Brazil-National_Olympiad-1992-58

1992

Day1

1The equation x3+px+q=0has three distinct real roots.Show that p<0

2Show that there is a positive integer n such that the?rst1992digits of n1992are1.

3Given positive real numbers x1,x2,...,x n?nd the polygon A0A1...A n with A i A i+1=x i+1 and which has greatest area.

This?le was downloaded from the AoPS Math Olympiad Resources Page Page1 https://www.wendangku.net/doc/b517251721.html,/

1992 Day2

5Let d(n)=

0

1.Show that,for any natural n>1,

2≤i≤n

1

i

≤

d(i)

n

≤

1≤i≤n

1

i

6Given a set of n elements,?nd the largest number of subsets such that no subset is contained in any other

7Find all4-tuples(a,b,c,n)of naturals such that

n a+n b=n c

8In a chess tournament each player plays every other player once.A player gets1point for a win,0.5point for a draw and0for a loss.Both men and women played in the tournament and each player scored the same total of points against women as against men.Show that the total number of players must be a square.

https://www.wendangku.net/doc/b517251721.html,/

This?le was downloaded from the AoPS Math Olympiad Resources Page Page2