1987年IMO中国国家队选拔考试试题

1987

Day 111a.)For all positive integer k find the smallest positive integer f (k )such that 5sets s 1,s 2,...,s 5exist satisfying:

I.each has k elements;II.s i and s i +1are disjoint for i =1,2,...,5(s 6=s 1)III.the union of the 5sets has exactly f (k )elements.

b.)Generalisation:Consider n ≥3sets instead of 5.

Corrected due to the courtesy of

[url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url]2A closed recticular polygon with 100sides (may be concave)is given such that it’s vertices have integer coordinates,it’s sides are parallel to the axis and all it’s sides have odd length.Prove that it’s area is odd.

Corrected due to the courtesy of

[url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url]3Let r 1=2and r n =

n −1 k =1r i +1,n ≥2.Prove that among all sets of positive integers such that n

k =11a i

<1,the partial sequences r 1,r 2,...,r n are the one that gets nearer to 1./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/

1987

Day 21Given a convex figure in the Cartesian plane that is symmetric with respect of both axis,we construct a rectangle A inside it with maximum area (over all posible rectangles).Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure.Find the smallest lamda such that it works for all convex figures.2Find all positive integer n such that the equation x 3+y 3+z 3=n ·x 2·y 2·z 2has positive integer solutions.3Let G be a simple graph with 2·n vertices and n 2+1edges,then there is a K 4-one edge,that is two triangles with a common edge./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 2http://www.mathlinks.ro/