Difference between revisions of "1987 IMO Problems"
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− | In an acute-angled triangle <math>ABC </math> the interior bisector of the angle <math>A </math> intersects <math>BC </math> at <math>L </math> and intersects the [[circumcircle]] of <math>ABC </math> again at <math>N </math>. From point <math>L </math> perpendiculars are drawn to <math>AB </math> and <math>AC </math>, the feet of these perpendiculars being <math>K </math> and <math>M </math> respectively. Prove that the quadrilateral <math>AKNM </math> and the triangle <math>ABC </math> have equal areas. | + | In an acute-angled triangle <math>ABC </math> the interior bisector of the angle <math>A </math> intersects <math>BC </math> at <math>L </math> and intersects the [[circumcircle]] of <math>ABC </math> again at <math>N </math>. From point <math>L </math> perpendiculars are drawn to <math>AB </math> and <math>AC </math>, the feet of these perpendiculars being <math>K </math> and <math>M </math> respectively. Prove that the quadrilateral <math>AKNM </math> and the triangle <math>ABC </math> have equal areas. |
Latest revision as of 14:24, 17 September 2023
Problems of the 1987 IMO Cuba.
Contents
Day I
Problem 1
Let be the number of permutations of the set
, which have exactly
fixed points. Prove that
.
(Remark: A permutation of a set
is a one-to-one mapping of
onto itself. An element
in
is called a fixed point of the permutation
if
.)
Problem 2
In an acute-angled triangle the interior bisector of the angle
intersects
at
and intersects the circumcircle of
again at
. From point
perpendiculars are drawn to
and
, the feet of these perpendiculars being
and
respectively. Prove that the quadrilateral
and the triangle
have equal areas.
Problem 3
Let be real numbers satisfying
. Prove that for every integer
there are integers
, not all 0, such that
for all
and
.
Day 2
Problem 4
Prove that there is no function from the set of non-negative integers into itself such that
for every
.
Problem 5
Let be an integer greater than or equal to 3. Prove that there is a set of
points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
Problem 6
Let be an integer greater than or equal to 2. Prove that if
is prime for all integers
such that
, then
is prime for all integers
such that
.
Resources
- 1987 IMO
- IMO 1987 problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1987 IMO (Problems) • Resources | ||
Preceded by 1986 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1988 IMO |
All IMO Problems and Solutions |