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Part of 2008 Moldova Team Selection Test

Problems(3)

Warm-up Diophantine Equation.

Source: Moldova 2008 IMO-BMO First TST Problem 1

p

be a prime number. Solve in

\mathbb{N}_0\times\mathbb{N}_0

the equation x^3\plus{}y^3\minus{}3xy\equal{}p\minus{}1.

number theory proposednumber theory

Solve a system in real numbers.

Source: Moldova 2008 IMO-BMO Second TST Problem 1

Find all solutions

(x,y)\in \mathbb{R}\times\mathbb R

of the following system: \begin{cases}x^3 \plus{} 3xy^2 \equal{} 49, \\ x^2 \plus{} 8xy \plus{} y^2 \equal{} 8y \plus{} 17x.\end{cases}

algebra proposedalgebra

Find a solution for a^2+b^2+c^2+d^2+e^2=abcde in integers>0.

Source: Moldova 2008 IMO-BMO Third TST Problem 1

Determine a subset

A\subset \mathbb{N}^*

5

different elements, so that the sum of the squares of its elements equals their product. Do not simply post the subset, show how you found it.

quadraticsnumber theory proposednumber theory