Drzavno tekmovanje (national round) 2011 · Lemma

1Number theoryFree answerDetermine all prime numbers \(p\) and \(q\) for which the number \[ 2^{2} + p^{2} + q^{2} \] is also prime.2AlgebraFree answerFind all pairs of real numbers \(x\) and \(y\) satisfying \[ x + y^{2} = xy + 1 \qquad\text{and}\qquad xy = 4 + y . \]3GeometryFree answerIn a hexagon \(ABCDEF\) the following hold: \(\angle BAF = 150^{\circ}\), \(\angle ACB = \angle ADC = 90^{\circ}\), \(|AC| = |BC|\), triangle \(ABC\) is similar to triangle \(ADE\), and triangle \(BCD\) …4CombinatoricsFree answerA \(4 \times 4\) table is divided into \(16\) unit cells. On this table we place tiles of the shape drawn alongside: two unit squares that meet at a single corner. A tile may be rotated, and each tile …

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