Opštinsko takmičenje (Municipal round) 2007 · Lemma

1CombinatoricsProofA car leaves town \(A\) and drives along a straight road at constant speed. Every \(15\) minutes it makes a turn of \(90\) degrees, to the left or to the right. Prove that the car can be back in \(A\) …2AlgebraFree answerSolve the system of equations \[ \begin{aligned} x - y &= 2005, \\ \lfloor x \rfloor + \lfloor y \rfloor &= 2007, \end{aligned} \] where \(\lfloor t \rfloor\) denotes the integer part of the real number …3GeometryProofOn the bisector of the angle \(\angle BAC\) of a triangle \(ABC\), points \(B_1\) and \(C_1\) are chosen so that \(BB_1 \perp AB\) and \(CC_1 \perp AC\). Let \(M\) be the midpoint of the segment \(B_1C_1\). …4Number theoryProofNatural numbers \(a\), \(b\) and \(c\) satisfy \[ a + \cfrac{1}{b + \cfrac{1}{c}} = \frac{4016}{2007} . \] Prove that \[ \cfrac{1}{c + \cfrac{1}{b + \cfrac{1}{a}}} = \frac{2007}{4016} . \]5Number theoryFree answerDetermine in how many ways the number \(441000\) can be written as a product of two factors \(m\) and \(n\) with \[ m > 1, \qquad n > 1, \qquad \gcd(m,n) = 1 , \] where the order of the factors is irrelevant, …

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