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Đề thi học sinh giỏi Toán toàn quốc năm
1971
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Dịch sang tiếng
Anh
| A1. m, n, r,
s are positive integers such that: (1) m < n
and r < s; (2) m and n are relatively prime,
and r and s are relatively prime; and (3)
tan-1m/n + tan-1r/s = π/4.
Given m and n, find r and s. Given n and s, find m
and r. Given m and s, find n and r.
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| B1.
ABCDA'B'C'D' is a cube (with ABCD and A'B'C'D'
faces, and AA', BB', CC', DD' edges). L is a line
which intersects or is parallel to the lines AA',
BC and DB'. L meets the line BC at M (which may be
the point at infinity). Let m = |BM|. The plane
MAA' meets the line B'C' at E. Show that |B'E| =
m. The plane MDB' meets the line A'D' at F. Show
that |D'F| = m. Hence or otherwise show how to
construct the point P at the intersection of L and
the plane A'B'C'D'. Find the distance between P
and the line A'B' and the distance between P and
the line A'D' in terms of m. Find a relation
between these two distances that does not depend
on m. Find the locus of M. Let S be the envelope
of the line L as M varies. Find the intersection
of S with the faces of the cube.
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