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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1966 Spain Mathematical Olympiad
5
5
Part of
1966 Spain Mathematical Olympiad
Problems
(1)
computational geo related to a right triangle
Source: Spanish Mathematical Olympiad 1966 P5
12/4/2022
The length of the hypotenuse
B
C
BC
BC
of a right triangle
A
B
C
ABC
A
BC
is
a
a
a
, and on it the points
M
M
M
and
N
N
N
are taken such that
B
M
=
N
C
=
k
BM = NC = k
BM
=
NC
=
k
, with
k
<
a
/
2
k < a/2
k
<
a
/2
. Assuming that (only) the data
a
a
a
and
k
k
k
are known, calculate: a) The value of the sum of the squares of the lengths
A
M
AM
A
M
and
A
N
AN
A
N
. b) The ratio of the areas of triangles
A
B
C
ABC
A
BC
and
A
M
N
AMN
A
MN
. c) The area enclosed by the circle that passes through the points
A
,
M
′
,
N
′
A, M' , N'
A
,
M
′
,
N
′
, where
M
′
M'
M
′
is the orthogonal projection of
M
M
M
onto
A
C
AC
A
C
and
N
′
N'
N
′
that of
N
N
N
onto
A
B
AB
A
B
.
geometry
right triangle