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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1942 Eotvos Mathematical Competition
1942 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
Hide problems
new triangle has area 1/7 of original equilateral
Let
A
′
A'
A
′
,
B
′
B'
B
′
and
C
′
C'
C
′
be points on the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively, of an equilateral triangle
A
B
C
ABC
A
BC
. If
A
C
′
=
2
C
′
B
AC' = 2C'B
A
C
′
=
2
C
′
B
,
B
A
′
=
2
A
′
C
BA' = 2A'C
B
A
′
=
2
A
′
C
and
C
B
′
=
2
B
′
A
CB' = 2B'A
C
B
′
=
2
B
′
A
, prove that the lines
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
enclose a triangle whose area is
1
/
7
1/7
1/7
that of
A
B
C
ABC
A
BC
.
2
1
Hide problems
ad - bc = + - 1 if ax + by = m and cx + dy = n
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
be integers such that for all integers m and n, there exist integers
x
x
x
and
y
y
y
such that
a
x
+
b
y
=
m
ax + by = m
a
x
+
b
y
=
m
, and
c
x
+
d
y
=
n
cx + dy = n
c
x
+
d
y
=
n
. Prove that
a
d
−
b
c
=
±
1
ad - bc = \pm 1
a
d
−
b
c
=
±
1
.
1
1
Hide problems
at most 1 side can be shorter than altitude from opposite vertex
Prove that in any triangle, at most one side can be shorter than the altitude from the opposite vertex.