MathDB

3

Part of 2011 Poland - Second Round

Problems(2)

Perfect square

Source: Polish MO second round 2011

2/19/2012
Prove that ∀x1,x2,…,x2011,y1,y2,…,y2011∈Z+\forall x_{1},x_{2},\ldots,x_{2011},y_{1},y_{2},\ldots,y_{2011}\in\mathbb{Z_{+}} product: (2x12+3y12)(2x22+3y22)…(2x20112+3y20112)(2x_{1}^{2}+3y_{1}^{2})(2x_{2}^{2}+3y_{2}^{2})\ldots(2x_{2011}^{2}+3y_{2011}^{2}) is not a perfect square.
modular arithmeticnumber theory unsolvednumber theory
Divisibility of polynomials

Source: Polish MO second round 2011

2/19/2012
There are two given different polynomials P(x),Q(x)P(x),Q(x) with real coefficients such that P(Q(x))=Q(P(x))P(Q(x))=Q(P(x)). Prove that ∀n∈Z+\forall n\in \mathbb{Z_{+}} polynomial: P(P(…P(P⏟n(x))…))−Q(Q(…Q(Q⏟n(x))…))\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots)) is divisible by P(x)−Q(x)P(x)-Q(x).
algebrapolynomialcalculusintegrationalgebra unsolved