MathDB

2

Part of 2012 Vietnam National Olympiad

Problems(2)

Sequences a_n, b_n and P_k(x)=x^2+a_kx+b_k

Source: Vietnamese National Mathematical Olympiad 2012-P2

2/8/2012
Let ⟨an⟩\langle a_n\rangle and ⟨bn⟩ \langle b_n\rangle be two arithmetic sequences of numbers, and let mm be an integer greater than 2.2. Define Pk(x)=x2+akx+bk, k=1,2,⋯ ,m.P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m. Prove that if the quadratic expressions P1(x),Pm(x)P_1(x), P_m(x) do not have any real roots, then all the remaining polynomials also don't have real roots.
quadraticsalgebrapolynomialarithmetic sequencealgebra proposed
Odd natural numbers a,b; a|b^2+2 and b|a^2+2

Source: Vietnamese National Mathematical Olympiad 2012-P6

2/8/2012
Consider two odd natural numbers aa and bb where aa is a divisor of b2+2b^2+2 and bb is a divisor of a2+2.a^2+2. Prove that aa and bb are the terms of the series of natural numbers ⟨vn⟩\langle v_n\rangle defined by v1=v2=1;vn=4vn−1−vn−2  for n≥3.v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.
inductionalgebrasystem of equationsnumber theory proposednumber theory