MathDB

p1. Show that the difference between the cubes of two consecutive naturals is the sum of a square of one natural plus three times the square of another.
p2. The area of a plane figure

F

is denoted by

A_F

. Given a circumference

C

, consider all pairs of triangles

(T, S)

, such that

C

is the incircle of

T

and the circumcircle of

S

. Between all these pairs, find a pair

(T_0,S_0)

such that the expression

\frac{A_{T_0}}{A_{S_0}}

has the minimum value.
p3. Let

C

be the set of all naturals that can be expressed as

m^2 + n^2

, with

m, n

natural. Consider

s, t \in C

. Prove that:

\bullet

st \in C

\bullet

t \ne 0

then there are rationals

x,y

such that

\frac{s}{t}= x^2 + y^2

.
p4. At the exit of a stadium I met my friends Carlos, Pedro and José, who are still to witness the classic match. I asked them what the result was, and their responses were:

\bullet

Carlos: U won and UC scored the first goal.

\bullet

Pedro: U won and scored the first goal.

\bullet

José: Pedro never tells the truth and the U lost. Also, I knew that each of them was telling either two truths or two lies. Could I deduce from this the final score of the match? How?
p5. A natural

n

is the product of three odd primes. The sum of the primes is

1993

, the sum of its squares is

1363347

, and the sum of the divisors of

n

, including

1

and

n

, is

280411488

. Determine

n

.
p6. To calculate the area of a quadrilateral, where

a

b

c

d

are the lengths of its sides (in that order), we have Humbertito's Formula:

A =\frac{a + c}{2}\cdot \frac{b + d}{2}.

https://cdn.artofproblemsolving.com/attachments/6/9/01a5812d194de4e31e4ce53643d99377925201.png Prove that this formula is correct only in the case of rectangles.
p7. For each natural

k

, let

d (k)

be the number of positive divisors of

k

, including

1

and

k

. Find the maximum value of

d (k)

, when

k

is an integer, with

1 \le k \le1993

Problem Statement