Okruzno takmicenje (Regional round) 2021 · Lemma

1AlgebraFree answerFind every real value of the parameter \(a\) for which the polynomial \[ P(x) = x^{2021} - 2x^{2} + x + a^{3} - a \] is divisible by the polynomial \[ Q(x) = x^{2} - (a+1)x + a. \]2Number theoryFree answerFind the smallest natural number \(n\) for which there exist natural numbers \(a\) and \(b\) whose digit sums are \(28\) and \(21\) respectively, and \[ a + b = \underbrace{11\ldots1}_{n}. \]3GeometryFree answerOn the sides \(AB\) and \(BC\) of an equilateral triangle \(ABC\), points \(Z\) and \(X\) are chosen so that \[ AZ : ZB = BX : XC = 2021 : 2020. \] The perpendicular bisector of the segment \(XZ\) meets …4CombinatoricsFree answerThe cells of a \(4 \times 4\) table are to be coloured with several colours so that in every figure congruent to the one shown below, all four cells have different colours. The figure may be rotated or …5Number theoryFree answerFind all natural numbers \(n\) for which the number \[ n \cdot 2^{n} + 4 \] is the square of an integer.

Work through the paper in order - each problem opens in the workspace with this paper as its trail.