MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1950 Polish MO Finals
1950 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
Hide problems
product of 1 to n-1 is divisible by n, when n>4 is not a prime ,
Prove that if a natural number
n
n
n
is greater than
4
4
4
and is not a prime number, then the produxt of the consecutive natural numbers from
1
1
1
to
n
−
1
n-1
n
−
1
is divisible by
n
n
n
.
5
1
Hide problems
sin^2 A+sin^2 B +sin^2 c=2 iff ABC is right triangle
Prove that if for angles
A
,
B
,
C
A,B,C
A
,
B
,
C
of a triangle holds
sin
2
A
+
sin
2
B
+
sin
2
C
=
2
\sin^2 A+\sin^2 B +\sin^2 C=2
sin
2
A
+
sin
2
B
+
sin
2
C
=
2
iff the triangle
A
B
C
ABC
A
BC
is right.
3
1
Hide problems
2 altitudes of tetrahedron intersect
Prove that if the two altitudes of a tetrahedron intersect, then the other two atltitudes intersect also.
4
1
Hide problems
unscrew a square nut with a wrench, whose hole is regular hexagon
Someone wants to unscrew a square nut with side
a
a
a
, with a wrench whose hole has the form of a regular hexagon with side
b
b
b
. What condition should the lengths
a
a
a
and
b
b
b
meet to make this possible?
2
1
Hide problems
square constuction on 2 concentric circles
We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.
1
1
Hide problems
factor x^8 + x^4 +1 to trinomials
Decompose the polynomial
x
8
+
x
4
+
1
x^8 + x^4 +1
x
8
+
x
4
+
1
to factors of at most second degree.