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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1926 Eotvos Mathematical Competition
1926 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
1
1
Hide problems
x + y + 2z + 2t = a, 2x - 2y + z- t = b , NT system
Prove that, if
a
a
a
and
b
b
b
are given integers, the system of equatìons
x
+
y
+
2
z
+
2
t
=
a
x + y + 2z + 2t = a
x
+
y
+
2
z
+
2
t
=
a
2
x
−
2
y
+
z
−
t
=
b
2x - 2y + z- t = b
2
x
−
2
y
+
z
−
t
=
b
has a solution in integers
x
,
y
,
z
,
t
x, y,z,t
x
,
y
,
z
,
t
.
2
1
Hide problems
product of 4 consecutive naturals cannot be perfect square
Prove that the product of four consecutive natural numbers cannot be the square of an integer.
3
1
Hide problems
a circle rolls along the inside of another circle with double radius
The circle
k
′
k'
k
′
rolls along the inside of circle
k
k
k
, the radius of
k
k
k
is twice the radius of
k
′
k'
k
′
. Describe the path of a point on
k
k
k
..