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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1896 Eotvos Mathematical Competition
1896 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
2
1
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Prove that the equations $$x^2-3xy+2y^2+x-y=0 \text{ and } x^2-2xy+y^2-5x+7y=0$$
Prove that the equations
x
2
−
3
x
y
+
2
y
2
+
x
−
y
=
0
and
x
2
−
2
x
y
+
y
2
−
5
x
+
7
y
=
0
x^2-3xy+2y^2+x-y=0 \text{ and } x^2-2xy+y^2-5x+7y=0
x
2
−
3
x
y
+
2
y
2
+
x
−
y
=
0
and
x
2
−
2
x
y
+
y
2
−
5
x
+
7
y
=
0
imply the equation
x
y
−
12
x
+
15
y
=
0
xy-12x+15y=0
x
y
−
12
x
+
15
y
=
0
.
1
1
Hide problems
If $k$ is the number of distinct prime divisors of a natural number $n$, prove t
If
k
k
k
is the number of distinct prime divisors of a natural number
n
n
n
, prove that log
n
≥
k
n \geq k
n
≥
k
log
2
2
2
.
3
1
Hide problems
Construct a triangle, given the feet of its altitudes. Express the sides of a tr
Construct a triangle, given the feet of its altitudes. Express the sides of a triangle
Y
Y
Y
in terms of the sides of the triangle
X
X
X
formed by the feet of the altitudes of
Y
Y
Y
.