IMO预选题1970
Proposed IMO Problems By Belgium 1970
IMO ShortList/LongList Project Group
June 19, 2004
1. (Belgium 1) Show that the equation
n∑
i=1
b
x− ai = c
with bi > 0 and a1 < a2 < a3 < . . . < an has n− 1 solutions x1, x2, x3, . . . , xn−1 satisfying
a1 < x1 <...
Proposed IMO Problems By Belgium 1970
IMO ShortList/LongList Project Group
June 19, 2004
1. (Belgium 1) Show that the equation
n∑
i=1
b
x− ai = c
with bi > 0 and a1 < a2 < a3 < . . . < an has n− 1 solutions x1, x2, x3, . . . , xn−1 satisfying
a1 < x1 < a2 < x2 < a3 < x3 < . . . < xn−1 < an−1.
2. (Belgium 2) On the sides [AB], [BC], [CD] and [DA] of a convex quadrilateral ABCD squares
are constructed outward this quadrilateral. M1,M2,M3,M4 are the midpoints of those squares.
Show that |M1M3| = |M2M4| and M1M3⊥M2M4.
Remark: In 1970 this was rather an unknown fact but it is not an original question. The theorem
is due to Von Aubel.
3. (Belgium 3) A regular convex polygon with polygon with 2n sides is given. Let n diagonals pass
through the center of the polygon, let P be a point of the inscribed circle and let aˆ1, aˆ2, . . . , aˆn
be the angles from which the n diagonals are seen from the point P. Show that
n∑
i=1
tan2(ai) = 2n ·
cos2
(
pi
2n
)
sin4
(
pi
2n
) .
4. (Belgium 4) Show that if n is a natural number we have:
n∑
k=1
(−1)k+1 1
k
= 2 ·
n
2∑
k=1
1
n+ 2k
.
5. (Belgium 5) Let A,B and C be angles of a plane triangle. Show that:
1 < cos(A) + cos(B) + cos(C) ≤ 3
2
.
6. (Belgium 6) Let ABCD and A′B′C ′D′ be two squares in the same plane and they are oriented
in the same way. Let A′′, B′′, C ′′ and D′′ be the midpoints of [AA′], [BB′], [CC ′] and [DD′]. Show
that the quadrilateral A′′B′′C ′′D′′ is also a square.
Global Remark: Questions 4,5 and 6 are not very original. These questions are typical ’handbook-
questions’. (Rene Laumen)
c© by Orlando Do¨hring, member of the IMO ShortList/LongList Project Group, page 1/1
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