Drzavno takmicenje (National round) 2008 · Lemma

1Number theoryFree answerIn how many ways can natural numbers \(a\), \(b\), \(c\) be chosen so that all of the following hold? \[ 1^{\circ} \quad a < b < c < 52 ; \] \[ 2^{\circ} \quad a \mid c ; \qquad 3^{\circ} \quad b \mid c ; \] …2GeometryFree answerIn a triangle \(ABC\) we have \(\angle ABC = 45^\circ\) and \(\angle CAB = 15^\circ\). Let \(M\) be the point of the ray \(BC\) for which \[ \overrightarrow{BM} = 3 \cdot \overrightarrow{BC} . \] Determine …3AlgebraFree answerDetermine the remainder left by the polynomial \[ x^{2008} - x^{2007} - 3x + 4 \] on division by the polynomial \((x - 1)^{3}\).4GeometryProofCircumscribe about a given triangle \(ABC\) an equilateral triangle \(PQR\) whose side is as long as possible. (Here \(\triangle PQR\) is called circumscribed about \(\triangle ABC\) when \(A \in QR\), …5Number theoryProofProve that there is a natural number \(n\) for which the number \[ 2p^{n} + 3 \] is composite for every prime number \(p\).

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