Difference between revisions of "1987 IMO Problems"
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=== Problem 1 === | === Problem 1 === | ||
− | Let <math> | + | Let <math>p_n (k) </math> be the number of permutations of the set <math>\{ 1, \ldots , n \} , \; n \ge 1 </math>, which have exactly <math>k </math> fixed points. Prove that |
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− | (Remark: A permutation <math> | + | (Remark: A permutation <math>f </math> of a set <math>S </math> is a one-to-one mapping of <math>S </math> onto itself. An element <math>i </math> in <math>S </math> is called a fixed point of the permutation <math>f </math> if <math>f(i) = i </math>.) |
[[1987 IMO Problems/Problem 1 | Solution]] | [[1987 IMO Problems/Problem 1 | Solution]] | ||
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=== Problem 2 === | === Problem 2 === | ||
− | In an acute-angled triangle <math> | + | 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. |
+ | |||
[[1987 IMO Problems/Problem 2 | Solution]] | [[1987 IMO Problems/Problem 2 | Solution]] | ||
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=== Problem 3 === | === Problem 3 === | ||
− | Let <math> | + | Let <math>x_1 , x_2 , \ldots , x_n </math> be real numbers satisfying <math>x_1^2 + x_2^2 + \cdots + x_n^2 = 1 </math>. Prove that for every integer <math>k \ge 2 </math> there are integers <math>a_1, a_2, \ldots a_n </math>, not all 0, such that <math>| a_i | \le k-1 </math> for all <math>i </math> and |
<center> | <center> | ||
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=== Problem 4 === | === Problem 4 === | ||
− | Prove that there is no function <math> | + | Prove that there is no function <math>f </math> from the set of non-negative integers into itself such that <math>f(f(n)) = n + 1987 </math> for every <math>n </math>. |
[[1987 IMO Problems/Problem 4 | Solution]] | [[1987 IMO Problems/Problem 4 | Solution]] | ||
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=== Problem 5 === | === Problem 5 === | ||
− | Let <math> | + | Let <math>n </math> be an integer greater than or equal to 3. Prove that there is a set of <math>n </math> 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. |
[[1987 IMO Problems/Problem 5 | Solution]] | [[1987 IMO Problems/Problem 5 | Solution]] | ||
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=== Problem 6 === | === Problem 6 === | ||
− | Let <math> | + | Let <math>n </math> be an integer greater than or equal to 2. Prove that if <math>k^2 + k + n </math> is prime for all integers <math>k </math> such that <math> 0 \leq k \leq \sqrt{n/3} </math>, then <math>k^2 + k + n </math> is prime for all integers <math>k </math> such that <math>0 \leq k \leq n - 2 </math>. |
[[1987 IMO Problems/Problem 6 | Solution]] | [[1987 IMO Problems/Problem 6 | Solution]] | ||
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* [[1987 IMO]] | * [[1987 IMO]] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1987 IMO 1987 problems on the Resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1987 IMO 1987 problems on the Resources page] | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] {{IMO box|year=1987|before=[[1986 IMO]]|after=[[1988 IMO]]}} |
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 |