MathDB

1

The greates-area polygon

x_1, x_2, \ldots , x_n

find the polygon

A_0A_1\ldots A_n

with

A_iA_{i+1} = x_{i+1}

and which has greatest area.

4

1

Find de such that [ade]=[debc] and de is minimal

Let

ABC

be a triangle. Find the point

D

on its side

AC

and the point

E

on its side

AB

such that the area of triangle

ADE

equals to the area of the quadrilateral

DEBC

, and the segment

DE

has minimum possible length.

8

1

Chess, women versus men

In a chess tournament each player plays every other player once. A player gets 1 point for a win, 0.5 point for a draw and 0 for 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.

6

1

No subset in any other

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

7

1

n^a + n^b = n^c

Find all 4-tuples

(a,b,c,n)

of naturals such that

n^a + n^b = n^c

5

1

A very good estimate in the sum of (number of divisors of n)

Let

d(n)=\sum_{0<d|n}{1}

. Show that, for any natural

n>1

,

\sum_{2 \leq i \leq n}{\frac{1}{i}} \leq \sum{\frac{d(i)}{n}} \leq \sum_{1 \leq i \leq n}{\frac{1}{i}}

2

1

1992 1's at the beginning of n^1992

Show that there is a positive integer n such that the first 1992 digits of

n^{1992}

are 1.

1

X^3+px+q=0 implies p<0

The equation

x^3+px+q=0

has three distinct real roots. Show that

p<0

1992 Brazil National Olympiad

Subcontests

The greates-area polygon

Find de such that [ade]=[debc] and de is minimal

Chess, women versus men

No subset in any other

n^a + n^b = n^c

A very good estimate in the sum of (number of divisors of n)

1992 1's at the beginning of n^1992

X^3+px+q=0 implies p<0

1992 Brazil National Olympiad

Subcontests

The greates-area polygon

Find de such that [ade]=[debc] and de is minimal

Chess, women versus men

No subset in any other

n^a + n^b = n^c

A very good estimate in the sum of (number of divisors of n)

1992 1's at the beginning of n^1992

X^3+px+q=0 implies p&lt;0

X^3+px+q=0 implies p<0