearrange them so that everyone has two new neighbors.
1
1
Hide problems
polynomial with integer coefficients has no integer zeros.
Prove that if a polynomial with integer coefficients takes a value equal to
1
1
1
in absolute value at three different integer points, then it has no integer zeros.