MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1959 Polish MO Finals
1959 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
Hide problems
sidelengths and angles of triangle form 2 arithmetic progressions
Given a triangle in which the sides
a
a
a
,
b
b
b
,
c
c
c
form an arithmetic progression and the angles also form an arithmetic progression. Find the ratios of the sides of this triangle.
5
1
Hide problems
locus of midpoints
In the plane of the triangle
A
B
C
ABC
A
BC
a straight line moves which intersects the sides
A
C
AC
A
C
and
B
C
BC
BC
in such points
D
D
D
and
E
E
E
that
A
D
=
B
E
AD = BE
A
D
=
BE
. Find the locus of the midpoint
M
M
M
of the segment
D
E
DE
D
E
.
4
1
Hide problems
rational root of integer ax^2 + bx + c =
Prove that if a quadratic equation
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
with integer coefficients has a rational root, then at least one of the numbers
a
a
a
,
b
b
b
,
c
c
c
is even.
3
1
Hide problems
min distance on syrface of pyramid with square base
Given a pyramid with square base
A
B
C
D
ABCD
A
BC
D
and vertex
S
S
S
. Find the shortest path whose starting and ending point is the point
S
S
S
and which passes through all the vertices of the base.
2
1
Hide problems
AP + BM+CN= fixed, equilateral related
In an equilateral triangle
A
B
C
ABC
A
BC
, point
O
O
O
is chosen and perpendiculars
O
M
OM
OM
,
O
N
ON
ON
,
O
P
OP
OP
are dropped to the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
, respectively. Prove that the sum of the segments
A
P
AP
A
P
,
B
M
BM
BM
,
C
N
CN
CN
does not depend on the position of point
O
O
O
.
1
1
Hide problems
\frac{a+b}{2} \frac{a^2+b^2}{2} \frac{a^3+b^3}{2} \leq \frac{a^6+b^6}{2}.
Prove that for any numbers
a
a
a
and
b
b
b
the inequality holds
a
+
b
2
⋅
a
2
+
b
2
2
⋅
a
3
+
b
3
2
≤
a
6
+
b
6
2
.
\frac{a+b}{2} \cdot \frac{a^2+b^2}{2} \cdot \frac{a^3+b^3}{2} \leq \frac{a^6+b^6}{2}.
2
a
+
b
⋅
2
a
2
+
b
2
⋅
2
a
3
+
b
3
≤
2
a
6
+
b
6
.