MathDB

1

Part of 2003 Moldova Team Selection Test

Problems(3)

Polynomial with positive real roots

Source: Moldova IMO-BMO TST 2003, day 2, problem 1.

8/15/2008
Let n>0 n>0 be a natural number. Determine all the polynomials of degree 2n 2n with real coefficients in the form P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1, if it is known that all the roots of them are positive reals. Proposer: Baltag Valeriu
algebrapolynomialVieta
Sides of triangle divided into 2002 congruent segments

Source: Moldova IMO-BMO TST 2003, day 1, problem 1.

8/14/2008
Each side of an arbitrarly triangle is divided into 2002 2002 congruent segments. After that, each vertex is joined with all "division" points on the opposite side. Prove that the number of the regions formed, in which the triangle is divided, is divisible by 6 6. Proposer: Dorian Croitoru
modular arithmeticrationumber theoryrelatively prime
Quadratic permutations

Source: Moldova IMO-BMO TST 2003, day 3, problem 1.

8/16/2008
Let n∈N∗ n\in N^*. A permutation (a1,a2,...,an) (a_1,a_2,...,a_n) of the numbers (1,2,...,n) (1,2,...,n) is called quadratic iff at least one of the numbers a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n is a perfect square. Find the greatest natural number n≤2003 n\leq 2003, such that every permutation of (1,2,...,n) (1,2,...,n) is quadratic.
quadraticsnumber theoryrelatively primeDiophantine equation